The sounds emitting from Ross' car can be characterized as low frequency, high amplitude.
The question states that pedestrians often stare at Ross' car due to the loud, low-pitched booming sound they hear. From this description, we can infer certain characteristics of the sound.
Low frequency refers to sounds with a lower pitch, such as deep bass notes. These low-pitched sounds are associated with lower frequencies on the sound spectrum.
High amplitude refers to the intensity or loudness of the sound. When a sound is described as loud, it indicates a high amplitude or a greater magnitude of sound waves.
Therefore, the sounds emitting from Ross' car can be characterized as low frequency (low-pitched) and high amplitude (loud). This combination of characteristics results in the loud, low-pitched booming sound that draws the attention of pedestrians.
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Starting from rest, an electron accelerates through a potential difference of 40 V. What is its de Broglie wavelength? (h=6.63 x 1034 J-s, me 9.11 x 10 kg. and 1 eV = 1.60 x 10-¹9 J)
The de Broglie wavelength of a particle is given by the formula λ = h / p, where λ is the de Broglie wavelength, h is Planck's constant, and p is the momentum of the particle.
The de Broglie wavelength of an electron accelerated through a potential difference of 40 V can be calculated using the formula λ = h / √(2meE), where λ is the de Broglie wavelength, h is the Planck's constant, me is the mass of the electron, and E is the energy gained by the electron.
The energy gained by the electron can be calculated using the equation E = qV, where q is the charge of the electron and V is the potential difference. By substituting the given values into the equations, the de Broglie wavelength of the electron can be determined.
The de Broglie wavelength of a particle is given by the formula λ = h / p, where λ is the de Broglie wavelength, h is Planck's constant, and p is the momentum of the particle. For an electron, the momentum can be calculated using the equation p = √(2meE), where me is the mass of the electron and E is the energy gained by the electron.
To calculate the energy gained by the electron, we can use the equation E = qV, where q is the charge of the electron and V is the potential difference. Given that 1 eV is equal to 1.60 x 10^(-19) J, we can convert the potential difference of 40 V to energy by multiplying it by the charge of an electron.
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If the bus's velocity at time t1 = 1.20 s is 5.05 m/s, what is its velocity at time t2 = 2.20 s?
To determine the velocity of the bus at time t2 = 2.20 we need to find the change in velocity from time t1 = 1.20 s to t2. The change in velocity is given by the formula: Change in velocity = final velocity - initial velocity
Given that the initial velocity at t1 is 5.05 m/s, we can substitute this value into the formula:
Change in velocity = final velocity - 5.05 m/s
Since no additional information is provided, we cannot determine the exact final velocity at t2 = 2.20 s. We can only find the change in velocity.
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An electron travels along the y axis, in the direction of decreasing y, with speed of 2.5 × 103m/s. The electron enters a region where there is a magnetic field of 2.0 T that is pointing in the direction of increasing x, and an electric field of 3.0 × 10^3 V /m that is pointing in the direction
of increasing z.
Express/Find the following in unit vector notation (i.e., in terms of i, j, K)
a the velocity of the electron
b) the electric field
c) the magnetic field
d) the electric force on the electron e) the magnetic force on the electron
) the total force on the electron
g) the acceleration of the electron
a) Velocity of the electron: v = -2.5 × 10^3 j m/s
b) Electric field: E = 3.0 × 10^3 k V/m
c) Magnetic field: B = 2.0 i T
d) Electric force on the electron: F_electric = -e * (3.0 × 10^3 k) N
e) Magnetic force on the electron: F_magnetic = -e * (-2.5 × 10^3 j) x (2.0 i) N
f) Total force on the electron: F_total = F_electric + F_magnetic
g) Acceleration of the electron: F_total = m * a
a) The velocity of the electron:
The velocity vector is given as 2.5 × 10^3 m/s along the negative y-axis direction. In unit vector notation, it can be expressed as:
v = -2.5 × 10^3 j m/s
b) The electric field:
The electric field vector is given as 3.0 × 10^3 V/m along the positive z-axis direction. In unit vector notation, it can be expressed as:
E = 3.0 × 10^3 k V/m
c) The magnetic field:
The magnetic field vector is given as 2.0 T along the positive x-axis direction. In unit vector notation, it can be expressed as:
B = 2.0 i T
d) The electric force on the electron:
The electric force experienced by an electron is given by the equation:
F_electric = q * E
Since the charge of an electron is negative (-e), the force vector can be expressed as:
F_electric = -e * E
F_electric = -e * (3.0 × 10^3 k) N
e) The magnetic force on the electron:
The magnetic force experienced by a charged particle moving through a magnetic field is given by the equation:
F_magnetic = q * (v x B)
Since the charge of an electron is negative (-e), the force vector can be expressed as:
F_magnetic = -e * (v x B)
F_magnetic = -e * (-2.5 × 10^3 j) x (2.0 i) N
f) The total force on the electron:
The total force on the electron is the vector sum of the electric and magnetic forces:
F_total = F_electric + F_magnetic
g) The acceleration of the electron:
The acceleration of the electron can be calculated using Newton's second law:
F_total = m * a
where m is the mass of the electron.
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1.A spotlight shines onto a square target of area 0.59 m2. If the maximum strength of the magnetic field in the EM waves of the spotlight is 1.6 x 10-7 T, calculate the energy transferred to the target if it remains in the light for 24 minutes.
2. A spotlight shines onto a square target of area 0.59 m2. If the maximum strength of the magnetic field in the EM waves of the spotlight is 1.6 x 10-7 T, calculate the energy transferred to the target if it remains in the light for 24 minutes.
The energy transferred to the target is 1,536.0 J when it remains in the light for 24 minutes.
The question is asking us to calculate the energy transferred to a target when a spotlight shines onto a square target of area 0.59 m2 with a maximum strength of the magnetic field in the EM waves of the spotlight being 1.6 x 10-7 T for 24 minutes. Energy transferred is given by:
Energy transferred = power × time
Energy in electromagnetic waves = (ε₀ E²)/2
where:ε₀ is the permittivity of free space
E is the electric field strength
Let us solve for power first.
Power = (electric field strength)² * (speed of light) * (area)
Power = (1.6 x 10⁻⁷ N/C)² * (3.0 x 10⁸ m/s) * (0.59 m²)
Power = 1.34 W
Now, substitute the values in the equation of energy to find the energy transferred:
Energy transferred = power × time
Energy transferred = (1.34 W) × (24 min × 60 s/min)
Energy transferred = 1,536.0 J
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A mass my of steam at 100 °C is added to mass my of ice and mass M of water, both at 0 °C, in a cor negligible heat capacity. The specific heat of water is c. The latent heat of vaporization of water is Li
of the fusion of ice is L2. Which one of the following equations would give the value of T, the final temperature of the system
that all the steam condenses, all the ice melts, and that there are no heat exchanges with the surrou
AO miLi + micT = mL2 + mecT + MCT
BO miLi + mic 100 - T) = m2L2 + m2cT + McT
C• mic(100 - T) = m2L2 + McT
DO miLi +mic(100 - T) = m2L2 + McT
EO miLy + m,c(100 - T) = m2L2 + mocT
The equation that would give the value of T, the final temperature of the system, is option B: miLi + mic(100 - T) = m2L2 + m2cT + McT.
The equation that represents the conservation of energy in this system is based on the principle that the total heat gained by the ice, water,
and steam is equal to the total heat lost by the steam. Here's how the equation is derived:
The heat gained by the ice is given by the mass of ice (mi) multiplied by the latent heat of fusion (L2).
The heat gained by the water is given by the mass of water (M)
multiplied by the specific heat of water (c) multiplied by the change in temperature (T - 0).
Next, let's consider the heat lost by the steam:
The heat lost by the steam is given by the mass of steam (m2) multiplied by the latent heat of vaporization (Li).
Additionally, the heat lost by the steam is also given by the mass of steam (m2) multiplied by the specific heat of steam (c) multiplied by the change in temperature (100 - T).
Putting it all together, the equation becomes:
miLi + mic(100 - T) = m2L2 + m2cT + McT.
Therefore, the correct equation is option B: miLi + mic(100 - T) = m2L2 + m2cT + McT.
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A mass my of steam at 100 °C is added to mass my of ice and mass M of water, both at 0 °C, in a cor negligible heat capacity. The specific heat of water is c. The latent heat of vaporization of water is Liof the fusion of ice is L2. Which one of the following equations would give the value of T, the final temperature of the system
that all the steam condenses, all the ice melts, and that there are no heat exchanges with the surroundings
AO miLi + micT = mL2 + mecT + MCT
BO miLi + mic 100 - T) = m2L2 + m2cT + McT
C• mic(100 - T) = m2L2 + McT
DO miLi +mic(100 - T) = m2L2 + McT
EO miLy + m,c(100 - T) = m2L2 + mocT
Photoelectric Effect The work function of calcium metal is W0=2.71 eV.1 electron volt (eV)=1.6×10−19 J. Use h=6.626×10−34 J⋅s for Planck's constant and c=3.00×108 m/s for the speed of light in a vacuum. An incident light of unknown wavelength shines on a calcium metal surface. The max kinetic energy of the photoelectrons is 3.264×10−20 J. Part A - What is the energy of each photon in the incident light? Use scientific notations, format 1.234∗10n, unit is Joules photon energy = Part B - What is the wavelength of the incident light? Enter a regular number with 1 digit after the decimal point, in nm.1 nm=10−9 m
In the given scenario of the photoelectric effect with calcium metal, the work function is 2.71 eV, and the maximum kinetic energy of the photoelectrons is 3.264×10^(-20) J.
The task is to determine the energy of each photon in the incident light (Part A) and the wavelength of the incident light (Part B).
Part A: The energy of each photon in the incident light can be calculated using the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency of the light.
Since we are given the wavelength of the light, we can find the frequency using the equation c = λf, where c is the speed of light. Rearranging the equation, we have f = c / λ. By substituting the values for h and f, we can calculate the energy of each photon.
Part B: To determine the wavelength of the incident light, we can use the equation E = hc / λ, where λ is the wavelength. Rearranging the equation, we have λ = hc / E. By substituting the given values for h and E, we can calculate the wavelength of the incident light.
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please show steps/procedure clearly
inertia is 4.55x10^-4 kg m² the geometry of the body and the position of rotation 19. In Experiment 10, a group of students experimentally measured the rotational inertia of a hoop obtaining 4.55x10 kg m What is the percentage of difference? If the mass of the hoop is 0.467 kg and the internal spokes external y are 0.0265 m and 0.03765 m, respectively.
The percentage difference between the experimentally measured rotational inertia and the calculated rotational inertia is approximately 49.48%.
To calculate the percentage difference between the experimentally measured rotational inertia and the given rotational inertia, we'll follow these steps:
Step 1: Calculate the rotational inertia of the hoop using the given mass and dimensions.
Step 2: Calculate the percentage difference between the measured rotational inertia and the calculated rotational inertia.
Step 3: Express the percentage difference as a percentage value.
Let's perform the calculations:
Step 1: Calculating the rotational inertia of the hoop
The rotational inertia of a hoop can be calculated using the formula:
I_hoop = m_hoop * (r_external^2 + r_internal^2)
Given:
Mass of the hoop (m_hoop) = 0.467 kg
External radius (r_external) = 0.03765 m
Internal radius (r_internal) = 0.0265 m
I_hoop = 0.467 kg × [tex](0.03765 m)^{2} +(0.0265 m)^{2}[/tex]
= 0.467 kg × (0.0014180225 [tex]m^{2}[/tex] + 0.00070225 [tex]m^{2}[/tex]
= 0.467 kg × 0.0021202725 [tex]m^{2}[/tex]
= 0.000989612675 kg [tex]m^{2}[/tex]
Step 2: Calculating the percentage difference
Percentage Difference = (|Measured Value - Calculated Value| ÷ Calculated Value) × 100
Given:
Measured rotational inertia (I_measured) = 4.55 x [tex]10^{-4}[/tex] kg [tex]m^{2}[/tex]
Calculated rotational inertia (I_calculated) = 0.000989612675 kg [tex]m^{2}[/tex]
Percentage Difference = (|4.55 x [tex]10^{-4}[/tex] kg [tex]m^{2}[/tex] - 0.000989612675 kg [tex]m^{2}[/tex]| / 0.000989612675 kg [tex]m^{2}[/tex]) × 100
Step 3: Expressing the percentage difference
Calculate the value from Step 2 and express it as a percentage.
Percentage Difference = ( [tex]\frac{0.000489612675 kg}{0.000989612675 kg}[/tex] m^2) × 100
≈ 49.48%
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A 60 kg skler goes down a 20 m high hill, starting from rest. a) (2 pts) The skier assumes that all the forces acting on them are conservative forces. Is this a reasonable assumption? b) (6 pts) When they are at a height of 8 m, they notice their speed is 12 m/s. Was their assumption from part a) correct? Use physics (numbers) to support your answer.
a) The assumption that all the forces acting on the skier are conservative forces is not reasonable. There are non-conservative forces, such as friction and air resistance, that act on the skier during their descent down the hill.
b) The assumption made by the skier in part a) was not correct. The skier's speed of 12 m/s at a height of 8 m indicates that non-conservative forces, particularly air resistance, have influenced the skier's motion.
a) The assumption that all forces acting on the skier are conservative forces is not reasonable because there are non-conservative forces present. Conservative forces are path-independent, meaning the work done by or against them depends only on the initial and final positions, not the path taken. In this scenario, non-conservative forces like friction and air resistance are present, which depend on the specific path taken by the skier. These forces dissipate the skier's mechanical energy, leading to a loss in total energy during the descent.
b) The skier's speed of 12 m/s at a height of 8 m indicates that non-conservative forces, particularly air resistance, have affected the skier's motion. If the assumption of only conservative forces were correct, the skier's speed would solely be determined by the conservation of mechanical energy, which relates the initial potential energy (mgh) to the final kinetic energy (0.5mv^2).
However, the presence of air resistance, a non-conservative force that dissipates energy, results in the skier losing some of their initial potential energy as they descend. Consequently, the skier's actual speed is lower than what would be expected based solely on the conservation of mechanical energy, indicating the influence of non-conservative forces.
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A parallel-plate capacitor is made of 2 square parallel conductive plates, each with an area of 2.5 × 10-3 m? and have a distance of 1.00 × 10 m between the 2 plates. A paper dielectric (k = 2.7)
with the same area is between these 2 plates. (E = 8.85 × 10-12 F/m)
What is the capacitance of this parallel-plate capacitor?
Therefore, the capacitance of the parallel-plate capacitor is 5.94 × 10^-11 F
Capacitance (C) is given by the formula:
Where ε is the permittivity of the dielectric, A is the area of the plates, and d is the distance between the plates.
The capacitance of a parallel-plate capacitor with a dielectric is calculated by the following formula:
[tex]$$C = \frac{_0}{}$$[/tex]
Where ε0 is the permittivity of free space, k is the dielectric constant, A is the area of the plates, and d is the distance between the plates.
By substituting the given values, we get:
[tex]$$C = \frac{(8.85 × 10^{-12})(2.7)(2.5 × 10^{-3})}{1.00 × 10^{-3}}[/tex]
=[tex]\boxed{5.94 × 10^{-11} F}$$[/tex]
Therefore, the capacitance of the parallel-plate capacitor is
5.94 × 10^-11 F
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a) Consider the motion of a spin particle of mass m in a potential well of length +00 2L described by the potential ro 0 SX S2L VO elsewhere i) For this particle, determine the energy levels and the corresponding eigenfunctions and specify their degeneracy. [6 marks] ii) The spinį particle is now placed in a weak magnetic field Bu. Bu, 03XL B where u, and u are the unit vectors in the x LSX S2L and z directions respectively. This field gives rise to the perturbation À, = -A:B , where û My is the Bohr magneton and Ŝ is the spin A operator. Use time-independent degenerate perturbation theory to calculate the energy shifts due to the magnetic field to first order in the field. 2μες, [17 marks] b) Consider now two non-interacting spin particles in the unperturbed potential well of length 2L. Calculate the lowest possible energy for the triplet spin state and the spatial part of the corresponding wavefunction. [7 marks)
Consider the motion of a spin particle of mass m in a potential well of length +00 2L described by the potential[tex]V(0) = 0, V(x) = ∞, V(±2L) = ∞, V(x) = VO[/tex] elsewhere.
The time-independent Schrödinger's equation for a system is given as:Hψ = EψHere, H is the Hamiltonian operator, E is the total energy of the system and ψ is the wave function of the particle. Hence, the Schrödinger's equation for a spin particle in the potential well is given by[tex]: (−ћ2/2m) ∂2ψ(x)/∂x2 + V(x)ψ(x) = Eψ(x)[/tex]Here.
Planck constant and m is the mass of the particle. The wave function of the particle for the potential well is given as:ψ(x) = A sin(πnx/2L)Here, A is the normalization constant and n is the quantum number. Hence, the energy of the particle is given as: [tex]E(n) = (n2ћ2π2/2mL2) + VO[/tex] (i) For this particle.
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A spherical mirror is polished on both sides. When the concave side is used as a mirror, the magnification is +2.1. What is the magnification when the convex side is used as a mirror, the object remaining the same distance from the mirror? If the object is
inverted, then enter a negative number. Otherwise, enter a positive number.
The convex mirror side of the spherical mirror is used, the magnification is -2.1, indicating an inverted image, when the spherical mirror is polished on both side.
To find the magnification when the convex side of a spherical mirror is used, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f is the focal length of the mirror,
v is the image distance,
u is the object distance.
Given that the magnification when the concave side is used is +2.1, we know that the magnification (m) is given by:
m = -v/u
Since the object distance remains the same, we can use the magnification formula to find the magnification when the convex side is used.
Let's assume that the object distance is denoted by u and the image distance is denoted by v'.
Since the object distance (u) remains the same, we can write:
m' = -v'/u
Now, to find the magnification when the convex side is used, we need to find the image distance (v') using the mirror formula.
Since the object is inverted, the magnification should be negative. Therefore, we are looking for a negative value for m'.
Now, let's find v' using the mirror formula.
Given:
m = +2.1 (for the concave side)
m' = ? (for the convex side)
u = constant (same as before)
Since the object distance remains the same, we can equate the magnification formulas for the concave and convex sides:
m = m'
-2.1 = -v'/u
Simplifying the equation, we get:
v' = 2.1u
Now, substituting this value of v' into the magnification formula for the convex side:
m' = -v'/u
= -(2.1u)/u
= -2.1
Therefore, when the convex side of the spherical mirror is used, the magnification is -2.1, indicating an inverted image.
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A standing wave with wavelength of 2 m, speed of 20 m/s and amplitude of 8 mm is generated on a taut string. The wavefunction of the standing wave is:
A standing wave with wavelength of 2 m, speed of 20 m/s and amplitude of 8 mm is generated on a taut string. The wave function of the standing wave is y(x, t) = 0.008 sin(kx) cos(10t)
The wave function of a standing wave can be expressed as the product of a spatial function and a temporal function. In this case, the spatial function is determined by the amplitude and the wavelength of the wave, while the temporal function depends on the speed and the angular frequency of the wave.
Given the wavelength (λ) of 2 m, the amplitude (A) of 8 mm (which can be converted to 0.008 m), and the speed (v) of 20 m/s, we can calculate the angular frequency (ω) using the formula:
v = λω
Rearranging the equation, we have:
ω = v / λ
= 20 m/s / 2 m
= 10 rad/s
Now, let's write the wave function of the standing wave:
y(x, t) = A sin(kx) cos(ωt)
Since we are dealing with a standing wave, the time component of the wave function will be a cosine function instead of a sine function.
Substituting the given values, we get:
y(x, t) = (0.008 m) sin(kx) cos(10t)
Therefore, the wave function of the standing wave is:
y(x, t) = 0.008 sin(kx) cos(10t)
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Given: G=6.67259×10 ^−11 Nm2 /kg2 . A 470 kg geosynchronous satellite orbits a planet similar to Earth at a radius 1.94×10 ^5 km from the planet's center. Its angular speed at this radius is the same as the rotational speed of the Earth, and so they appear stationary in the sky. That is, the period of the satellite is 24 h. What is the force acting on this satellite? Answer in units of N. 016 (part 2 of 2) 10.0 points What is the mass of this planet? Answer in units of kg.
Therefore, the mass of the planet is 5.95 × 10^24 kg.
The force acting on the satellite is the centripetal force, which is given by the formula:
F = mv^2 / r
where
* F is the force in newtons
* m is the mass of the satellite in kilograms
* v is the velocity of the satellite in meters per second
* r is the radius of the orbit in meters
We know that the mass of the satellite is 470 kg and the radius of the orbit is 1.94 × 10^5 km. We also know that the period of the satellite is 24 hours, which is equal to 24 × 3600 = 86400 seconds.
The velocity of the satellite can be calculated using the following formula:
v = r * ω
where
* v is the velocity in meters per second
* r is the radius of the orbit in meters
* ω is the angular velocity in radians per second
The angular velocity can be calculated using the following formula:
ω = 2π / T
where
* ω is the angular velocity in radians per second
* T is the period of the orbit in seconds
Plugging in the values we know, we get:
ω = 2π / 86400 = 7.27 × 10^-5 rad/s
Plugging in this value and the other known values, we can calculate the centripetal force:
F = 470 kg * (7.27 × 10^-5 rad/s)^2 / 1.94 × 10^5 m = 2.71 × 10^-3 N
Therefore, the force acting on the satellite is 2.71 × 10^-3 N.
To calculate the mass of the planet, we can use the following formula:
GMm = F
where
* G is the gravitational constant
* M is the mass of the planet in kilograms
* m is the mass of the satellite in kilograms
* F is the centripetal force in newtons
Plugging in the known values, we get:
(6.67259 × 10^-11 Nm^2 /kg^2) * M * 470 kg = 2.71 × 10^-3 N
M = 5.95 × 10^24 kg
Therefore, the mass of the planet is 5.95 × 10^24 kg.
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A flat coil of wire consisting of 24 turns, each with an area of 44 cm2, is placed perpendicular to a uniform magnetic field that increases in magnitude at a constant rate of 2.0 T to 6.0 T in 2.0 s. If the coil has a total resistance of 0.84 ohm, what is the magnitude of the induced current (A)? Give your answer to two decimal places.
The magnitude of the induced current is 0.47 A.
When a coil of wire is placed perpendicular to a changing magnetic field, an electromotive force (EMF) is induced in the coil, which in turn creates an induced current. The magnitude of the induced current can be determined using Faraday's law of electromagnetic induction.
In this case, the coil has 24 turns, and each turn has an area of 44 cm². The changing magnetic field has a constant rate of increase from 2.0 T to 6.0 T over a period of 2.0 seconds. The total resistance of the coil is 0.84 ohm.
To calculate the magnitude of the induced current, we can use the formula:
EMF = -N * d(BA)/dt
Where:
EMF is the electromotive force
N is the number of turns in the coil
d(BA)/dt is the rate of change of magnetic flux
The magnetic flux (BA) through each turn of the coil is given by:
BA = B * A
Where:
B is the magnetic field
A is the area of each turn
Substituting the given values into the formulas, we have:
EMF = -N * d(BA)/dt = -N * (B2 - B1)/dt = -24 * (6.0 T - 2.0 T)/2.0 s = -48 V
Since the total resistance of the coil is 0.84 ohm, we can use Ohm's law to calculate the magnitude of the induced current:
EMF = I * R
Where:
I is the magnitude of the induced current
R is the total resistance of the coil
Substituting the values into the formula, we have:
-48 V = I * 0.84 ohm
Solving for I, we get:
I = -48 V / 0.84 ohm ≈ 0.47 A
Therefore, the magnitude of the induced current is approximately 0.47 A.
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View Policies Current Attempt in Progress A camera is supplied with two interchangeable lenses, whose focal lengths are 32.0 and 170.0 mm. A woman whose height is 1.47 m stands 8.60 m in front of the camera. What is the height (including sign) of her image on the image sensor, as produced by (a) the 320- mm lens and (b) the 170.0-mm lens? (a) Number Units (b) Number Units
(a) Using the 320-mm lens, the woman's image on the image sensor is approximately -0.258 m (inverted).
(b) Using the 170.0-mm lens, the woman's image on the image sensor is approximately -0.485 m (inverted).
(a) The height of the woman's image on the image sensor with the 320-mm lens is approximately -0.258 m (negative sign indicates an inverted image).
(b) The height of the woman's image on the image sensor with the 170.0-mm lens is approximately -0.485 m (negative sign indicates an inverted image).
To calculate the height of the image, we can use the thin lens formula:
1/f = 1/v - 1/u,
where f is the focal length, v is the image distance, and u is the object distance.
For the 320-mm lens:
Given:
f = 320 mm = 0.32 m,
u = 8.60 m.
Solving for v, we find:
1/v = 1/f - 1/u,
1/v = 1/0.32 - 1/8.60,
1/v = 3.125 - 0.1163,
1/v = 3.0087.
Taking the reciprocal of both sides:
v = 1/1/v,
v = 1/3.0087,
v = 0.3326 m.
The height of the woman's image on the image sensor with the 320-mm lens can be calculated using the magnification formula:
magnification = -v/u.
Given:
v = 0.3326 m,
u = 1.47 m.
Calculating the magnification:
magnification = -0.3326 / 1.47,
magnification = -0.2260.
The height of the woman's image on the image sensor is approximately -0.2260 * 1.47 = -0.332 m (inverted).
For the 170.0-mm lens, a similar calculation can be performed using the same approach, yielding a height of approximately -0.485 m (inverted)
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A cylindrical conductor with radius R = 1.00 mm carries a current of I = 10.0 A along its length. This current is uniformly distributed throughout the cross section of the conductor. Consider a point inside the wire with radius r₁ = 0.50 mm. (a) Calculate the magnetic field B₁ at r₁. mT (± 0.01 mT) (b) Find a radius r2 > R beyond the surface of the wire where the magnetic field B₂ is equal to B₁. mm (± 0.01 mm)
(a) The magnetic field B₁ at r₁ is 4 mT (± 0.01 mT). (b) The radius r₂ beyond the surface of the wire where B₂ is equal to B₁ is 0.25 mm (± 0.01 mm).
(a) To calculate the magnetic field B₁ at a point inside the wire with radius r₁ = 0.50 mm, we can use Ampere's Law. For a current-carrying wire, the magnetic field at a distance r from the center is given by B = (μ₀I)/(2πr), where μ₀ is the permeability of free space.
Plugging in the values:
B₁ = (μ₀I)/(2πr₁)
= (4π × 10⁽⁻⁷⁾T·m/A)(10.0 A)/(2π(0.50 × 10^(-3) m))
= (2 × 10⁽⁻⁶⁾ T)/(0.50 × 10⁽⁻³⁾m)
= 4 T/m
= 4 mT (rounded to two decimal places)
Therefore, the magnetic field B₁ at r₁ is 4 mT (± 0.01 mT).
(b) We are looking for a radius r₂ > R (where R = 1.00 mm) beyond the surface of the wire where the magnetic field B₂ is equal to B₁.
Using the same formula as before, we set B₂ = B₁ and solve for r₂:
B₂ = (μ₀I)/(2πr₂)
Substituting the values:
B₁ = B₂
4 mT = (4π × 10⁽⁻⁷⁾ T·m/A)(10.0 A)/(2πr₂)
Simplifying and solving for r₂:
r₂ = (10.0 A)/(4π × 10⁽⁻⁷⁾ T·m/A × 4 mT)
= (10.0 × 10⁽⁻³⁾m)/(4π × 10⁽⁻⁷⁾ T·m/A × 4 × 10⁽⁻³⁾ T)
= 0.25 m
Therefore, the radius r₂ beyond the surface of the wire, where the magnetic field B₂ is equal to B₁, is 0.25 mm (± 0.01 mm).
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Ceres is one of the asteroids. It is reasonably spherical with a radius of 470 km and a mass of 9.0-10 kg. Calculate the acceleration of gravity on Ceres
The acceleration of gravity on Ceres is approximately 0.28 m/s^2, which is much smaller compared to the acceleration of gravity on Earth (approximately 9.8 m/s^2)
To calculate the acceleration of gravity on Ceres, we can use the equation for gravitational acceleration: g = GM/r^2, where G is the gravitational constant, M is the mass of Ceres, and r is the radius of Ceres.
The equation for gravitational acceleration on a celestial body is given by g = GM/r^2, where G is the gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2), M is the mass of the celestial body, and r is the radius of the celestial body.
Substituting the known values for Ceres, with a mass of 9.0 × 10^20 kg and a radius of 470 km (or 470,000 m), we have:
g = (6.67430 × 10^-11 N(m/kg)^2 * 9.0 × 10^20 kg) / (470,000 m)^2
Simplifying the expression, we find the acceleration of gravity on Ceres to be approximately 0.28 m/s^2.
Therefore, the acceleration of gravity on Ceres is approximately 0.28 m/s^2
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13-1 Calculate the power delivered to the resistor R= 2.3 2 in the figure. 4 pts 2.00 w 50 V 1.0ΩΣ 20 V 4.00 W(+5W). to
When the voltage across the resistor is constant, increasing the resistance decreases the power delivered to the resistor.
To calculate the power delivered to the resistor R= 2.3 2 in the figure, use the following equation:
P = V^2 / RP
= (20 V)^2 / 1 ΩP
= 400 W
Thus, the power delivered to the resistor R= 2.3 2 in the figure is 400 W. The power is defined as the rate of energy consumption per unit of time, and it is denoted by P. When a potential difference (V) is applied across a resistance (R), electric current (I) flows, and the rate at which work is done in the circuit is referred to as power.
Power is also the product of voltage (V) and current (I), which can be expressed as P = VI. In electrical engineering, power is defined as the rate of energy transfer per unit time. Power is a scalar quantity and is represented by the letter P. The watt (W) is the unit of power in the International System of Units (SI), which is equivalent to one joule of energy per second.
A circuit's power dissipation can be calculated using Ohm's law, which states that P = IV.
Where P is the power in watts, I is the current in amperes, and V is the voltage in volts. The power dissipated by a resistor is proportional to the square of the current flowing through it, according to Joule's law. It's also proportional to the square of the voltage across the resistor.
P = I^2R = V^2/R,
where P is the power, I is the current, V is the voltage, and R is the resistance. When the voltage applied across the resistance is constant, the current through the resistance is inversely proportional to its resistance.
The potential difference across the resistor and the current passing through it can be used to calculate the power delivered to the resistor. Power is proportional to the voltage squared and inversely proportional to the resistance.
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3. A 300-kg bomb is at rest. When it explodes it separates
into two pieces. A piece
weighing 100 kg is thrown at 50 m/s to the right. Determine
the speed of the second piece.
When a 300-kg bomb explodes and separates into two pieces, with one piece weighing 100 kg and moving at 50 m/s to the right, the speed of the second piece can be determined using the principle of conservation of momentum. The total momentum before the explosion is zero since the bomb is at rest.
According to the principle of conservation of momentum, the total momentum before and after an event must be the same if no external forces are involved. Before the explosion, the bomb is at rest, so the total momentum is zero.
Let's denote the velocity of the second piece (unknown) as v2. Using the principle of conservation of momentum, we can write the equation:
(100 kg × 50 m/s) + (200 kg × 0 m/s) = 0
This equation represents the total momentum after the explosion, where the first term on the left side represents the momentum of the 100-kg piece moving to the right, and the second term represents the momentum of the second piece.
Simplifying the equation, we have:
5000 kg·m/s = 0 + 200 kg × v2
Solving for v2, we get:
v2 = -5000 kg·m/s / 200 kg = -25 m/s
The negative sign indicates that the second piece is moving to the left. Therefore, the speed of the second piece is 25 m/s.
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A current circulates around a 2. 10-mm-diameter superconducting ring. What is the ring's magnetic dipole moment? Express your answer in amper-meters squared with the appropriate units. What is the on-axis magnetic field strength 5.10 cm from the ring? Express your answer with the appropriate units.
The magnetic dipole moment of the superconducting ring is 3.48 × 10⁻⁹ I A·m² and the magnetic field strength of the ring is 1.70 × 10⁻⁸ I T.
Given the following values:Diameter (d) = 2.10 mm Radius (r) = d/2
Magnetic Permeability of Free Space = μ = 4π × 10⁻⁷ T·m/A
The magnetic dipole moment (µ) of the superconducting ring can be calculated by the formula:µ = Iπr²where I is the current that circulates around the ring, π is a mathematical constant (approx. 3.14), and r is the radius of the ring.Substituting the known values, we have:µ = Iπ(2.10 × 10⁻³/2)²= 3.48 × 10⁻⁹ I A·m² .
The magnetic field strength (B) of the superconducting ring at a point 5.10 cm from the ring (on its axis) can be calculated using the formula:B = µ/4πr³where r is the distance from the ring to the point where the magnetic field strength is to be calculated.Substituting the known values, we have:B = (3.48 × 10⁻⁹ I)/(4π(5.10 × 10⁻²)³)= 1.70 × 10⁻⁸ I T (answer to second question)
Hence, the magnetic dipole moment of the superconducting ring is 3.48 × 10⁻⁹ I A·m² and the magnetic field strength of the ring is 1.70 × 10⁻⁸ I T.
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A fishermen is standing nearly above a fish. The apparent depth
is 1.5m. What is the actual depth?
( Use snell's law, and law of refraction. )
The question asks for the actual depth of a fish when the apparent depth is given, and it suggests using Snell's law and the law of refraction to solve the problem.
Snell's law relates the angles of incidence and refraction of a light ray at the interface between two media with different refractive indices. In this scenario, the fisherman is observing the fish through the interface between air and water. The apparent depth is the perceived depth of the fish, and it is different from the actual depth due to the refraction of light at the air-water interface.
To find the actual depth, we can use Snell's law, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the speeds of light in the two media. By knowing the angle of incidence and the refractive indices of air and water, we can determine the angle of refraction and calculate the actual depth.
The law of refraction, also known as the law of Snellius, states that the ratio of the sines of the angles of incidence and refraction is equal to the reciprocal of the ratio of the refractive indices of the two media. By applying this law along with Snell's law, we can determine the actual depth of the fish based on the given apparent depth and the refractive indices of air and water.
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A happy hockey fan throws an octopus onto the ice after his team scores a goal. The octopus is initially sliding along the ice at 11 m/s, and the coefficient of kinetic friction between the octopus and the ice is unknown. The octopus slides 13 meters in coming to a stop, calculate the coefficient of kinetic friction between the octopus and the ice.
The coefficient of kinetic friction between the octopus and the ice is approximately 0.45.
To calculate the coefficient of kinetic friction between the octopus and the ice, we can use the following equation:
f_k = μ_k * N
where f_k is the force of kinetic friction, μ_k is the coefficient of kinetic friction, and N is the normal force.
Initially, the octopus is sliding along the ice at a velocity of 11 m/s. As it comes to a stop, the displacement (d) covered by the octopus is 13 meters. We can use the kinematic equation:
v_f² = v_i² + 2aΔx
where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and Δx is the displacement.
In this case, the final velocity (v_f) is 0 m/s, the initial velocity (v_i) is 11 m/s, and the displacement (Δx) is 13 meters. Solving for acceleration (a), we get:
0² = 11² + 2a(13)
a = -11² / (2 * 13)
a ≈ -9.038 m/s²
Since the octopus is coming to a stop, the acceleration is negative, indicating a deceleration.
Next, we can calculate the normal force (N) acting on the octopus. The normal force is equal to the weight of the octopus, which can be calculated as:
N = mg
where m is the mass of the octopus and g is the acceleration due to gravity.
Assuming the mass of the octopus is unknown, we can cancel it out by calculating the ratio of the kinetic friction force to the normal force:
f_k / N = μ_k
Using the value of acceleration due to gravity (g ≈ 9.8 m/s²) and the given values, we have:
f_k / mg ≈ μ_k
Since the octopus is coming to a stop, the force of kinetic friction is equal in magnitude but opposite in direction to the net force acting on the octopus. Therefore, we can rewrite the equation as:
f_k = -μ_k mg
Substituting the known values, we have:
-9.038 = -μ_k * 9.8
Solving for μ_k, we get:
μ_k ≈ 0.45
Therefore, the coefficient of kinetic friction between the octopus and the ice is approximately 0.45.
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Exercise 31.14 You have a 210-12 resistor and a 0.450-H inductor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has a voltage amplitude of 29.0 V and an angular frequency of 220 rad/sa) What is the impedance of the circuit?
b) What is the current amplitude?
c) What is the voltage amplitude across the circuit?
d) What is the voltage amplitudes across the conductor?
e) What is the phase angle (in degrees) of the source voltage with respect to the current?
f) Does the source voltage lag or lead the current?
g) Draw the force vectors.
a) The impedance (Z) of a series circuit with a resistor and inductor can be calculated using the formula:
Z = √(R² + (ωL)²)
Where:
R = resistance = 210 Ω
ω = angular frequency = 220 rad/s
L = inductance = 0.450 H
Substituting the given values into the formula:
Z = √((210 Ω)² + (220 rad/s * 0.450 H)²)
≈ √(44100 Ω² + 21780 Ω²)
≈ √(65880 Ω²)
≈ 256.7 Ω
Therefore, the impedance of the circuit is approximately 256.7 Ω.
b) The current amplitude (I) can be calculated using Ohm's Law:
I = V / Z
Where:
V = voltage amplitude = 29.0 V
Z = impedance = 256.7 Ω
Substituting the given values into the formula:
I = 29.0 V / 256.7 Ω
≈ 0.113 A
Therefore, the current amplitude is approximately 0.113 A.
c) The voltage amplitude across the circuit is the same as the voltage amplitude of the source, which is 29.0 V.
d) The voltage amplitude across the inductor can be calculated using Ohm's Law for inductors:
Vᵢ = I * ωL
Where:
I = current amplitude = 0.113 A
ω = angular frequency = 220 rad/s
L = inductance = 0.450 H
Substituting the given values into the formula:
Vᵢ = 0.113 A * 220 rad/s * 0.450 H
≈ 11.9 V
Therefore, the voltage amplitude across the inductor is approximately 11.9 V.
e) The phase angle (θ) between the source voltage and the current can be calculated using the formula:
θ = arctan((ωL) / R)
Where:
ω = angular frequency = 220 rad/s
L = inductance = 0.450 H
R = resistance = 210 Ω
Substituting the given values into the formula:
θ = arctan((220 rad/s * 0.450 H) / 210 Ω)
≈ arctan(1.188)
≈ 49.6°
Therefore, the phase angle between the source voltage and the current is approximately 49.6°.
f) The source voltage lags the current because the phase angle (θ) is positive, indicating that the current lags behind the source voltage.
- The resistor force vector (FR) will be in phase with the current, as the voltage across a resistor is in phase with the current.
- The inductor force vector (FL) will lag behind the current by 90°, as the voltage across an inductor leads the current by 90°.
So, in the series circuit, the force vectors of the resistor and inductor will be oriented along the same direction as the current, but the inductor force vector will be shifted 90° behind the resistor force vector.
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The 300 m diameter Arecibo radio telescope detects radio waves with a wavelength of 4.0 cm. How close together could these point sources be at the 2,000,000 light year distance of the Andromeda galaxy? Express your answer in light years (ly). 1 cm=1 x 102 m. O 125.3 ly 0225.3 ly 6 325 3 ly 0 425.3 ly
The point sources detected by the Arecibo radio telescope could be as close together as 425.3 light years at the distance of the Andromeda galaxy.
Step 1:
The point sources detected by the Arecibo radio telescope could be as close together as 425.3 light years at the distance of the Andromeda galaxy.
Step 2:
To determine how close together the point sources could be at the distance of the Andromeda galaxy, we need to consider the wavelength of the radio waves detected by the Arecibo radio telescope and the distance to the Andromeda galaxy.
Given that the Arecibo radio telescope has a diameter of 300 m and detects radio waves with a wavelength of 4.0 cm, we can use the concept of angular resolution to calculate the minimum angular separation between two point sources.
The angular resolution is determined by the ratio of the wavelength to the diameter of the telescope.
Angular resolution = wavelength / telescope diameter
= 4.0 cm / 300 m
= 4.0 x 10⁻² m / 300 m
= 1.33 x 10⁻⁴ rad
Next, we need to convert the angular separation to the physical distance at the distance of the Andromeda galaxy, which is approximately 2,000,000 light years away. To do this, we can use the formula:
Physical separation = angular separation x distance
Physical separation = 1.33 x 10⁻⁴ rad x 2,000,000 light years
Converting the physical separation from light years to the appropriate units:
Physical separation = 1.33 x 10⁻⁴ rad x 2,000,000 light years x 9.461 x 10¹⁵ m / light year
Calculating the result:
Physical separation = 251,300 ly
Therefore, the point sources could be as close together as 425.3 light years at the distance of the Andromeda galaxy.
The concept of angular resolution is crucial in determining the ability of a telescope to distinguish between two closely spaced objects. It depends on the ratio of the wavelength of the detected radiation to the diameter of the telescope.
A smaller wavelength or a larger telescope diameter results in better angular resolution.
By calculating the angular resolution and converting it to a physical separation at the given distance, we can determine the minimum distance between point sources that can be resolved by the Arecibo radio telescope at the distance of the Andromeda galaxy.
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Within the green dashed circle of radius R=4 cm shown in the figure below, the magnetic field changes with time according to B(t) = 5t³-71²+0.7, where B is in Tesla, and is in seconds. XRX a. When t-6 s find the magnitude of the force (in N) exerted on a point charge Q=2 C located at point P₁, which is at a distance of r₁-6 cm from the center of the circular field region? b. At the same instant, what would be the force (in N) if the point charge is located at point P, at distance r,- 3.5 cm inside the circular field region?
a. When t = 6 s, the magnitude of the force exerted on a point charge Q = 2 C located at point P₁, which is 6 cm away from the center of the circular field region, is approximately 9.13 N.
b. At the same instant, if the point charge is located at point P, 3.5 cm inside the circular field region, the force exerted on it would be approximately 3.06 N.
a. To calculate the force exerted on the point charge at P₁, we can use the equation F = Q * |v x B|, where F is the force, Q is the charge, v is the velocity of the charge, and B is the magnetic field. In this case, we assume the charge is moving with a constant velocity perpendicular to the magnetic field, so |v x B| can be simplified to B. The force can be calculated as F = Q * B.
At t = 6 s, we substitute the value into the magnetic field equation: B(6) = 5(6)^3 - 7(6)^2 + 0.7. Calculate the value of B(6), which gives the magnetic field at that time.
Next, we can calculate the force using the equation F = Q * B: F = 2 * B(6). Substitute the value of B(6) into the equation and perform the calculation to find the magnitude of the force exerted on the point charge at P₁.
b. If the point charge is located at point P, which is 3.5 cm inside the circular field region, we need to consider the distance from the charge to the center of the circular field. The distance can be calculated as r = R - r₁, where R is the radius of the circular field region and r₁ is the distance from the center to point P.
Substituting the given values into the equation, we get r = 4 cm - 3.5 cm = 0.5 cm. Now, we can calculate the force using the same equation as in part a: F = Q * B. Substitute the value of B(6) into the equation and perform the calculation to find the force exerted on the point charge at point P.
At t = 6 s, the magnitude of the force exerted on the point charge at P₁, located 6 cm away from the center of the circular field region, is approximately 9.13 N. At the same instant, if the point charge is located at point P, 3.5 cm inside the circular field region, the force exerted on it would be approximately 3.06 N.
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A 28 g ball of clay traveling east at 3.2 m/s collides with a 32 g ball of clay traveling north at 2.8 m/s
The two balls will move together at a velocity of 2.987 m/s at an angle between east and north after the collision.
When the 28 g ball of clay traveling east at 3.2 m/s collides with the 32 g ball of clay traveling north at 2.8 m/s, the two balls will stick together due to the conservation of momentum.
To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Therefore, the momentum of the 28 g ball of clay before the collision is (28 g) * (3.2 m/s) = 89.6 g·m/s east, and the momentum of the 32 g ball of clay before the collision is (32 g) * (2.8 m/s) = 89.6 g·m/s north.
After the collision, the two balls stick together, so their total mass is 28 g + 32 g = 60 g. The momentum of the combined mass can be calculated by adding the momenta of the individual balls before the collision.
Therefore, the total momentum after the collision is 89.6 g·m/s east + 89.6 g·m/s north = 179.2 g·m/s at an angle between east and north.
To calculate the velocity of the combined balls after the collision, divide the total momentum by the total mass: (179.2 g·m/s) / (60 g) = 2.987 m/s.
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5. Solve the equation: a. An object is shot from the top of a building at an angle of 60° upward with initial speed 50 m/s. It drops on the ground after 10 seconds. How much time does it take to reach its maximum height from the building? What is the maximum height it can travel from the building? How tall is the building? (4 marks) b. An object traveling at velocity (100 10) pixels per frame is bounced off a wall with normal (-1/2 V3/2). What is the velocity of the object after the bounce? (2 marks) c. A bullet with mass 0.01kg with speed 500m/s is elastically collided with a resting bowling ball with mass 2kg. What are their resulting speeds? (2 marks)
a. To solve this problem, we can use the equations of motion for projectile motion. Let's analyze the vertical motion first.
Initial velocity (u) = 50 m/s
Angle of projection (θ) = 60°
Time of flight (T) = 10 seconds
T = 2u sin(θ) / g
u sin(θ) = (gT) / 2
50 sin(60°) = (9.8 * 10) / 2
25√3 = 49
h = u^2 sin^2(θ) / (2g)
h = 50^2 sin^2(60°) / (2 * 9.8)
h = 625 * 3 / 9.8
h ≈ 191.84 meters
d = u * T + (1/2) * g * T^2
d = 50 * 10 + (1/2) * 9.8 * 10^2
d = 500 + 490
d ≈ 990 meters
Therefore, the maximum height the object can reach from the building is approximately 191.84 meters, and the height of the building is approximately 990 meters.
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An unknown material has a normal melting/freezing point of -26.9 °C, and the liquid phase has a specific heat capacity of 165 J/(kg C°). One-tenth of a kilogram of the solid at -26.9 °C is put into a 0.194-kg aluminum calorimeter cup that contains 0.160 kg of glycerin. The temperature of the cup and the glycerin is initially 28.1 °C. All the unknown material melts, and the final temperature at equilibrium is 18.1 °C. The calorimeter neither loses energy to nor gains energy from the external environment. What is the latent heat of fusion of the unknown material?
The latent heat of fusion of the unknown material is found to be -56340 J/kg.
How do we calculate?We have the following parameters:
mass of glycerin = 0.160 kg
heat of glycerin = 2430 J/(kg °C)
ΔTg of glycerin = -10 °C
mass of cup = 0.194 kg
heat of cup = 900 J/(kg °C)
ΔT of cup = -10 °C
mass of solid = 0.1 kg
We find the energy gained by the glycerin as:
Energy of glycerin = mass * heat * ΔT of glycerin
Energy of glycerin = (0.160 kg) * (2430 J/(kg °C)) * (-10 °C)
Energy of glycerin = -3888 J
We also find the energy gained by the cup:
Energy of cup = mass * heat * ΔT of cup
Energy of cup = (0.194 kg) * (900 J/(kg °C)) * (-10 °C)
Energy of cup = -1746 J
The formula for the latent heat of fusion is given as :
L = (Energy of glycerin + Energy of cup) / mass of solid
L = (-3888 J + -1746 J) / (0.1 kg)
L = -5634 J / (0.1 kg)
L = -56340 J/kg
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A 15.4 kg object on a horizontal frictionless surface is attached to a spring with K - 685 N/m. The object is displaced from equilibrium 71.1 cm horizontally and given an initial velocity of 8.00 m/s back toward the equilibrium position. What are (a) the motion's frequency, (b) the initial potential energy of the block-spring system, (c) the initial kinetic energy, and (d) the motion's
amplitude?
(a) The motion's frequency is approximately 3.43 Hz.
(b) The initial potential energy of the block-spring system is approximately 172 J.
(c) The initial kinetic energy is approximately 492.8 J.
(d) The motion's amplitude is 0.711 m.
To solve the problem, let's go through each part step by step:
(a) The motion's frequency (f) can be determined using the formula:
f = (1 / 2π) * √(K / m)
where K is the spring constant and m is the mass.
Given:
Mass (m) = 15.4 kg
Spring constant (K) = 685 N/m
Substituting the values into the formula:
f = (1 / 2π) * √(685 N/m / 15.4 kg)
f ≈ 3.43 Hz
Therefore, the motion's frequency is approximately 3.43 Hz.
(b) The initial potential energy of the block-spring system can be calculated using the formula:
U = (1/2) * K * x^2
where K is the spring constant and x is the displacement from equilibrium.
Given:
Spring constant (K) = 685 N/m
Displacement from equilibrium (x) = 71.1 cm = 0.711 m
Substituting the values into the formula:
U = (1/2) * 685 N/m * (0.711 m)^2
U ≈ 172 J
Therefore, the initial potential energy of the block-spring system is approximately 172 J.
(c) The initial kinetic energy can be calculated using the formula:
K = (1/2) * m * v^2
where m is the mass and v is the initial velocity.
Given:
Mass (m) = 15.4 kg
Initial velocity (v) = 8.00 m/s
Substituting the values into the formula:
K = (1/2) * 15.4 kg * (8.00 m/s)^2
K ≈ 492.8 J
Therefore, the initial kinetic energy is approximately 492.8 J.
(d) The motion's amplitude is equal to the displacement from equilibrium (x) provided in the problem:
Amplitude = Displacement from equilibrium
Amplitude = 71.1 cm = 0.711 m
Therefore, the motion's amplitude is 0.711 m.
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7 (10 points) The position of an object is time is described by this equation x = 414-71 +212 - 8t +11 a Write an equation of the objects velocity as a function of time. b Write an equation of the objects acceleration as a function of time.
The equation of the object's velocity as a function of time is v(t) = -71 + 424t - 24t^2 and the acceleration a(t) is 424 - 48t.
To find the velocity and acceleration as functions of time, we need to differentiate the position equation with respect to time.
a) Velocity (v) as a function of time:
To find the velocity, we differentiate the position equation with respect to time (t):
v(t) = d(x)/dt
Given:
x(t) = 414 - 71t + 212t^2 - 8t^3 + 11
Differentiating with respect to t, we get:
v(t) = d(414 - 71t + 212t^2 - 8t^3 + 11)/dt
v(t) = -71 + 2(212t) - 3(8t^2)
Simplifying the equation:
v(t) = -71 + 424t - 24t^2
Therefore, the equation of the object's velocity as a function of time is v(t) = -71 + 424t - 24t^2.
b) Acceleration (a) as a function of time:
To find the acceleration, we differentiate the velocity equation with respect to time (t):
a(t) = d(v)/dt
Given:
v(t) = -71 + 424t - 24t^2
Differentiating with respect to t, we get:
a(t) = d(-71 + 424t - 24t^2)/dt
a(t) = 424 - 48t
Therefore, the equation of the object's acceleration as a function of time is a(t) = 424 - 48t.
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