Final kinetic energy,Ek2 = 1/2 × m × v2²Ek2 = 1/2 × (68 kg) × (v2)²Ek2 = 34m²/s². The weight of the skier, mg = (68 kg)(9.8 m/s²)mg = 666.4 N. Therefore, the frictional force will be able to balance the weight of the skier and prevent her from sliding down the hill.
(a) Maximum height the skier will reach. The work-energy principle of physics states that the total work done on a system is equal to the change in its kinetic energy.
In other words, the work-energy principle says that the initial kinetic energy plus the work done on the system equals the final kinetic energy.
When a skier is skiing down a hill, he is losing gravitational potential energy and gaining kinetic energy. So, if we can determine the initial and final kinetic energies, we can find the maximum height reached by the skier.
Work done by frictional force, Wfriction = fs×m×g×cosθ×dwhere fs = 0.75 is the coefficient of static friction between skis and snow,m = 68 kg is the mass of the skier, g = 9.8 m/s² is the acceleration due to gravity,θ = 40.0° is the angle of the slope, d = L/sinθ is the length of the slope,L = vt = (15 m/s)(10 s) = 150 m is the length of the slope that the skier covers in 10 seconds. Wfriction = (0.75)(68 kg)(9.8 m/s²) cos 40° (150 m/sin 40°)W friction = 21917 J Initial kinetic energy,Ek1 = 1/2 × m × v1²Ek1 = 1/2 × (68 kg) × (15 m/s)²Ek1 = 15300 J
Conservation of energy states that the sum of initial kinetic energy and initial potential energy is equal to the sum of final kinetic energy and final potential energy, where potential energy comes in the form of gravitational potential energy when we deal with vertical motions. Mathematically, it can be written asInitial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energySince the skier starts from rest, the initial kinetic energy is zero.
Hence, Initial potential energy at the foot of the hill = Final kinetic energy + Final potential energywhere potential energy is given bymgh where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above some reference point (usually the ground).
Final kinetic energy,Ek2 = 1/2 × m × v2²Ek2 = 1/2 × (68 kg) × (v2)²Ek2 = 34m²/s²
Final potential energy at the maximum height h = Final potential energy at the foot of the hill + Work done by frictional force-mgh = 0 + Ek1 - Ek2 - Wfriction-mgh = (15300 J) - (34 m²/s²) - (21917 J)-mgh = -66617 Jh = 33.81 mTherefore, the maximum height that the skier will reach is 33.81 m.
(b)The skier will remain at rest once she stops since the coefficient of static friction between skis and snow is 0.75, which is greater than the coefficient of kinetic friction, 0.25.
When the skier stops, the force of friction between skis and snow will be the maximum value of static friction, which is given byfs × m × gfs × m × g = (0.75)(68 kg)(9.8 m/s²)fs × m × g = 477.48 N
The weight of the skier,mg = (68 kg)(9.8 m/s²)mg = 666.4 N
Therefore, the frictional force will be able to balance the weight of the skier and prevent her from sliding down the hill.
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A plastic rod of length 1.88 meters contains a charge of 6.8nC. The rod is formed into semicircle What is the magnitude of the electric field at the center of the semicircle? Express your answer in NiC
A plastic rod of length 1.88 meters contains a charge of 6.8nC.The magnitude of the electric field at the center of the semicircle is approximately [tex]1.19 * 10^6 N/C[/tex]
To find the magnitude of the electric field at the center of the semicircle formed by a plastic rod, we can use the concept of electric field due to a charged rod.
The electric field at the center of the semicircle can be calculated by considering the contributions from all the charges along the rod. Since the rod is uniformly charged, we can divide it into infinitesimally small charge elements and integrate their contributions.
The formula for the electric field due to a charged rod at a point along the perpendicular bisector of the rod is:
E = (kλ / R) * (1 - cosθ)
Where E is the electric field, k is the electrostatic constant (9 x 10^9 Nm²/C²), λ is the linear charge density (charge per unit length), R is the distance from the rod to the point, and θ is the angle between the perpendicular bisector and a line connecting the point to the rod.
In this case, the rod is formed into a semicircle, so the angle θ is 90 degrees (or π/2 radians). The linear charge density λ can be calculated by dividing the total charge Q by the length of the rod L:
λ = Q / L
Plugging in the values:
λ = 6.8 nC / 1.88 m
Converting nC to C and m to meters:
λ = 6.8 x 10^(-9) C / 1.88 m
Now, we can calculate the electric field at the center of the semicircle by plugging in the values into the equation:
E = ([tex]9 * 10^9[/tex] Nm²/C²) * [tex]6.8 x 10^(-9)[/tex])C / 1.88 m) * (1 - cos(π/2))
Simplifying the equation:
E ≈ [tex]1.19 * 10^6 N/C[/tex]
Therefore, the magnitude of the electric field at the center of the semicircle is approximately [tex]1.19 * 10^6 N/C[/tex]
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Figure 1 Two opposing speakers are shown in Figure 1. A standing wave is produced from two sound waves traveling in opposite directions; each can be described as follows: y 1
=(5 cm)sin(4x−2t),
y 2
=(5 cm)sin(4x+2t)
where x and y, are in centimeters and t is in seconds. Find i. amplitude of the simple harmonic motion of a medium element lying between the two speakers at x=2.5 cm. ii. amplitude of the nodes and antinodes. iii. maximum amplitude of an element at an antinode
The amplitude of the simple harmonic motion of a medium element lying between the two speakers at x=2.5 cm is 0. Ans: Part i: amplitude of the simple harmonic motion of a medium element lying between the two speakers at x=2.5 cm.
First, let's determine the wave function of the medium element y at point x=2.5 cm. We have;y=y1+y2 =(5 cm)sin(4x−2t)+(5 cm)sin(4x+2t)y=5 sin(4x−2t)+5sin(4x+2t)Now we find the amplitude of y when x=2.5 cm.
We have;y=5 sin(4(2.5)−2t)+5sin(4(2.5)+2t)y=5 sin(10−2t)+5sin(14+2t)We need to find the amplitude of this equation by taking the maximum value and subtracting the minimum value of this equation. However, we notice that the equation oscillates between maximum and minimum values of equal magnitude, so the amplitude is 0. Part ii: amplitude of the nodes and antinodesNodes and antinodes correspond to the points where the displacement amplitude is zero and maximum, respectively.
The nodes are located halfway between the speakers while the antinodes occur at the positions of the speakers themselves. Hence, the amplitude of the nodes is 0 while the amplitude of the antinodes is 5 cm. Part iii: maximum amplitude of an element at an antinodeThe maximum amplitude of an element at an antinode is 5 cm.
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A car with a mass of 405 kg is driving in circular path with a radius of 120 m at a constant speed of 5.5 m/s. What is the magnitude of the net force on the car? Round to the nearest whole number. 102 N 14182 N 6600 N 78000 N 558 N You throw a ball horizontally with an initial speed of 20 m/s from a height of 7.2 meters. How long does it take for the ball to land? Round to two decimal places. 0.55 seconds 0.39 seconds 6.53 seconds 0.15 seconds 1.20 seconds A car is initially traveling due South at 20 m/s. The driver hits the brake pedal and 1 second later, the car is traveling due South at 7 m/s. What is the magnitude of the average acceleration of the car during this 1 second interval? 13 m/s^2 27 m/s^2 7 m/s^2 60 m/s^2 25 m/s^2 Your friend (mass 60 kg) is wearing frictionless roller skates on a horizontal surface and is initially at rest. If you push your friend with a constant force of 1200 N, over what distance must you exert the force so they reach a final speed of 10 m/s? 0.25 meters 0.5 meters 1.25 meters: 2.5 meters 5 meters
1. the magnitude of the net force on the car is 558 N. Hence, the correct option is (e) 558 N.
2. it will take 1.20 seconds for the ball to land. Hence, the correct option is (e) 1.20 seconds.
3. the magnitude of the average acceleration of the car is 13 m/s². Hence, the correct option is (a) 13 m/s².
4. the distance over which the force must be exerted is 0.5 meters. Hence, the correct option is (b) 0.5 meters.
1. Calculation of the magnitude of the net force on the car:
We know that,
Formula used for the calculation of net force is:
F = m * v²/r
F = (405 kg) * (5.5 m/s)²/120 m
F = 558 N
2. Calculation of time taken by the ball to land:
Given,
V₀ = 20 m/s, h = 7.2 m, and g = 9.81 m/s². Formula used for the calculation of time taken by the ball to land is:
t = (sqrt(2h/g))
t = sqrt(2 * 7.2/9.81)
t = 1.20 s (rounded to two decimal places)
3. Calculation of the magnitude of the average acceleration of the car:
Given,
Vᵢ = 20 m/s, Vf = 7 m/s, and t = 1 s. Formula used for the calculation of the magnitude of the average acceleration of the car is:
a = (Vf - Vᵢ)/t
a = (7 - 20)/1
a = -13 m/s²
4. Calculation of the distance over which the force must be exerted:
Given,m = 60 kg, F = 1200 N, Vf = 10 m/s, and V₀ = 0 m/s. Formula used for the calculation of the distance over which the force must be exerted is:
Vf² = V₀² + 2*a*d10² = 0 + 2*(F/m)*d10² = (2400/60)*dd = 0.5 m
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Charles Cansado launched a 100 g dart upwards from a height of 150 cm using a toy gun. The stiffness of the gun's spring is 1 000 N/m which was compressed 10 cm. Determine the impact velocity of the dart the instant it reaches its target at a height of 450 cm if the heat loss was 0.588 J. Determine the percentage efficiency of the shot.
The impact velocity of the dart when it reaches its target at a height of 450 cm is 5.20 m/s. The percentage efficiency of the shot is 95.2%.
In order to determine the impact velocity of the dart, we can use the principle of conservation of mechanical energy. The initial potential energy of the dart is given by mgh, where m is the mass of the dart (0.1 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the initial height (1.5 m). The final potential energy of the dart is mgh, where h is the final height (4.5 m). The initial kinetic energy of the dart is zero, as it was launched from rest. Therefore, the final kinetic energy of the dart is equal to the difference between the initial potential energy and the heat loss (0.588 J). Using these values, we can calculate the final velocity of the dart using the equation KE = 0.5mv^2, where KE is the kinetic energy, m is the mass of the dart, and v is the velocity.
The percentage efficiency of the shot can be determined by calculating the ratio of the actual energy output (final kinetic energy) to the theoretical maximum energy output (initial potential energy). The efficiency is then multiplied by 100 to express it as a percentage. In this case, the efficiency is 95.2%. This means that 95.2% of the energy stored in the spring was transferred to the dart as kinetic energy, while the remaining 4.8% was lost as heat.
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Find the electric potential induced by an uniformly polarized sphere (radius R, R polarization P). (15 marks)
The electric potential induced by a uniformly polarized sphere with radius R and polarization P is given by the formula V = (1/4πε₀) * (P/R).
The electric potential induced by a uniformly polarized sphere can be calculated using the formula V = (1/4πε₀) * (P/R).
The polarization of a sphere is a measure of the dipole moment per unit volume. It indicates the extent to which the charges in the sphere are displaced from their equilibrium positions. When a sphere is uniformly polarized, the dipole moment is constant throughout the volume of the sphere.
By using this formula, you can calculate the electric potential induced by a uniformly polarized sphere for a given radius and polarization. This provides a useful tool for understanding the electrical behavior of polarized spheres and their impact on the surrounding electric field.
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A 200 g metal container, insulated on the outside, holds 100 g of water in thermal equilibrium at 22.0°C. A 9 g ice cube, at temperature -20.0°C, is dropped into the water, and when thermal equilibrium is reached the temperature is 12.0°C. Assume there is no heat exchange with the surroundings. Find the specific heat of the metal the container is made from. cwater = 4190 J/kg∙C°
cice = 2090 J/kg∙C°
Lf = 3.34×105 J/kg
The specific heat of the metal container is approximately 2095 J/kg∙C°.
The specific heat of the metal container can be determined by applying the principle of conservation of energy and considering the heat transfer that occurs during the process.
To find the specific heat of the metal container, we need to calculate the amount of heat transferred during the process. We can start by calculating the heat transferred from the water to the ice, which causes the water's temperature to drop from 22.0°C to 12.0°C.
The heat transferred from the water to the ice can be calculated using the formula:
Qwater → ice = mcΔT
where:
m is the mass of the water (100 g),
c is the specific heat of water (4190 J/kg∙C°), and
ΔT is the change in temperature (22.0°C - 12.0°C = 10.0°C).
Substituting the given values into the equation, we have:
Qwater → ice = (0.1 kg) * (4190 J/kg∙C°) * (10.0°C) = 4190 J
The heat transferred from the water to the ice is equal to the heat gained by the ice, causing it to melt. The heat required to melt the ice can be calculated using the formula:
Qmelting = mLf
where:
m is the mass of the ice (9 g),
Lf is the latent heat of fusion for ice (3.34×10^5 J/kg).
Substituting the given values into the equation, we have:
Qmelting = (0.009 kg) * (3.34×10^5 J/kg) = 3010 J
Since the metal container is insulated and there is no heat exchange with the surroundings, the heat transferred from the water to the ice and the heat required to melt the ice must be equal. Therefore, we can equate the two equations:
Qwater → ice = Qmelting
4190 J = 3010 J
Now, we can solve for the specific heat of the metal container (cm) by rearranging the equation:
cm = Qwater → ice / (mwater * ΔTwater)
Substituting the known values, we get:
cm = (4190 J) / ((0.2 kg) * (10.0°C)) ≈ 2095 J/kg∙C°
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8. You observe a star through a telescope.
What happens to the apparent wavelength of the star's light as it moves toward you?
a) It gets shorter.
b) It gets longer.
c) It stays the same.
9. Explain your answer.
8. The apparent
wavelength
of the star's light gets shorter when it moves towards you.
Explanation:The wavelength of light is a measure of the distance between two successive peaks (or troughs) of a wave. As an object, such as a star, moves towards an observer, the
distance
between each successive peak of light waves appears to be shortened. This causes the apparent wavelength of the star's light to decrease, resulting in what is called blue shift.In contrast, when an object such as a star is moving away from an observer, the distance between each
successive peak
of light waves appears to be lengthened, causing the apparent wavelength of the star's light to increase. This is known as redshift.9. As an object moves towards an observer, its wavelength appears to decrease, leading to a shorter apparent wavelength of light. This is a phenomenon known as blue shift, which is caused by the Doppler effect.
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The correct answer is Option a) It gets shorter. When the apparent wavelength of the star's light as it moves toward you It gets shorter.
8. The apparent wavelength of the star's light gets shorter as it moves toward you. This phenomenon is known as "Doppler effect." When an object emitting waves, such as light or sound, moves toward an observer, the waves become compressed or "squeezed" together. This causes a shift towards the shorter wavelengths, resulting in a "blue shift." The opposite occurs when the object moves away from the observer, causing a shift towards longer wavelengths or a "red shift."
To better understand this, imagine a car passing by while honking its horn. As the car approaches, the pitch of the sound appears higher because the sound waves are compressed. Similarly, when a star moves toward us, its light waves are compressed, causing a blue shift in the spectrum. This shift can be observed in the laboratory and is a crucial tool for astronomers to study the motion of stars and galaxies.
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A hydrogen atom is in its ground state (nᵢ = 1) when a photon impinges upon it. The atom absorbs the photon, which has precisely the energy required to raise the atom to the nf = 3 state. (a) What was the photon's energy (in eV)? _________eV (b) Later, the atom returns to the ground state, emitting one or more photons in the process. Which of the following energies describes photons that might be emitted thus? (Select all that apply.) O 1.89 ev O 12.1 eV O 10.2 ev O 13.6 ev
A hydrogen atom is in its ground state (nᵢ = 1) when a photon impinges upon it. The atom absorbs the photon, which has precisely the energy required to raise the atom to the nf = 3 state. (a) The photon's energy that was absorbed is approximately 1.51 eV (negative sign indicates absorption).(b)option B and C are correct.
To determine the photon's energy and the energies of photons that might be emitted when the hydrogen atom returns to the ground state, we can use the energy level formula for hydrogen atoms:
E = -13.6 eV / n^2
where E is the energy of the electron in the atom, and n is the principal quantum number.
(a) To find the energy of the photon that was absorbed by the hydrogen atom to raise it from the ground state (nᵢ = 1) to the nf = 3 state, we need to calculate the energy difference between the two states:
ΔE = Ef - Ei = (-13.6 eV / 3^2) - (-13.6 eV / 1^2)
Calculating the value of ΔE:
ΔE = -13.6 eV / 9 + 13.6 eV
= -1.51 eV
Therefore, the photon's energy that was absorbed is approximately 1.51 eV (negative sign indicates absorption).
(b) When the hydrogen atom returns to the ground state, it can emit photons with energies corresponding to the energy differences between the excited states and the ground state. We need to calculate these energy differences and check which values are present among the given options.
ΔE1 = (-13.6 eV / 1^2) - (-13.6 eV / 3^2) = 10.20 eV
ΔE2 = (-13.6 eV / 1^2) - (-13.6 eV / 4^2) = 10.20 eV
ΔE3 = (-13.6 eV / 1^2) - (-13.6 eV / 5^2) = 12.10 eV
ΔE4 = (-13.6 eV / 1^2) - (-13.6 eV / 6^2) = 12.10 eV
ΔE5 = (-13.6 eV / 1^2) - (-13.6 eV / 7^2) = 13.55 eV
ΔE6 = (-13.6 eV / 1^2) - (-13.6 eV / 8^2) = 13.55 eV
ΔE7 = (-13.6 eV / 1^2) - (-13.6 eV / 9^2) = 13.55 eV
Comparing the calculated energy differences with the given options:
(A) 1.89 eV: This energy difference does not match any of the calculated values.
(B) 12.1 eV: This energy difference matches ΔE3 and ΔE4.
(C) 10.2 eV: This energy difference matches ΔE1 and ΔE2.
(D) 13.6 eV: This energy difference does not match any of the calculated values.
Therefore option B and C are correct.
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A tunnel diode can be connected to a microwave circulator to make a negative resistance amplifier. Support this statement with your explanations and a sketch
A tunnel diode can indeed be connected to a microwave circulator to create a negative resistance amplifier. This configuration takes advantage of the unique characteristics of a tunnel diode to amplify microwave signals effectively. The negative resistance property of the tunnel diode compensates for the losses in the circulator, resulting in overall signal amplification.
A tunnel diode is a semiconductor device that exhibits a negative resistance region in its current-voltage (I-V) characteristic curve. This negative resistance region allows the diode to amplify signals. When connected to a microwave circulator, which is a three-port device that directs microwave signals in a specific direction, the negative resistance property of the tunnel diode can compensate for the inherent losses in the circulator.
In the configuration, the microwave signal is input to one port of the circulator, and the tunnel diode is connected to another port. The negative resistance of the diode counteracts the losses in the circulator, resulting in signal amplification. The amplified signal can then be extracted from the third port of the circulator.
The combination of the tunnel diode and microwave circulator creates a stable and efficient negative resistance amplifier, suitable for microwave applications. This setup is commonly used in microwave communication systems, radar systems, and other high-frequency applications.
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If the exposure rate constant is 0. 87 Rcm2/mCi-hr and the average patient transmission factor is 0. 2, the exposure rate mR/hr. At 12. 5 cm for a patient who has been injected with 20 mCi of Tc-99m is 22 21 20 19
Answer:
To find the exposure rate (in mR/hr) at a distance of 12.5 cm, we can use the following equation:
Exposure Rate (mR/hr) = Exposure Rate Constant (Rcm²/mCi-hr) × Activity (mCi) × Transmission Factor / Distance² (cm²)
Plugging in the given values:
Exposure Rate (mR/hr) = 0.87 Rcm²/mCi-hr × 20 mCi × 0.2 / (12.5 cm)²
Exposure Rate (mR/hr) = 17.4 Rcm²/hr × 0.2 / 156.25 cm²
Exposure Rate (mR/hr) = 3.48 Rcm²/hr / 156.25 cm²
Exposure Rate (mR/hr) ≈ 0.0223 R/hr
Since 1 R (Roentgen) is equal to 1000 mR (milliroentgen), we can convert the exposure rate to mR/hr:
Exposure Rate (mR/hr) ≈ 0.0223 R/hr × 1000 mR/R
Exposure Rate (mR/hr) ≈ 22.3 mR/hr
The closest answer choice is:
A) 22
Imagine you're an astronaut working on the new space station in orbit around Mars While working a distance 154 m from the station, your cool little jet pack goes out and you have no way to get back to safely. Fortunately, you're a physics fan so you calmly and cooly use that knowledge and loss your 18 kg jetpack at a speed of 19 m/s directly away from the station to make your way back to safety Part A How long does it take you to reach the space station after the jetpack leaves your hands? Assume that the combined mass of you and your space suite is 100 kg NoteBe sure to round to the appropriate number of significant figures as the final step of your calculation before submitting your response unde vado reset keyboard shortcuts help Value Units
After losing your 18 kg jetpack at a speed of 19 m/s away from the space station, it will take approximately 45.0 seconds for you to reach the station.
To calculate the time it takes for you to reach the space station, we can apply the principle of conservation of momentum. Initially, the total momentum of the system (you and your jetpack) is zero since you are at rest relative to the space station.
When you release the jetpack, it gains momentum in one direction, causing you to gain an equal amount of momentum in the opposite direction.
The conservation of momentum equation can be written as:
m1 * v1 = m2 * v2
where m1 and v1 are the mass and velocity of the jetpack, and m2 and v2 are the mass and velocity of you and your space suit.
Substituting the given values (m1 = 18 kg, v1 = -19 m/s, m2 = 100 kg), we can solve for v2, the velocity of you and your space suit after releasing the jetpack. Rearranging the equation, we have:
v2 = (m1 * v1) / m2
v2 = (18 kg * -19 m/s) / 100 kg
v2 = -3.42 m/s
Since you and your space suit are initially at rest, the final velocity is equal to the relative velocity between you and the space station. The distance between you and the station is 154 m, and to find the time it takes to cover this distance, we use the equation:
time = distance / velocity
time = 154 m / 3.42 m/s
time ≈ 45.0 seconds
Rounding to the appropriate number of significant figures, it will take approximately 45.0 seconds for you to reach the space station after the jetpack leaves your hands.
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current of 10.0 A, determine the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them. Tries 4/10 Previous Tries
Therefore, the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them is 5.42 × 10⁻⁵ T.
Two circular coils are placed one over the other such that they share a common axis. The radius of the top coil is 0.120 m and it carries a current of 2.00 A. The radius of the bottom coil is 0.220 m and it carries a current of 10.0 A.
Determine the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them.Step-by-step solution:Here, N1 = N2 = 1 (because they haven't given the number of turns for the coils)Radius of top coil, r1 = 0.120 m, current in the top coil, I1 = 2.00 ARadius of bottom coil, r2 = 0.220 m, current in the bottom coil, I2 = 10.0 AWe have to determine the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them,
such that,B = μ0(I1 / 2r1 + I2 / 2r2)Putting the given values in the above equation, we get,B = 4π × 10⁻⁷ (2 / 2 × 0.120 + 10 / 2 × 0.220)B = 4π × 10⁻⁷ (1 / 0.12 + 5 / 0.22)B = 5.42 × 10⁻⁵ TTherefore, the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them is 5.42 × 10⁻⁵ T.
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During a very quick stop, a car decelerates at 6.8 m/s2. Assume the forward motion of the car corresponds to a positive direction for the rotation of the tires (and that they do not slip on the pavement).
Randomized Variablesat = 6.8 m/s2
r = 0.255 m
ω0 = 93 rad/s
Part (a) What is the angular acceleration of its tires in rad/s2, assuming they have a radius of 0.255 m and do not slip on the pavement?
Part (b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is 93 rad/s ?
Part (c) How long does the car take to stop completely in seconds?
Part (d) What distance does the car travel in this time in meters?
Part (e) What was the car’s initial speed in m/s?
Part (a). the angular acceleration of the tires is 26.67 rad/s².Part (b)the tires make approximately 80.85 revolutions before coming to rest.Part (c)the car takes 3.49 seconds to stop completely.Part (d) the car travels 83.85 meters.Part (e)the initial speed of the car was 23.7 m/s.
Part (a)Angular acceleration, α can be calculated using the formula α = at/r.Substituting at = 6.8 m/s² and r = 0.255 m, we getα = 6.8/0.255α = 26.67 rad/s²Therefore, the angular acceleration of the tires is 26.67 rad/s².
Part (b)To calculate the number of revolutions the tires make before coming to rest, we can use the formulaω² - ω0² = 2αθwhere ω0 = 93 rad/s, α = 26.67 rad/s², and ω = 0 (since the tires come to rest).Substituting these values in the above equation and solving for θ, we getθ = ω² - ω0²/2αθ = (0 - (93)²)/(2(26.67))θ = 129.97 radThe number of revolutions the tires make can be calculated as follows:Number of revolutions, n = θ/2πrwhere r = 0.255 mSubstituting the values of θ and r, we getn = 129.97/(2π(0.255))n = 80.85 revTherefore, the tires make approximately 80.85 revolutions before coming to rest.
Part (c)Time taken by the car to stop, t can be calculated as follows:t = ω/αwhere ω = 93 rad/s and α = 26.67 rad/s²Substituting these values in the above equation, we gett = 3.49 sTherefore, the car takes 3.49 seconds to stop completely.
Part (d)Distance traveled by the car, s can be calculated using the formula,s = ut + 1/2 at²where u = initial velocity = final velocity, a = deceleration = -6.8 m/s² and t = 3.49 s.Substituting the values of u, a, and t in the above equation, we get,s = ut + 1/2 at²s = ut + 1/2 (-6.8)(3.49)²s = us = 83.85 mTherefore, the car travels 83.85 meters during this time.
Part (e)Initial speed of the car, u can be calculated using the formulau = ω0 ru = 93(0.255)u = 23.7 m/sTherefore, the initial speed of the car was 23.7 m/s.
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A spacecraft is moving through a vaccum. It changes its velocity from 9050 ft/sec to 5200 ft/sec in 48 seconds. Calculate the power required to accomplished this if the spacecraft mass is 13,000 slugs.
When the spacecraft moving through a vaccum, changes its velocity from 9050 ft/sec to 5200 ft/sec in 48 seconds then the power required to change the velocity of the spacecraft is -5,491,500,000 ft·lb²/sec³.
The power required to change the velocity of a spacecraft can be calculated using the formula P = Fv, where P is power, F is the force applied, and v is the velocity change.
First, we need to find the force applied to the spacecraft.
The force can be determined using Newton's second law of motion, F = ma, where F is the force, m is the mass of the spacecraft, and a is the acceleration.
To find the acceleration, we can use the formula a = (v_final - v_initial) / t, where v_final is the final velocity, v_initial is the initial velocity, and t is the time taken to change the velocity.
Given that the initial velocity (v_initial) is 9050 ft/sec, the final velocity (v_final) is 5200 ft/sec, and the time (t) is 48 seconds, we can calculate the acceleration:
a = (5200 - 9050) / 48 = -81.25 ft/sec²
Since the spacecraft is decelerating, the acceleration is negative.
Now we can calculate the force:
F = ma = 13000 slugs * -81.25 ft/sec² = -1,056,250 ft·lb/sec²
Finally, we can calculate the power:
P = Fv = (-1,056,250 ft·lb/sec²) * 5200 ft/sec = -5,491,500,000 ft·lb²/sec³
Therefore, the power required to change the velocity of the spacecraft is -5,491,500,000 ft·lb²/sec³.
The negative sign indicates that work is being done on the spacecraft to decelerate it.
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If an AC generator is provides a voltage given by ΔV=1.20×10 2
V " sin(30πt), and the current passes thru and Inductor with value 0.500H. Calculate the following parameters:
The rms value of current in the inductor is 169.7 A.The frequency of the generator is 15 Hz.The inductive reactance of the inductor is 47.1 Ω.
Given, ΔV=1.20×10^2V sin(30πt), and L=0.500H
We know that V = L di/dt
Here, ΔV = V = 1.20×10^2V sin(30πt)
By integrating both sides, we get∫di = (1/L)∫ΔV dt
Integrating both sides with respect to time, we get:i(t) = (1/L) ∫ΔV dt
The integral of sin(30πt) will be - cos(30πt) / (30π)
Let's substitute the values:∫ΔV dt = ∫1.20×10^2 sin(30πt) dt = -cos(30πt) / (30π)
Therefore, i(t) = (1/L) (-cos(30πt) / (30π))
Now, we can calculate the following parameters:
Peak value of current, I0= (1/L) × Vmax= (1/0.5) × 120= 240 A
So, the peak value of current is 240 A.
The rms value of current is given by Irms= I0/√2= 240/√2= 169.7 A
Therefore, the rms value of current in the inductor is 169.7 A.
The given voltage equation is ΔV=1.20×10^2 V sin(30πt)
The voltage equation is given by Vmax sinωt
Here, Vmax = 1.20×10^2V and ω = 30π
The frequency of the generator is given by f = ω / (2π) = 15 Hz
Therefore, the frequency of the generator is 15 Hz.
The inductive reactance of an inductor is given by XL= 2πfL= 2 × 3.14 × 15 × 0.5= 47.1 Ω
Therefore, the inductive reactance of the inductor is 47.1 Ω.
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At the last stage of stellar evolution, a heavy star can collapse into an extremely dense object made mostly of neutrons. The star is called a neutron star. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was the solar radius 8.5×10 5
km; its final radius is 7.1 km. If the original star rotated with the solar rotation period 19 days, find the rotation period of the collapsed neutron star in the unit of millisecond.
At the last stage of stellar evolution, a heavy star can collapse into an extremely dense object made mostly of neutrons. the rotation period of the collapsed neutron star is approximately 0.5 milliseconds.
To find the rotation period of the collapsed neutron star, we can apply the principle of conservation of angular momentum. Since the neutron star is a rigid object, its angular momentum will remain constant before and after the collapse.
The formula for angular momentum (L) is given by the product of moment of inertia (I) and angular velocity (ω):
L = I * ω
Since the neutron star is assumed to be a uniform, solid, rigid sphere, its moment of inertia can be calculated using the formula for a solid sphere:
I = (2/5) * M * R²
Where M is the mass of the neutron star and R is its radius.
Now, let's consider the initial star and the collapsed neutron star:
For the initial star:
Initial radius (R_initial) = 8.5 × 10^5 km
Initial rotation period (T_initial) = 19 days
For the neutron star:
Final radius (R_final) = 7.1 km
Final rotation period (T_final) = unknown (to be calculated)
The mass (M) of the star remains the same before and after the collapse.
Using the conservation of angular momentum, we can equate the initial and final angular momenta:
I_initial * ω_initial = I_final * ω_final
Substituting the expressions for moment of inertia and angular velocity:
[(2/5) * M * R_initial²] * (2π / T_initial) = [(2/5) * M * R_final²] * (2π / T_final)
Simplifying the equation and canceling common factors:
(R_initial² / T_initial) = (R_final² / T_final)
Substituting the known values:
[(8.5 × 10^5 km)² / (19 days)] = [(7.1 km)² / T_final]
Converting the units to a common form:
[(8.5 × 10^5 km)² / (19 days)] = [(7.1 km)² / (T_final * 86,400 seconds/day)]
Solving for T_final:
T_final = [(7.1 km)² * (19 days) * (86,400 seconds/day)] / [(8.5 × 10^5 km)²]
Calculating the value:
T_final ≈ 0.5 milliseconds
Therefore, the rotation period of the collapsed neutron star is approximately 0.5 milliseconds.
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b) Obtain Tc, the temperature at point c.
To obtain the temperature at point C, we need to analyze the given information or equations related to the system.
The specific method or equations required to determine the temperature at point C will depend on the specific context or problem at hand. In order to provide a more specific answer on how to obtain the temperature at point C, additional information or context is needed. The approach to determining the temperature at point C can vary depending on the nature of the problem, such as whether it involves heat transfer, thermodynamics, or a specific system or process. If you can provide more details about the problem or context in which point C is mentioned, I can provide a more tailored explanation of how to obtain the temperature at that point.
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Without plagiarizing. Write a meaningful Thesis paragraph about
Einstein's life and Contribution to quantum physics
Here is a Thesis about Einstein's life and Contribution to quantum physics.
Albert Einstein, widely regarded as the most brilliant scientist of the twentieth century, was one of the pioneering figures in the field of quantum physics.
He was a theoretical physicist who is best known for developing the theory of relativity and for his contributions to the development of quantum mechanics. Einstein's work in quantum physics helped to revolutionize our understanding of the nature of reality and the behavior of matter at the atomic and subatomic levels. His contributions to the field have had a profound impact on modern physics, and his ideas continue to influence research in this area to this day.
This paper will explore Einstein's life and his significant contribution to quantum physics.
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A 1.40-cm-tall object is placed along the principal axis of a thin convex lens of 13.0 cm focal length. If the object distance is 19.2 cm, which of the following best describes the image distance and height, respectively? a. 7.75 cm and 4.34 cm b. 40.3 cm and 2.94 cm c. 7.75 cm and 7.27 cm d. 9.16 cm and 4.34 cm e. 41.4 cm and 0.668 cm
The best description for the image distance and height, respectively, is: Image distance: Approximately 7.75 cm; Image height: Approximately 0.561 cm. To determine the image distance and height, we can use the lens equation and magnification formula.
The lens equation is given by:
1/f = 1/do + 1/di
Where:
f = focal length of the lens
do = object distance
di = image distance
Substituting the given values:
f = 13.0 cm
do = 19.2 cm
1/13.0 = 1/19.2 + 1/di
To find the image distance, we rearrange the equation:
1/di = 1/13.0 - 1/19.2
di = 1 / (1/13.0 - 1/19.2)
di ≈ 7.75 cm
Now, let's calculate the image height using the magnification formula:
m = -di/do
Where:
m = magnification
do = object distance
di = image distance
m = -7.75 cm / 19.2 cm
m ≈ -0.4036
The negative sign indicates that the image is inverted.
The image height can be calculated using the formula:
hi = |m| *
Where:
hi = image height
h o = object height
Given:
hi = |-0.4036| * 1.40 cm
hi ≈ 0.561 cm
Therefore, the best description for the image distance and height, respectively, is:
Image distance: Approximately 7.75 cm
Image height: Approximately 0.561 cm
The closest option to these values is option e. 41.4 cm and 0.668 cm, although the calculated values do not exactly match this option.
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A scuba tank, when fully submerged, displaces 14.1 L of seawater. The tank itself has a mass of 13.5 kg and, when "full," contains 1.25 kg of air. Assuming only a weight and buoyant force act, determine the net force (magnitude) on the fully submerged tank at the beginning of a dive (when it is full of air). Express your answer with the appropriate units. X Incorrect; Try Again; 2 attempts remaining Express your answer with the appropriate units.
The net force on the tank is 10.13 Newtons (N). So, the coorect anser is 10.13 N.
To determine the net force, we need to consider the weight of the tank and the buoyant force acting on it.
1. Weight of the tank:
Weight = mass * acceleration due to gravity
Weight = 13.5 kg * 9.8 m/s^2
The weight of the tank is approximately 132.3 N.
2. Buoyant force:
Buoyant force = density of fluid * volume displaced * acceleration due to gravity
First, let's convert the volume of seawater displaced by the tank to cubic meters:
Volume = 14.1 L * 0.001 m^3/L
The volume is approximately 0.0141 m^3.
Now, let's calculate the buoyant force using the density of seawater, which is approximately 1025 kg/m^3:
Buoyant force = 1025 kg/m^3 * 0.0141 m^3 * 9.8 m/s^2
The buoyant force is approximately 142.43 N.
3. Net force:
Net force = Buoyant force - Weight
Net force = 142.43 N - 132.3 N
The net force on the fully submerged scuba tank at the beginning of a dive is approximately 10.13 N.
Therefore, the net force on the tank is 10.13 Newtons (N).
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In a particular application, the current in the inner conductor is 1.30 A out of the page, and the current in the outer conductor is 2.52 A into the page. Determine the magnitude of the magnetic field at point Tries 0/10 Determine the magnitude of the magnetic field at point b. Tries 0/10
Therefore, the magnitude of the magnetic field at point T due to the inner conductor is 5.49 x 10^-6 Tesla, and the magnitude of the magnetic field at point T due to the outer conductor is 1.94 x 10^-6 Tesla. Note that the direction of the magnetic field is out of the page for the inner conductor and into the page for the outer conductor.
Given the following information:Current flowing through inner conductor = 1.30 A (out of the page)Current flowing through outer conductor = 2.52 A (into the page)To determine the magnitude of the magnetic field at point T, we use the right-hand thumb rule, which states that if we grip a wire with our right hand and point our thumb in the direction of current flow, our fingers will curl in the direction of the magnetic field (i.e. counter-clockwise or clockwise).
Since the current is out of the page in the inner conductor, the magnetic field is also directed out of the page. For the outer conductor, the current is flowing into the page, so the magnetic field is directed into the page.Using Ampere's circuital law, we can find the magnitude of the magnetic field at point T.
Ampere's law states that the line integral of the magnetic field around a closed path is equal to the current enclosed by the path times the permeability of free space (μ0).B = μ0I / 2πrWhere,I = Current enclosed by the pathμ0 = Permeability of free space = 4π x 10^-7 Tesla meter per ampere2πr = Circumference of the circular path at point TFor the inner conductor, the current enclosed by the path is 1.30 A, soB = (4π x 10^-7) x 1.30 / (2π x 0.15) = 5.49 x 10^-6 Tesla
For the outer conductor, the current enclosed by the path is 2.52 A - 1.30 A = 1.22 A, soB = (4π x 10^-7) x 1.22 / (2π x 0.25) = 1.94 x 10^-6 Tesla
Therefore, the magnitude of the magnetic field at point T due to the inner conductor is 5.49 x 10^-6 Tesla, and the magnitude of the magnetic field at point T due to the outer conductor is 1.94 x 10^-6 Tesla. Note that the direction of the magnetic field is out of the page for the inner conductor and into the page for the outer conductor.
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Calculate the following: a) A point charge q is located at distance z above a grounded conducting plane. Find the net force exerted by the conducting plane on the charge. b) Calculate the induced charge density on the conducting plane.
The net force exerted by the conducting plane on the charge, Net force = -q² / [2ε(h+z)²].
Induced charge density on the conducting plane is, Induced charge density = -q / (2πh) where q is the charge and h is the distance of charge q from the grounded conducting plane.
a. The net force exerted on the point charge by the grounded conducting plane:
Given that a point charge q is located at a distance z above a grounded conducting plane, we want to find the net force exerted by the conducting plane on the charge.
We define h as the distance of charge q from the grounded conducting plane. The net force exerted on the point charge by the grounded conducting plane is given by the equation:
F = -q² / [2ε(h+z)²]
where ε represents the permittivity of free space. The negative sign in the expression indicates that the net force exerted by the conducting plane is opposite to the direction of the charge q.
b. The induced charge density on the conducting plane:
The induced charge density can be calculated by,
Induced charge density = -q / (2πh)
This formula provides the charge density induced on the conducting plane as a result of the presence of the point charge q, where q is the charge and h is the distance of charge q from the grounded conducting plane.
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A 278 kg crate hangs from the end of a rope of length L = 13.3 m. You push horizontally on the crate with a varying force F to move it distance d = 4.94 m to the side (see the figure). (a) What is the magnitude of F when the crate is in this final position? During the crate's displacement, what are (b) the total work done on it, (c) the work done by the gravitational force on the crate, and (d) the work done by the pull on the crate from the rope? (e) Knowing that the crate is motionless before and after its displacement, use the answers to (b), (c), and (d) to find the work your force F does on the crate. (a) Number ________Units ____________
(b) Number ________Units ____________
(c) Number ________Units ____________
(d) Number ________Units ____________
(e) Number ________Units ____________
A 278 kg crate hangs from the end of a rope of length L = 13.3 m. You push horizontally on the crate with a varying force F to move it distance d = 4.94 m to the side .(a)Magnitude of F: 2671 N(b) Total work done: 13,186 J(c) Work done by gravity: -12,868 J(d) Work done by the rope: 12,868 J(e) Work done by force F: 12,186 J
To solve this problem, we need to analyze the forces involved and calculate the work done. Let's break it down step by step:
(a) To find the magnitude of force F when the crate is in its final position, we need to consider the equilibrium of forces. In this case, the horizontal force you apply (F) must balance the horizontal component of the gravitational force. Since the crate is motionless before and after displacement, the net force in the horizontal direction is zero.
Magnitude of F = Magnitude of the horizontal component of the gravitational force
= Magnitude of the gravitational force × cosine(theta)
The angle theta can be determined using trigonometry. It can be calculated as:
theta = arccos(d / L)
where d is the displacement (4.94 m) and L is the length of the rope (13.3 m).
Once we have the value of theta, we can calculate the magnitude of F using the given information about the crate's mass.
(b) The total work done on the crate can be calculated as the product of the force applied (F) and the displacement (d):
Total work done = F × d
(c) The workdone by the gravitational force on the crate can be calculated using the formula:
Work done by gravity = -m × g × d ×cos(theta)
where m is the mass of the crate (278 kg), g is the acceleration due to gravity (9.8 m/s²), d is the displacement (4.94 m), and theta is the angle calculated earlier.
(d) The work done by the pull on the crate from the rope is given by:
Work done by the rope = F × d × cos(theta)
(e) Knowing that the crate is motionless before and after its displacement, the net work done on the crate by all forces should be zero. Therefore, the work done by your force F can be calculated as:
Work done by force F = Total work done - Work done by gravity - Work done by the rope
Now let's calculate the values:
(a) To find the magnitude of F:
theta = arccos(4.94 m / 13.3 m) = 1.222 rad
Magnitude of F = (278 kg × 9.8 m/s²) ×cos(1.222 rad) ≈ 2671 N
(b) Total work done = F × d = 2671 N × 4.94 m ≈ 13,186 J
(c) Work done by gravity = -m × g × d × cos(theta) = -278 kg × 9.8 m/s² × 4.94 m × cos(1.222 rad) ≈ -12,868 J
(d) Work done by the rope = F × d × cos(theta) = 2671 N * 4.94 m * cos(1.222 rad) ≈ 12,868 J
(e) Work done by force F = Total work done - Work done by gravity - Work done by the rope
= 13,186 J - (-12,868 J) - 12,868 J ≈ 12,186 J
The answers to the questions are:
(a) Magnitude of F: 2671 N
(b) Total work done: 13,186 J
(c) Work done by gravity: -12,868 J
(d) Work done by the rope: 12,868 J
(e) Work done by force F: 12,186 J
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You have an 8 -pole DC machine with a lap winding. The emf generated by the machine is 118 V. What would the emf of a similar machine with a wave winding be?
The emf of a similar machine with a wave winding would also be 118 V.
The emf (electromotive force) generated by a DC machine depends on various factors such as the number of poles, the speed of rotation, the magnetic field strength, and the winding configuration.
In this case, we have an 8-pole DC machine with a lap winding. Lap winding is a winding configuration where each armature coil overlaps with adjacent coils in a parallel manner.
When we consider a similar machine with a wave winding, it means the winding configuration changes to a wave winding. In a wave winding, the armature coils are connected in a wave-like pattern, where each coil is connected to the adjacent coil in a series manner.
Changing the winding configuration from lap winding to wave winding does not affect the number of poles or the magnetic field strength. Therefore, the only significant difference between the two machines is the winding configuration.
Since the emf generated by a machine depends on the speed of rotation, magnetic field strength, and winding configuration, and these factors remain the same in this scenario, the emf of a similar machine with a wave winding would still be 118 V, the same as the original machine.
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Describe how the scientific approach is different than other
ways of understanding.
Mathematical quantitative formulas to get answers.
The scientific approach is different from other ways of understanding in that it is based on empirical evidence and the use of the scientific method. Unlike other approaches that rely on intuition, tradition, or authority, the scientific approach is objective and systematic, and it uses empirical evidence to test hypotheses and theories.
A scientific approach uses observation, experimentation, and data analysis to answer questions and solve problems. It involves developing a hypothesis, testing the hypothesis through experiments, collecting and analyzing data, and drawing conclusions based on the evidence collected. The scientific approach is designed to minimize biases and errors, and it is constantly open to revision based on new evidence.
The scientific approach is also different from other approaches in that it emphasizes the importance of replication and independent verification of findings. This helps to ensure that scientific findings are reliable and not the result of chance or errors in the research process.
The use of mathematical quantitative formulas is an important part of the scientific approach, as it allows researchers to measure and analyze data in a rigorous and systematic way. Mathematical formulas help to provide precise answers to research questions, and they can help to identify patterns and relationships in data that might not be apparent through qualitative analysis.
In summary, the scientific approach is different from other ways of understanding in that it is based on empirical evidence, uses the scientific method, and is designed to minimize biases and errors. It emphasizes the importance of replication and independent verification of findings, and it makes use of mathematical quantitative formulas to get answers.
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The molar mass of argon is M = 40 x 10⁻³ kg/mol, and the molar mass of helium is M = 4 x 10⁻³ kg/mol. a) Find vᵣ ₘₛ for an argon atom if 1 mol of the gas is confined to a 1-liter container at a pressure of 10 atm. b) Find vᵣ ₘₛ for a helium atom under the same conditions and compare it to the value you calculated for argon. c) How much heat is removed when 100 g of steam at 150°C is cooled and frozen into 100 g of ice at 0°C. Note that the specific heat of ice is 2,010 J/kg·K and the specific heat of liquid water is 4,186 J/kg·K.
The root mean square velocity of an argon atom under the given conditions is approximately 226.23 m/s. The root mean square velocity for a helium atom under the given conditions is also approximately 226.23 m/s. The amount of heat removed when 100 g of steam at 150°C is cooled and frozen into 100 g of ice at 0°C is 661,300 J.
a) To find vᵣ ₘₛ for an argon atom if 1 mol of the gas is confined to a 1-liter container at a pressure of 10 atm, use the ideal gas law formula:
vᵣ ₘₛ = RT/P
where R is the gas constant, T is the temperature, and P is the pressure.
Given:
R = 8.31 J/(mol·K)
T = 273 K (room temperature)
P = 10 atm
vᵣ ₘₛ = (8.31 J/(mol·K) * 273 K) / (10 atm) ≈ 226.23 m/s
Therefore, the root mean square velocity of an argon atom under the given conditions is approximately 226.23 m/s.
b) For a helium atom under the same conditions, use the same formula:
vᵣ ₘₛ = RT/P
Substituting the values:
vᵣ ₘₛ = (8.31 J/(mol·K) * 273 K) / (10 atm) ≈ 226.23 m/s
The root mean square velocity for a helium atom under the given conditions is also approximately 226.23 m/s.
Comparing the values, it is seen that the root mean square velocities of argon and helium are the same.
c) To calculate the amount of heat removed when 100 g of steam at 150°C is cooled and frozen into 100 g of ice at 0°C, we need to consider two processes: cooling the steam and freezing the water.
Cooling the steam:
Q1 = m1 * c1 * ΔT1
where m1 is the mass, c1 is the specific heat capacity, and ΔT1 is the change in temperature.
Given:
m1 = 100 g
c1 (specific heat of steam) = 4,186 J/(kg·K)
ΔT1 = 150°C - 0°C = 150 K
Q1 = 100/1000 * 4,186 J/(kg·K) * 150 K = 627,900 J
Freezing the water:
Q2 = m2 * L
where m2 is the mass and L is the latent heat of fusion.
Given:
m2 = 100 g
L (latent heat of fusion) = 334,000 J/kg
Q2 = 100/1000 * 334,000 J/kg = 33,400 J
The total heat removed is the sum of Q1 and Q2:
Q = Q1 + Q2 = 627,900 J + 33,400 J = 661,300 J
Therefore, the amount of heat removed when 100 g of steam at 150°C is cooled and frozen into 100 g of ice at 0°C is 661,300 J.
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If a runners power is 400 W as runs, how much chemical energy does she convert into other forms in 10.0 minutes?
Answer:
If a runner's power is 400 watts as she runs , then the chemical energy she converts into other forms in 10.0 minutes would be 240,000 Joules . This information may be found in several of the search results provided, including result numbers 1, 2, 4, 5, 6, 8, and 9.
Explanation:
Newton's theory of gravity consists of Select all that apply. the law of gravitational force the three laws of motion the law of conservation of angular momentum the principle of equivalence the principle of energy
Newton's theory of gravity consists of the law of gravitational force and the three laws of motion.
Newton's theory of gravity, formulated by Sir Isaac Newton in the 17th century, encompasses several key principles. One of the fundamental components of this theory is the law of gravitational force, which states that every particle in the universe attracts every other particle with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them.
Additionally, Newton's theory of gravity includes the three laws of motion. The first law, known as the law of inertia, states that an object at rest will remain at rest, and an object in motion will continue to move at a constant velocity unless acted upon by an external force. The second law describes how the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. The third law states that for every action, there is an equal and opposite reaction.
However, the law of conservation of angular momentum, the principle of equivalence, and the principle of energy are not specific components of Newton's theory of gravity. The law of conservation of angular momentum pertains to the conservation of angular momentum in rotational systems. The principle of equivalence is a fundamental concept in Einstein's theory of general relativity, stating that the effects of gravity are indistinguishable from the effects of acceleration. The principle of energy, though a fundamental concept in physics, is not exclusively associated with Newton's theory of gravity but applies to various aspects of the physical world.
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The work function of a metal surface is 4.80 x 10⁻¹⁹ J. The maximum speed of the emitted electrons is va = 730 km/s when the wavelength of the light is λA. However, a maximum speed of vB = 500 km/s is observed when the wavelength is λB. Find the wavelengths.
The wavelengths of the electrons at maximum speed 730km/s and 500 km/s are 1.008 × 10^-12 km and 6.9× 10^-13 km respectively.
What is wavelength?Wavelength is the distance between identical points (adjacent crests) in the adjacent cycles of a waveform signal propagated in space or along a wire.
Wavelength can also be defined as the distance between two successive crest or trough.
Work function of a surface is the minimum energy required to a free electrons to come out of the metal surface.
W = h( v/λ)
where h is the Planck constant = 6.63 × 10^-34 J/s
Therefore;
4.80 × 10^-19 = 6.63 × 10^-34 × 730/λ
λ = 6.63 × 10^-34 × 730)/4.80 × 10^-19
λ = 1.008 × 10^-12 km
Also
4.80 × 10^-19 = 6.63 × 10^-34 × 500/λ
λ = 6.63 × 10^-34 × 500)/4.80 × 10^-19
λ = 6.9× 10^-13 km
Therefore the wavelengths are 1.008 × 10^-12 km and 6.9× 10^-13 km
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A 9500 kg spacecraft leaves the surface of the Earth for a mission in deep space. What is the change in the gravitational potential energy of the Earth+spacecraft system between when it was at the surface and when it reaches a location that is 5 times the radius of the Earth away from the Earth's center? If needed, use 6 x 10²⁴ kg as the mass of the Earth, 6.4 x 10⁶ m as the radius of the Earth, and 6.7×10⁻¹¹ N-m²/kg² as the universal gravitational constant.
The change in gravitational potential energy is - 3.31 x 10¹⁹ J.
Mass of the Earth, m = 6 x 10²⁴ kg
Radius of the Earth, r = 6.4 x 10⁶ m
Universal gravitational constant, G = 6.7×10⁻¹¹ N-m²/kg²
Mass of spacecraft, m = 9500 kg
At the surface of the Earth, the gravitational potential energy of the Earth+spacecraft system is given by;
U₁ = - GMm/R
Here,
M = mass of the Earth = 6 x 10²⁴ kg
m = mass of the spacecraft = 9500 kg
R = radius of the Earth = 6.4 x 10⁶ m
G = Universal gravitational constant = 6.7×10⁻¹¹ N-m²/kg²
U₁ = - (6.7×10⁻¹¹) x (6 x 10²⁴) x (9500) / (6.4 x 10⁶)
U₁ = - 8.407 x 10¹⁰ J
At a distance of 5 times the radius of the Earth from the Earth's center, the gravitational potential energy of the Earth+spacecraft system is given by;
U₂ = - GMm/2r
Here,
r = 5 x r = 5 x 6.4 x 10⁶ = 32 x 10⁶ m
U₂ = - (6.7×10⁻¹¹) x (6 x 10²⁴) x (9500) / (2 x 32 x 10⁶)
U₂ = - 1.171 x 10¹⁰ J
The change in gravitational potential energy of the Earth+spacecraft system between when it was at the surface and when it reaches a location that is 5 times the radius of the Earth away from the Earth's center is;
ΔU = U₂ - U₁
ΔU = - 1.171 x 10¹⁰ - (- 8.407 x 10¹⁰)
ΔU = - 3.31 x 10¹⁹ J
Therefore, the change in gravitational potential energy of the Earth+spacecraft system between when it was at the surface and when it reaches a location that is 5 times the radius of the Earth away from the Earth's center is - 3.31 x 10¹⁹ J.
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