1. Given
R= 8.31 J/mol K
kb = 1.38 x 10-23 J/K0°C = 273.15 KNA = 6.02 x 1023 atoms/mol
Density of Water, p=1000 kg/m³
Atmospheric Pressure, P = 101300 Pa
g= 9.8 m/s²
We know that PV = nRTOr
V = (nRT)/PN = given mass/molar mass
= 100/39.9
= 2.5063 moles
V = (2.5063 mol x 8.31 J/mol K x (20 + 273.15) K)/101300
Pa= 0.50 m³At -50°C or 223.15 K,
V = nRT/PV = 2.5063 mol x 8.31 J/mol K x 223.15 K/0.50 m³ x 1.38 x 10-23 J/K= 8.83 x 105 Pa
Therefore, the volume of gas at 20°C is 0.50 m³, and the pressure of gas at -50°C is 8.83 × 10⁵ Pa.2.
Given Area of the wall,
A = 5.0 m x 5.0 m = 25.0 m²
Thickness of the wall, L = 6.0 cm = 0.06 m
Temperature inside the building, Ti = 20°C = 293.15 K
Temperature outside the building, To = 5°C = 278.15 K
Thermal conductivity of brick, k = 0.84 W/(m·K)
Thermal conductivity of Styrofoam, k` = 0.010 W/(m·K)
A) Heat lost through the brick wall
Rate of heat transfer through the brick wall is given byQ = k A (Ti - To) / L= 0.84 W/(m·K) x 25.0 m² x (20 - 5) K / 0.06 m= 7.00 x 10⁴ W or 70 kW.
B) Temperature at the boundary between the Styrofoam and the brick wallLet
T be the temperature at the boundary between the Styrofoam and the brick wall.
Q = k A (Ti - T) / L1 + Q = k` A (T - To) / L2So (k A / L1) Ti - (k A / L1 + k` A / L2) T + (k` A / L2) To = 0On
solving this equation, we getT = (k` A / L2) To / (k A / L1 + k` A / L2)= (0.010 W/(m·K) x 25.0 m² x 278.15 K) / (0.84 W/(m·K) / 0.06 m + 0.010 W/(m·K) / 0.040 m)= 282.22 K = 9.07 °C
Therefore, the temperature at the boundary between the Styrofoam and the brick wall is 9.07 °C.
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you to analyse a single phase inverter utilizing thyristors that supply an RL load (R=1092 and L-25mH). Given that the supply voltage is from 12 Vpc PV solar systems which is then boosted to 125 Vpc and finally inverted to give the output of 110 Vrms, 60 Hz. Find: (i) the thyristors firing angle (ii) the inverter Total Harmonic Distortion (THD) (iii) a new firing angle for the thyristors to reduce the inverter THD (iv) the new THD of the inverter (10 marks) Assume: the inverter only carry odd number harmonics, and only harmonic up to n=11 are deemed significant.
The thyristors firing angle is 0°. The inverter Total Harmonic Distortion (THD) is 0%. Since the THD is already 0%, there is no need to adjust the firing angle. The new THD of the inverter remains 0%.
Supply voltage: 12 Vdc from PV solar systems
Boosted voltage: 125 Vdc
Inverted output voltage: 110 Vrms, 60 Hz
Load: RL load, where R = 1092 Ω and L = 25 mH
(i) Thyristors firing angle:
The firing angle of the thyristors in a single-phase inverter can be determined using the formula:
α = cos^(-1)((R/L)(Vdc/Vm))
Substituting the given values:
α = cos^(-1)((1092/25 × 10^(-3))(125/110))
= cos^(-1)(4.88)
≈ 0°
Note: The calculated firing angle of 0° indicates that the thyristors are triggered at the beginning of each half-cycle.
(ii) Inverter Total Harmonic Distortion (THD):
The THD of the inverter can be calculated using the formula:
THD = √[(V2^2 + V3^2 + V5^2 + ...)/(V1^2)]
Since the question assumes that the inverter carries only odd-numbered harmonics up to n = 11, we can calculate the THD considering the significant harmonics.
THD = √[(V2^2 + V3^2 + V5^2 + ...)/(V1^2)]
= √[(0^2 + 0^2 + 0^2 + ...)/(110^2)]
= 0
Note: The calculated THD of 0% indicates that there are no significant harmonics present in the inverter output.
(iii) New firing angle to reduce the inverter THD:
Since the THD was already 0% in the previous calculation, there is no need to adjust the firing angle to further reduce the THD.
(iv) New THD of the inverter:
As mentioned in the previous calculation, the THD is already 0% in this case, so there is no change in the THD.
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A loop of wire with velocity 3 m/s moves through a magnetic field whose strength increases with distance at a rate of 5T/m. If the loop has area 0.75 m² and internal resistance 5 Ω, what is the current in the wire?
A. I=3 A
B. I=56A
C. I=11.25 A
D. I=2.25A
The current in the wire is option is A, I = 3A.
The rate of increase of the magnetic field is 5 T/m and the velocity of the wire is 3 m/s.
Therefore, the change in the magnetic field per unit time, that is, the emf induced in the wire is;
emf = Bvl
where
B is the magnetic field,
v is the velocity,
l is the length of the wire, in this case, the length of the wire is equal to the perimeter of the loop.
The area of the loop is 0.75 m²;
therefore, the perimeter is;
P = √(4 × 0.75 m² / π) = 0.977m
Substituting the values given;
emf = (5 T/m × 3.08 m) × 3 m/s = 14.655 V
The current in the wire is given by;
I = emf / R
where
R is the internal resistance of the wire, in this case, it is 5 Ω.
Substituting the values in the equation,
I = 14.655 V / 5 Ω = 2.931 A = 3A(approx)
Therefore, the correct option is A. I = 3A.
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If the magnetic field at the center of a single loop wire with radius of 8.08cm in 0.015T, calculate the magnetic field if the radius would be 3.7cm with the same current. Express your result in units of T with 3 decimals.
Answer:
The magnetic field if the radius would be 3.7cm with the same current is 0.0069T.
Let B1 be the magnetic field at the center of a single loop wire with radius of 8.08cm and B2 be the magnetic field if the radius would be 3.7cm with the same current.
Now,
The magnetic field at the center of a single loop wire is given by;
B = (μ₀I/2)R
Where μ₀ is the magnetic constant,
I is the current and
R is the radius.
The magnetic field at the center of a single loop wire with radius of 8.08cm is given as,
B1 = (μ₀I/2)R1 …(i)
Similarly, the magnetic field at the center of a single loop wire with radius of 3.7cm is given as,
B2 = (μ₀I/2)R2 …(ii)
As given, current I is same in both the cases,
i.e., I1 = I2 = I
Also, μ₀ is a constant, hence we can write equation (i) and (ii) as, B1 ∝ R1 and B2 ∝ R2
Thus, the ratio of magnetic field for the two different radii can be written as;
B1/B2 = R1/R2
On substituting the values, we get;
B1/B2 = (8.08)/(3.7)
B2 = B1 × (R2/R1)
B2 = 0.015 × (3.7/8.08)
B2 = 0.00686061947
B2= 0.0069 (approx)
Therefore, the magnetic field if the radius would be 3.7cm with the same current is 0.0069T.
Hence, the answer is 0.0069T.
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A loop of wire with a diameter of 20 cm is located in a uniform magnetic field. The loop is perpendicular to the field. The field has a strength of 2.0 T. If the loop is removed completely from the field in 1.75 ms, what is the average induced emf? If the loop is connected to a 150 kohm resistor what is the current in the resistor?
Answer: The current in the resistor is 0.00024 A.
The average induced emf can be determined by Faraday's law of electromagnetic induction which states that the emf induced in a loop of wire is proportional to the rate of change of the magnetic flux passing through the loop.
Mathematically: ε = -N(ΔΦ/Δt)
where,ε is the induced emf, N is the number of turns in the loop, ΔΦ is the change in the magnetic flux, Δt is the time interval.
The magnetic flux is given as,Φ = BA
where, B is the magnetic field strength, A is the area of the loop.
Since the loop has been completely removed from the field, the change in magnetic flux (ΔΦ) is given by,ΔΦ = BA final - BA initial. Where,
BA initial = πr²
B = π(0.1m)²(2.0 T)
= 0.0628 Wb.
BA final = 0 Wb (As the loop has been removed completely from the field).
Therefore,ΔΦ = BA final - BA initial
= 0 - 0.0628
= -0.0628 Wb.
Since the time interval is given as Δt = 1.75 ms
= 1.75 × 10⁻³ s, the induced emf can be calculated as,
ε = -N(ΔΦ/Δt)
= -N × (-0.0628/1.75 × 10⁻³)
= 35.94 N.
The average induced emf is 35.94 V (approx).
Now, if the loop is connected to a 150 kΩ resistor, the current in the resistor can be determined using Ohm's law, which states that the current passing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. Mathematically, it can be represented as,
I = V/R Where, I is the current flowing through the resistor V is the voltage across the resistor R is the resistance of the resistor. From the above discussion, we know that the induced emf across the loop of wire is 35.94 V, and the resistor is 150 kΩ = 150 × 10³ Ω
Therefore, I = V/R
= 35.94/150 × 10³
= 0.00024 A.
The current in the resistor is 0.00024 A.
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A generator connectod to an RLC circuit has an ms voltage of 160 V and an ims current of 36 m . Part A If the resistance in the circuit is 3.1kΩ and the capacitive reactance is 6.6kΩ, what is the inductive reactance of the circuit? Express your answers using two significant figures. Enter your answers numerically separated by a comma. Item 14 14 of 15 A 1.15-k? resistor and a 585−mH inductor are connoctod in series to a 1150 - Hz generator with an rms voltage of 12.2 V. Part A What is the rms current in the circuit? Part B What capacitance must be inserted in series with the resistor and inductor to reduce the rms current to half the value found in part A?
a) The inductive reactance of the circuit IS 3.8KΩ and the rms current in the circuit is 1.68 mA
b) Capacitance that must be inserted in series with the resistor and inductor to reduce the rms current to half is 62.8μF
a) To calculate the inductive reactance [tex](\(X_L\))[/tex] of the circuit, we'll use the formula:
[tex]\[X_L = \sqrt{{X^2 - R^2}}\][/tex]
where X is the total reactance and R is the resistance in the circuit. Given that [tex]\(X_C = 6.6 \, \text{k}\Omega\)[/tex] and [tex]\(R = 3.1 \, \text{k}\Omega\),[/tex] we can calculate X:
[tex]\[X = X_C - R = 6.6 \, \text{k}\Omega - 3.1 \, \text{k}\Omega = 3.5 \, \text{k}\Omega\][/tex]
Substituting the values into the formula:
[tex]\[X_L = \sqrt{{(3.5 \, \text{k}\Omega)^2 - (3.1 \, \text{k}\Omega)^2}}\][/tex]
Calculating the expression:
[tex]\[X_L \approx 3.8 \, \text{k}\Omega\][/tex]
b) For the second problem, with a 1.15 k\(\Omega\) resistor, a 585 mH inductor, a 1150 Hz generator, and an rms voltage of 12.2 V:
a) To find the rms current I in the circuit, we'll use Ohm's law:
[tex]\[I = \frac{V}{Z}\][/tex]
The total impedance Z can be calculated as:
[tex]\[Z = \sqrt{{R^2 + (X_L - X_C)^2}}\][/tex]
Substituting the given values:
[tex]\[Z = \sqrt{{(1.15 \, \text{k}\Omega)^2 + (3.8 \, \text{k}\Omega - 6.6 \, \text{k}\Omega)^2}}\][/tex]
Calculating the expression:
[tex]\[Z \approx 7.24 \, \text{k}\Omega\][/tex]
Then, using Ohm's law:
[tex]\[I = \frac{12.2 \, \text{V}}{7.24 \, \text{k}\Omega} \approx 1.68 \, \text{mA}\][/tex]
b) To reduce the rms current to half the value found in part A, we need to insert a capacitor in series with the resistor and inductor. Using the formula for capacitive reactance [tex](\(X_C\))[/tex]:
[tex]\[X_C = \frac{1}{{2\pi fC}}\][/tex]
Rearranging the equation to solve for C:
[tex]\[C = \frac{1}{{2\pi f X_C}}\][/tex]
Substituting the values:
[tex]\[C = \frac{1}{{2\pi \times 1150 \, \text{Hz} \times (0.5 \times 1.68 \, \text{mA})}}\][/tex]
Calculating the expression:
[tex]\[C \approx 62.8 \, \mu\text{F}\][/tex]
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A radionsonde was launched at an elevation 200 m with observed surface air temperature 20∘ Cnd surface pressure 1002mb. The radiosonde data show that temperatures are 18∘ C at 980mb,15∘ C at 950mb, etc. Calculate geopotential heights at 980mb and 950mb
Answer:A radiosonde is a battery-powered telemetry instrument carried into the atmosphere usually by a weather balloon that measures various atmospheric parameters and transmits them by radio to a ground receiver. Modern radiosondes measure or calculate the following variables: altitude, pressure, temperature, relative humidity, wind (both wind speed and wind direction), cosmic ray readings at high altitude and geographical position (latitude/longitude). Radiosondes measuring ozone concentration are known as ozonesondes.[1]
sorry if this is to much
Explanation:
An annulus with inner radius a=1.6 m and outer radius b=3.8 m lies in the x−y plane. There is a constant electric field with magnitude 9.9 m
V
, that makes an angle θ=65.9 ∘
with the horizontal. What is the electric flux through the annulus? V⋅m 1 point possible (graded) An annulus with inner radius a=1.6 m and outer radius b=3.8 m lies in the x−y plane. There is a constant electric field with magnitude 9.9 m
V
, that makes an angle θ=65.9 ∘
with the horizontal. What is the electric flux through the annulus? V⋅m
the electric flux through the annulus is 34.3 V m.
Given that the inner radius of the annulus, a = 1.6 m, the outer radius of the annulus, b = 3.8 m, the magnitude of the electric field, E = 9.9 m V, and the angle between the horizontal and electric field, θ = 65.9°.
The formula to calculate the electric flux is given by,Φ = E.A cosθWhere E is the magnitude of the electric field, A is the area of the surface, and θ is the angle between the electric field and the normal to the surface.
The area of the annulus is given by,A = π(b² - a²)Substituting the given values in the above equation, we get,A = π(3.8² - 1.6²)A = 12.2 π m²Now substituting the values of E, A, and θ in the electric flux formula, we get,Φ = E.A cosθΦ = 9.9 × 12.2π × cos 65.9°Φ = 34.3 V mHence,
the electric flux through the annulus is 34.3 V m.
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f B⇀ represents a magnetic field and A represents the total area of the surface, what does the equation B→·A→=0 describe?
A magnetic field that is everywhere parallel to the surface.
A magnetic field that is uniform in magnitude and everywhere horizontal.
The equation is false because it describes a magnetic monopole, which does not exist.
The equation describes any magnetic field that can exist in nature.
The equation B→·A→=0 accurately describes a magnetic field that is everywhere parallel to the surface, indicating that the magnetic field lines are not intersecting or penetrating the surface but are instead running parallel to it.
The equation B→·A→=0 describes a magnetic field that is everywhere parallel to the surface. Here, B→ represents the magnetic field vector, and A→ represents the vector normal to the surface with a magnitude equal to the total area of the surface A. When the dot product B→·A→ equals zero, it means that the magnetic field vector B→ is perpendicular to the surface vector A→. In other words, the magnetic field lines are parallel to the surface.This scenario suggests that the magnetic field is not penetrating or intersecting the surface, but rather running parallel to it. This can occur, for example, when a magnetic field is generated by a long straight wire placed parallel to a surface. In such a case, the magnetic field lines would be perpendicular to the surface.
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Water is poured into a U-shaped tube. The right side is much wider than the left side. Once the water comes to rest, the water level on the right side is: Select one: a. the same as the water level on the left side. b. higher than the water level on the left side. c. lower than the water level on the left side.
The correct answer is the same as the water level on the left side. When water comes to rest in a U-shaped tube, it reaches equilibrium, which means that the pressure at any given level is the same on both sides of the tube.
The pressure exerted by a fluid depends on the depth of the fluid and the density of the fluid. In this case, since the right side of the U-shaped tube is wider than the left side, the water level on the right side will spread out over a larger area compared to the left side. However, the depth of the water is the same on both sides, as they are connected and in equilibrium.
Since the pressure is the same on both sides, and the pressure depends on the depth and density of the fluid, the water level on the right side will be the same as the water level on the left side.
Therefore, option a. "the same as the water level on the left side" is the correct answer.
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A cube is 2.0 cm on a side when at rest. (a) What shape does it
take on when moving past an observer at 2.5 x 10^8 m/s, and (b)
what is the length of each side?
Answer: The length of each side of the cube when moving past an observer at 2.5 x 10^8 m/s is 1.22 cm.
The question is asking us to consider the relativistic effect of time dilation and length contraction, which affect the measurement of distance and time by a moving observer. Therefore, the apparent length and shape of the cube will differ from the actual measurements as seen by an observer at rest.
a) When the cube moves past an observer at a velocity of 2.5 x 10^8 m/s, it takes on a shape that is flattened in the direction of motion. This is because of the relativistic effect of length contraction. This effect states that the length of an object appears shorter to an observer in motion than to an observer at rest.
The degree of length contraction increases with velocity and is given by the formula: L' = L₀ / γ
where L₀ is the length at rest, L' is the apparent length observed by a moving observer, and γ is the Lorentz factor given by :
γ = 1 / √(1 - v²/c²) where v is the velocity of the cube and c is the speed of light.
Substituting the values, we have:
L' = 2.0 cm / γL'
= 2.0 cm / √(1 - (2.5 x 10^8 m/s)²/(3.0 x 10^8 m/s)²)L'
= 0.47 cm.
b) The length of each side of the cube when moving past an observer at 2.5 x 10^8 m/s is given by: L' = L₀ / γL = L' x γSubstituting the values, we have:
L = L' x γL
= 0.47 cm x √(1 - (2.5 x 10^8 m/s)²/(3.0 x 10^8 m/s)²)L
= 1.22 cm.
Thus, the length of each side of the cube when moving past an observer at 2.5 x 10^8 m/s is 1.22 cm.
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A gun is fired vertically into a block of wood (mass ml) at rest directly above it. If the bullet has a mass of m2 and a speed of v, how high will the block rise into the air after the bullet becomes embedded in it?
Answer: the height to which the block will rise into the air after the bullet becomes embedded in it is given by
H = (m₂v)² / 2(m₁ + ml)g.
When a gun is fired vertically into a block of wood at rest directly above it, the velocity of the block can be calculated by applying the law of conservation of momentum. Here, the bullet of mass m₂ is fired into the block of wood of mass ml. According to the law of conservation of momentum, the initial momentum of the bullet and the final momentum of the bullet and the block combined must be equal, and it can be expressed as:m₂v = (m₁ + ml)VWhere V is the velocity of the bullet and the block combined.
From the equation, we have: V = m₂v / (m₁ + ml)As the bullet and the block rise to a maximum height H, their total energy is equal to their initial kinetic energy, given as: 1/2 (m₁ + m₂) V² = (m₁ + m₂)gh. Where g is the acceleration due to gravity. Solving for H, we get: H = V² / 2g
Substituting the value of V in the above equation, we have: H = (m₂v)² / 2(m₁ + ml)g.
Therefore, the height to which the block will rise into the air after the bullet becomes embedded in it is given by H = (m₂v)² / 2(m₁ + ml)g.
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A cart with mass 200 g moving on a friction-less linear air track at an initial speed of 1.2 m/s undergoes an elastic collision with an initially stationary cart of unknown mass. After the collision, the first cart continues in its original direction at 1.00 m/s. What is the mass of the second cart?
The mass of the second cart is 0 kg, indicating that it is an object with negligible mass or a stationary object.
In an elastic collision, the total momentum before and after the collision remains constant. We can express this principle using the equation:
(m1 * v1) + (m2 * v2) = (m1 * u1) + (m2 * u2)
Where m1 and m2 are the masses of the first and second carts, v1 and v2 are their initial velocities, and u1 and u2 are their velocities after the collision.
In this scenario, the initial velocity of the first cart is given as 1.2 m/s, and its velocity after the collision is 1.00 m/s. The mass of the first cart is 200 g, which is equivalent to 0.2 kg.
We can rearrange the equation and solve for the mass of the second cart:
(m1 * v1) + (m2 * v2) = (m1 * u1) + (m2 * u2)
(0.2 * 1.2) + (m2 * 0) = (0.2 * 1.2) + (m2 * 1.00)
0.24 = 0.24 + m2
By subtracting 0.24 from both sides, we find that m2 = 0 kg.
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Which will not be affected by the induced e.m.f when a magnet is in motion relative to a coil? A. Motion of the magnet B. Resistance of the coil C. Number of turns of the coil D. The strength of the magnet pole
The strength of the magnet pole (option D) will not be affected by the induced electromotive force (e.m.f) when a magnet is in motion relative to a coil.
When a magnet is in motion relative to a coil, it induces an electromotive force (e.m.f) in the coil due to the changing magnetic field. This induced e.m.f. can cause various effects, but it does not directly affect the strength of the magnet pole (option D). Option A, the motion of the magnet, is directly related to the induction of the e.m.f. When the magnet moves, the magnetic field through the coil changes, inducing the e.m.f.
Option B, the resistance of the coil, affects the amount of current flowing through the coil when the e.m.f is induced. Higher resistance can limit the current flow. Option C, the number of turns of the coil, affects the magnitude of the induced e.m.f. More turns increase the induced voltage.
However, the strength of the magnet pole (option D) itself is independent of the induced e.m.f. It is determined by the properties of the magnet, such as its magnetization and magnetic material. The induced e.m.f does not alter the intrinsic strength of the magnet pole.
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A coil of inductance 130 mH and unknown resistance and a 1.1 μF capacitor are connected in series with an alternating emf of frequency 790 Hz. If the phase constant between the applied voltage and the current is 60° what is the resistance of the coil? Number Units
The resistance of the coil is 349.5 ohms when the phase constant between the applied voltage and the current is 60°.
Inductance = 130 mH
capacitance (C) = 1.1 μF
Frequency = 790 Hz.
The given units of inductance and capacitance must be converted into base SI units.
Inductance = 130 mH = 0.130 H
capacitance (C) = 1.1 μF = 1.1 μF = [tex]1.1 * 10^{(-6)} F[/tex]
The reactance of an inductor (XL) and a capacitor (XC) in an AC circuit is given by the following formulas:
The reactance of an inductor = XL = 2πfL
Capacitor = 1/(2πfC)
Next, we can calculate the values of reactance:
XL = 2π × 790 × 0.130 = 645.4 Ω (ohms)
XC = 1/(2π × 790 × [tex]1.1 * 10^{(-6)} F[/tex])
XC = 181.2 Ω (ohms)
The impedance can be calculated as:
[tex]Z = \sqrt{(R^2 + (XL - XC)^2)}[/tex]
tan(θ) = (XL - XC) / R
θ = 60° × π/180
θ = 1.047 radians
tan(1.047) = (645.4 - 181.2) / R
R = (645.4 - 181.2) / tan(1.047)
R = 349.5 Ω
Therefore, we can infer that the resistance of the coil is 349.5 ohms.
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How much larger is the dameter of the sun compared to the
diameter of jupiter?
The diameter of the sun is about 109 times larger than the diameter of Jupiter.
How much larger is the diameter of the sun compared to the diameter of Jupiter?The diameter of the sun is about 109 times larger than the diameter of Jupiter. The diameter of the sun is approximately 1.39 million kilometers (864,938 miles), while the diameter of Jupiter is around 139,822 kilometers (86,881 miles).
Therefore, the difference between the diameter of the sun and the diameter of Jupiter is about 1,390,178 kilometers (864,938 - 86,881 x 2), which is over one million kilometers. Jupiter is the largest planet in our solar system, but it's still small compared to the sun. Jupiter has a diameter that is roughly 11 times greater than the diameter of Earth.
The sun and Jupiter are both celestial objects in our solar system. While they share certain characteristics, such as their spherical shape and their immense size, they also differ in many ways. One significant difference between the sun and Jupiter is their size, as evidenced by their diameters. The diameter of the sun is around 109 times greater than the diameter of Jupiter, which means that the sun is much larger than Jupiter. The diameter of the sun is roughly 1.39 million kilometers (864,938 miles), while the diameter of Jupiter is about 139,822 kilometers (86,881 miles). The difference between the two is over 1,390,000 kilometers (864,938 - 86,881 x 2), which is a difference of over one million kilometers. As the largest planet in our solar system, Jupiter is still quite small when compared to the sun.
The diameter of the sun is about 109 times larger than the diameter of Jupiter, making it much larger than Jupiter.
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1. We saw how hydrostatic equilibrium can be used to determine the conditions in the interior of the Sun, but it can also be applied to the Earth's ocean. The major difference is that water, to a good approximation, is incompressible-you can take its density to be constant. Furthermore, we can take the acceleration of gravity to be constant, since the depth of the ocean is thin compared to the radius of the Earth.
Using this approximation, find the pressure in the ocean 1 km beneath the surface.
Side note: the reason that we can assume that water is incompressible is that it does not obey the ideal gas law, but rather a different relation where pressure is proportional to density to a high power.
Hydrostatic equilibrium
can be used to determine the conditions in the interior of the sun, and it can also be applied to the Earth's ocean.
The major difference between the two is that water, to a good approximation, is incompressible; you can take its
density
to be constant. We can also take the acceleration of gravity to be constant because the depth of the ocean is thin compared to the radius of the Earth.The reason we can assume that water is incompressible is that it does not obey the ideal gas law but rather a different relation in which
pressure
is proportional to density to a high power. The pressure in the ocean 1 km beneath the surface can be calculated using hydrostatic equilibrium.Pressure is proportional to density and depth. Since the density of water is almost constant, we can use the expression pressure = ρgh to calculate the pressure at any depth h in the ocean, where ρ is the density of water and g is the acceleration due to gravity. Using this equation, we can calculate the pressure 1 km beneath the
surface
of the ocean.ρ = 1,000 kg/m³, g = 9.8 m/s², and h = 1,000 mUsing the expression pressure = ρgh, we get the following:Pressure = 1,000 x 9.8 x 1,000 = 9,800,000 PaThus, the pressure 1 km beneath the surface of the ocean is 9.8 MPa.
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Calculating this, we find that the pressure in the ocean 1 km beneath the surface is approximately 9,800,000 Pascals (Pa).
To find the pressure in the ocean 1 km beneath the surface, we can use the concept of hydrostatic equilibrium. In this case, we assume that water is incompressible, meaning its density remains constant. Additionally, we can consider the acceleration due to gravity as constant, since the depth of the ocean is much smaller compared to the radius of the Earth.
In hydrostatic equilibrium, the pressure at a certain depth is given by the equation P = P0 + ρgh, where P is the pressure, P0 is the pressure at the surface, ρ is the density of the fluid (water), g is the acceleration due to gravity, and h is the depth.
Since the density of water is constant, we can ignore it in our calculations. Given that the depth is 1 km (1000 m) and assuming the acceleration due to gravity as [tex]9.8 m/s^2[/tex], we can plug these values into the equation to find the pressure:
P = P0 + ρgh
P = P0 + (density of water) * (acceleration due to gravity) * (depth)
P = P0 + (1000 kg/m^3) * ([tex]9.8 m/s^2[/tex]) * (1000 m)
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An elevator is hoisted by its cables at constant speed. Is the total work done on the elevator positive, negative, or zero? Explain your reasoning.
The force applied by the cables to lift the elevator is equal to the weight of the elevator, which is mg. Since the elevator is moving at a constant speed, the net force acting on the elevator is zero.
When an elevator is hoisted by its cables at a constant speed, the total work done on the elevator is zero.
The work done on an object is defined as the product of the force applied on it and the displacement caused by that force.
Work done can be positive or negative depending on the direction of the force and the displacement caused by it.
In this case, the elevator is hoisted by its cables at a constant speed. Since the speed is constant, the net force acting on the elevator is zero. This means that no work is being done on the elevator by the cables, and hence the total work done on the elevator is zero.
Let's take an example to understand this better. Suppose an elevator of mass m is being hoisted by its cables with a constant speed v.
The force applied by the cables to lift the elevator is equal to the weight of the elevator, which is mg.
Since the elevator is moving at a constant speed, the net force acting on the elevator is zero.
Therefore, the work done on the elevator by the cables is zero.
In conclusion, when an elevator is hoisted by its cables at a constant speed, the total work done on the elevator is zero.
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What velocity would a proton need to circle Earth 1,050 km above the magnetic equator, where Earth's magnetic field is directed horizontally north and has a magnitude of
4.00 ✕ 10−8 T?
(Assume the raduis of the Earth is 6,380 km.)
Magnitude:
The velocity required for a proton to circle the Earth 1,050 km above the magnetic equator, given Earth's magnetic field of magnitude 4.00 x 10^-8 T, is approximately [tex]5.44 * 10^6 m/s[/tex]
To determine the velocity required for a proton to circle the Earth 1,050 km above the magnetic equator, we can use the concept of centripetal force and the Lorentz force.
The centripetal force required for the proton to move in a circular path is provided by the magnetic force exerted by Earth's magnetic field. The Lorentz force is given by the formula:
F = q * v * B
where F is the magnetic force, q is the charge of the proton, v is its velocity, and B is the magnitude of Earth's magnetic field.
Since the proton is moving in a circular orbit, the centripetal force required is:
F = (m * v^2) / r
where m is the mass of the proton and r is the radius of the proton's orbit.
Setting the Lorentz force equal to the centripetal force, we have:
q * v * B = (m * v^2) / r
Rearranging the equation, we find:
v = (q * B * r) / m
Substituting the given values:
q = charge of a proton = 1.6 x 10^-19 C
B = 4.00 x 10^-8 T
r = radius of orbit = radius of Earth + altitude = (6,380 km + 1,050 km) = 7,430 km = 7,430,000 m
m = mass of a proton = 1.67 x 10^-27 kg
Plugging in these values, we get:
v = [tex](1.6 * 10^{-19} C * 4.00 * 10^-8 T * 7,430,000 m) / (1.67 * 10^{-27} kg)[/tex]
Calculating the expression, we find:
v ≈ [tex]5.44 * 10^6 m/s[/tex]
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As a woman walks, her entire weight is momentarily placed on one heel of her high-heeled shoes. Calculate the pressure exerted on the floor by the heel if it has an area of 1 cm2cm2 and the woman's mass is 52.5 kg. Express the pressure in Pa. (In the early days of commercial flight, women were not allowed to wear high-heeled shoes because aircraft floors were too thin to withstand such large pressures.)
P=
The pressure exerted on the floor by the heel is 5.15025 × 10⁷ Pa.
Given data,Mass of the woman, m = 52.5 kgArea of the heel, A = 1 cm² = 1 × 10⁻⁴ m²We can calculate the pressure exerted on the floor by the heel using the formula:
Pressure, P = F/A, where F is the force exerted by the heel on the floor.To find F, we first need to calculate the weight of the woman, which can be found using the formula: Weight, W = mg, where g is the acceleration due to gravity, g = 9.81 m/s²Weight of the woman, W = mg = 52.5 × 9.81 = 515.025 N.
When the woman places her entire weight on one heel, the force exerted by the heel on the floor is equal to the weight of the woman.Force exerted by the heel, F = 515.025 NPressure, P = F/A = 515.025/1 × 10⁻⁴ = 5.15025 × 10⁷ Pa.
Therefore, the pressure exerted on the floor by the heel is 5.15025 × 10⁷ Pa.
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found at 18.3 cm and 58.2 cm. Since this distance is half a wavelength, what is the wavelength of the 426.7 hertz sound wave in meters? found at 15.4 cm and 49.7 cm. Since this distance is half a wavelength, what is the wavelength of the 500 hertz sound wave in meters? found at 15.3 cm and 48.7 cm. Since this distance is half a wavelength, what is the wavelength of the 512 hertz sound wave in meters? and 58.2 cm. Given this wavelength and frequency, what is the speed of the sound wave?
The wavelength of a 426.7 Hz sound wave is 39.9 cm, the wavelength of a 500 Hz sound wave is 34.3 cm, and the wavelength of a 512 Hz sound wave is 33.4 cm. Additionally, the speed of the sound wave is 171.008 m/s.
To find the wavelength of a sound wave, formula used
wavelength = velocity / frequency.
Given that the distance is half a wavelength, the wavelength can be calculated by doubling the given distance.
For the sound wave with a frequency of 426.7 Hz, the distances are 18.3 cm and 58.2 cm. Since the total distance is 2 times the wavelength, the wavelength is:
58.2 cm - 18.3 cm = 39.9 cm.
For the sound wave with a frequency of 500 Hz, the distances are 15.4 cm and 49.7 cm. The wavelength is:
49.7 cm - 15.4 cm = 34.3 cm.
For the sound wave with a frequency of 512 Hz, the distances are 15.3 cm and 48.7 cm. The wavelength is:
48.7 cm - 15.3 cm = 33.4 cm.
For finding the speed of the sound wave, the obtained wavelength of 33.4 cm and the frequency of 512 Hz can be use.
The formula for speed is:
velocity = wavelength * frequency.
Converting the wavelength to meters (1 cm = 0.01 m), the wavelength is
33.4 cm * 0.01 m/cm = 0.334 m
Therefore, the speed of the sound wave is:
0.334 m * 512 Hz = 171.008 m/s.
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A block of mass m=2.90 kg initially slides along a frictionless horizontal surface with velocity t 0
=1.50 m/s. At position x=0, it hits a spring with spring constant k=49.00 N/m and the surface becomes rough, with a coefficient of kinctic friction cqual to μ=0.300. How far Δx has the spring compressed by the time the block first momentanily contes to rest? Assame the pakative. direction is to the right.
Therefore, the spring has compressed 2.5 cm before the block comes momentarily to rest.
In this case, the kinetic energy of the block is dissipated into the spring energy and friction. The spring equation is given by,0 = m * v²/2 + k * x - f * x,where,m = mass of the block,v = velocity of the block before it collides with the spring,k = spring constant,x = compression of the spring,f = friction force.μ = friction coefficientf = μ * (mass of the block) * (acceleration due to gravity) = μ * m * gFrom this expression, the compression of the spring can be calculated as: x = (v²/2 + f * x) / k. For this particular case, the velocity of the block before it collides with the spring (v) is given by 1.5 m/s. The mass (m) is 2.9 kg and the spring constant (k) is 49 N/m. The coefficient of kinetic friction (μ) is 0.3. The acceleration due to gravity (g) is 9.8 m/s².Then, the friction force f is given by,f = μ * m * g = 0.3 * 2.9 * 9.8 = 8.514 NSubstitute all the values in the above expression, x = (1.5²/2 + 8.514 * x) / 49.Then, solving for x, we get x = 0.025 m = 2.5 cm. Therefore, the spring has compressed 2.5 cm before the block comes momentarily to rest.
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Hydroelectric generators at Hoover Dam produce a maximum current of 8.05 x 10³ A at 251 kV. a) What is the power output? ___________________ W b) The water that powers the generators enters and leaves the system at low speed (thus its kinetic energy does not change) but loses 155 m in altitude. How many cubic meters per second are needed, assuming 86 % efficiency? __________ m³/s
The power output Hydroelectric generators at Hoover Dam produce a maximum current of 8.05 x 10³ A at 251 kV is 2.022 x 10⁹W. Cubic meters per second needed by assuming 86 % efficiency is 1547.83 m³/s.
a) The formula to calculate the power output is,
Power (P) = Current (I) x Voltage (V)
It is given that, Current (I) = 8.05 x 10³ A and Voltage (V)= 251 kV= 251,000 V
Substituting these values into the formula:
Power = (8.05 x 10³ A) x (251,000 V)
Power = 2.022 x 10⁹ W
Therefore, the power output is 2.022 x 10⁹ watts.
b) To calculate the flow rate of water needed, we can use the formula:
Power (P) = Efficiency (η) x Density (ρ) x Acceleration due to gravity (g) x Flow rate (Q) x Height (h)
It is given that, Power (P) = 2.022 x 10⁹ W, Efficiency (η) = 0.86 (86% efficiency), Density of water (ρ) = 1000 kg/m³, Acceleration due to gravity (g) = 9.8 m/s², Height (h) = 155 m
Substituting these values into the formula:
2.022 x 10⁹ W = 0.86 x (1000 kg/m³) x (9.8 m/s²) x Q x 155 m
Simplifying the equation:
Q= (2.022 x 10⁹ W) / (0.86 x 1000 kg/m³ x 9.8 m/s² x 155 m)
Q=1547.83 m³/s
Therefore, 1547.83 cubic meters per second of water are needed, assuming 86% efficiency.
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A 0.250 kg mass is attached to a horizontal spring of spring constant 140 N/m, supported by a frictionless table. A physics student pulls the mass 0.12 m from equilibrium, and the mass is then let go. Assume no air resistance and that it undergoes simple harmonic motion.
a) Calculate the work done by the student on the mass in pulling it a distance of 0.12 m.
b) Using conservation of energy principles, calculate the maximum speed of the mass.
a) The work done by the student on the mass in pulling it a distance of 0.12 m is 0.10 J.b) The maximum speed of the mass is 0.79 m/s.
a) Work done by the student on the mass in pulling it a distance of 0.12 m.The amount of work done by the student is equal to the amount of potential energy stored in the spring.Potential energy stored in the spring = 1/2 kx²where, k is the spring constant and x is the displacement from the equilibrium position.Now, the displacement of the mass is given as 0.12 m.Substituting the given values,1/2 × 140 N/m × (0.12 m)² = 0.10 JTherefore, the work done by the student on the mass in pulling it a distance of 0.12 m is 0.10 J.
b) Maximum speed of the massUsing the law of conservation of energy, the potential energy stored in the spring is equal to the kinetic energy of the mass at the maximum speed.Potential energy stored in the spring = Kinetic energy of the mass at maximum speed1/2 kA² = 1/2 mv²where, A is the amplitude, m is the mass, and v is the maximum velocity of the mass.Substituting the given values,1/2 × 140 N/m × (0.12 m)² = 1/2 × 0.250 kg × v²Solving for v, v = 0.79 m/sTherefore, the maximum speed of the mass is 0.79 m/s.
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Water runs into a fountain, filling all the pipes, at a steady rate of 0.753 m3/s. How fast will it shoot out of a hole 4.42 cm in diameter? Express your answer in meters per second
At what speed will it shoot out if the diameter of the hole is three times as large? Express your answer in meters per second.
Water runs into a fountain, filling all the pipes, at a steady rate of 0.753 m3/s.(a)The speed of water shooting out of a hole with a diameter of 4.42 cm is 4.43 m/s.(b) The speed of water shooting out of a hole with a diameter that is three times as large is 7.07 m/s.
(a)The gravitational constant is 9.8 m/s^2, so the velocity of efflux is equal to:
v = sqrt(2 × 9.8 m/s^2) = 4.43 m/s
The diameter of the hole is 4.42 cm, which is 0.0442 m. The area of the hole is then equal to:
A = pi× r^2 = pi × (0.0442 m / 2)^2 = 5.27 × 10^-5 m^2
The volume flow rate is equal to the area of the hole multiplied by the velocity of efflux, so the volume flow rate is:
Q = A × v = 5.27 × 10^-5 m^2 × 4.43 m/s = 2.37 × 10^-4 m^3/s
Therefore, the speed of water shooting out of a hole with a diameter of 4.42 cm is 4.43 m/s.
(b)If the diameter of the hole is three times as large, then the area of the hole will be nine times as large. The volume flow rate will then be nine times as large, or 2.14 × 10^-3 m^3/s.
Therefore, the speed of water shooting out of a hole with a diameter that is three times as large is 7.07 m/s.
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a 27 cm wrench is used to generate a torque at a bolt. a force of 43 N is applied at the end of the wrench at an angle of 52 to the wrench. the torque generated at the bolt is
A 27 cm wrench is used to generate a torque at a bolt. a force of 43 N is applied at the end of the wrench at an angle of 52 to the wrench.The torque generated at the bolt is approximately 9.147 N·m.
Let's proceed with the calculation:
Given:
Length of the wrench (L) = 27 cm = 0.27 m
Force applied at the end of the wrench (F) = 43 N
Angle between the force and the wrench (θ) = 52°
To calculate the torque, we need to find the perpendicular distance between the point of application of the force and the axis of rotation. This can be done using trigonometry.
Perpendicular distance (d) = L × sin(θ)
= 0.27 m × sin(52°)
Calculating the value of d:
d ≈ 0.27 m × 0.788 = 0.21276 m
Now we can calculate the torque:
Torque (τ) = F × d
= 43 N × 0.21276 m
≈ 9.14668 N·m
Therefore, the torque generated at the bolt is approximately 9.147 N·m.
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Two pistons of a hydraulic lift have radii of 2.67 cm and 20.0 cm. A mass of 2.00×10^3 kg is placed on the larger piston. Calculate the minimum downward force needed to be exerted on the smaller piston to hold the larger piston level with the smaller piston.
------------- N
The minimum downward force needed to be exerted on the smaller piston to hold the larger piston level with the smaller piston is 348.8 N.
A hydraulic lift works on Pascal’s principle which states that pressure applied to an enclosed fluid is transmitted equally in all directions. The pressure applied to the fluid is equal to the force applied per unit area. A hydraulic lift system consists of two pistons of different sizes connected by a pipe filled with fluid. The force applied on one piston gets transmitted to the other piston with a force that is multiplied by the ratio of the area of the two pistons.
The area of the smaller piston is given as follows:A = πr²where r = 2.67 cm = 0.0267 mTherefore, A = π(0.0267)² = 0.002232 m²The area of the larger piston is given as follows:A = πr²where r = 20.0 cm = 0.20 mTherefore, A = π(0.20)² = 0.1257 m²Since the force exerted on the larger piston is due to the weight of the mass placed on it, we can calculate the force as follows:F = mgwhere m = 2.00×10³ kg, and g = 9.81 m/s²Therefore, F = (2.00×10³)(9.81) = 19.62 kN = 1.962×10⁴ N.To calculate the minimum downward force needed to hold the larger piston level with the smaller piston, we can use the ratio of the area of the two pistons. Let F₁ be the force needed to be exerted on the smaller piston.
Therefore, the force exerted on the larger piston is given as:F₂ = F₁ × (A₂ / A₁)where A₁ is the area of the smaller piston, and A₂ is the area of the larger piston.Since the two pistons are at the same level, the force exerted on the larger piston is equal and opposite to the force exerted on the smaller piston. Therefore, we can write:F₁ = F₂ / (A₂ / A₁)F₁ = (1.962×10⁴) / (0.1257 / 0.002232)F₁ = 348.8 NTherefore, the minimum downward force needed to be exerted on the smaller piston to hold the larger piston level with the smaller piston is 348.8 N.
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Alkaline batteries have the advantage of putting out constant voltage until very nearly the end of their life. How long will an alkaline battery rated at 1.04 A⋅h and 1.4 V keep a 0.92 W flashlight bulb burning? _____________ hours
The alkaline battery rated at 1.04 A⋅h and 1.4 V will keep the 0.92 W flashlight bulb burning for about 0.996 hours.
Alkaline battery rated at 1.04 A⋅h and 1.4 V
Power required for flashlight bulb to burn = 0.92 W
Power is given by P = VI, where P is the power, V is the voltage, and I is the current.
Rearranging the above equation, we get I = P/V.
The current required for the flashlight bulb to burn is:
I = 0.92/1.4 = 0.657 A
The total charge in the battery is Q = It.
Charge is given in the unit of Coulombs (C).
1 A flows when 1 C of charge passes a point in 1 second.
Hence, 1 A flows when 3600 C of charge passes a point in 1 hour.
Therefore, 1 Coulomb = 1 A × 1 s
1 Ah = 1 A × 3600 s
So, 1 A⋅h = 3600 C
Charge in the battery Q = It = 0.657 A × (1.04 A ⋅ h) × (3600 s/h) = 2.36 × 10⁶ C
The time for which the battery will last is t = Q/I = (2.36 × 10⁶ C)/(0.657 A) = 3.59 × 10³ s
The time in hours is 3.59 × 10³ s/(3600 s/h) = 0.996 h
Therefore, the alkaline battery rated at 1.04 A⋅h and 1.4 V will keep the 0.92 W flashlight bulb burning for about 0.996 hours.
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For an object moving with a constant velocity, what is the slope of a straight line in its position versus time graph? O velocity displacement acceleration
The slope of a straight line in a position versus time graph for an object moving with a constant velocity represents the object's velocity.
In a position versus time graph, the vertical axis represents the object's position or displacement, while the horizontal axis represents time. When the object is moving with a constant velocity, its position changes linearly with time, resulting in a straight line on the graph.
The slope of a straight line is defined as the change in the vertical axis (position) divided by the change in the horizontal axis (time). In this case, since the object is moving with a constant velocity, the change in position per unit change in time remains constant. Therefore, the slope of the line represents the object's velocity, which is the rate of change of position with respect to time.
Hence, for an object moving with a constant velocity, the slope of a straight line in its position versus time graph represents its velocity.
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During the transient analysis of an RLC circuit, if the response is V(s) = (16s-20)/(s+1)(s+5), it is:
A. Step response of a series RLC circuit
B. Natural response of a parallel RLC circuit
C. Natural response of a series RLC circuit
D. None of the other choices are correct
E. Step response of a parallel RLC circuit
The response V(s) = (16s-20)/(s+1)(s+5) belongs to natural response of a series RLC circuit. Therefore, option C is correct.
Explanation:
The response V(s) = (16s-20)/(s+1)(s+5) belongs to natural response of a series RLC circuit.
In an RLC circuit, the transient analysis relates to the study of circuit responses during time transitions before attaining the steady state. Here, the response of the circuit to a step input or impulse input is analyzed, which is known as step response or natural response.
The natural response of a circuit depends upon the initial conditions, which means it is an undamped oscillation.
The response V(s) = (16s-20)/(s+1)(s+5) does not belong to the step response of a series RLC circuit, nor the natural response of a parallel RLC circuit.
Therefore, option C is correct.
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Among other things, the angular speed of a rotating vortex (such as in a tornado) may be determined by the use of Doppler weather radar. A Doppler weather radar station is broadcasting pulses of radio waves at a frequency of 2.85 GHz, and it is raining northeast of the station. The station receives a pulse reflected off raindrops, with the following properties: the return pulse comes at a bearing of 51.4° north of east; it returns 180 ps after it is emitted; and its frequency is shifted upward by 262 Hz. The station also receives a pulse reflected off raindrops at a bearing of 52.20 north of east, after the same time delay, and with a frequency shifted downward by 262 Hz. These reflected pulses have the highest and lowest frequencies the station receives. (a) Determine the radial-velocity component of the raindrops (in m/s) for each bearing (take the outward direction to be positive). 51.4° north of east ________
52.2° north of east ________ m/s (b) Assuming the raindrops are swirling in a uniformly rotating vortex, determine the angular speed of their rotation (in rad/s). _____________ rad/s
(a) The radial-velocity component of the raindrops 51.4° north of east is -7.63 m/s
The radial-velocity component of the raindrops 52.2° north of east is 7.63 m/s.
(b) The angular speed of their rotation (in rad/s) is 1.68 × 10^3 rad/s.
(a) The radial velocity of raindrops (in m/s) for each bearing is determined as follows:
Bearing 51.4° north of east
The radial velocity is given by:
v_r = (f/f_0 - 1) * c
where
v_r is the radial velocity
f is the received frequency
f_0 is the emitted frequency
c is the speed of light
f_0 = 2.85 GHz = 2.85 × 10^9 Hz
f + 262 = highest frequency
f - 262 = lowest frequency
Adding both gives:
f = (highest frequency + lowest frequency)/2
Substituting the values gives:
f = (f + 262 + f - 262)/2
This simplifies to:
f = f
which is not useful
v_r = (f/f_0 - 1) * c
Substituting the values gives:
v_r = ((f + 262)/f_0 - 1) * c
v_r = ((262 + f)/2.85 × 10^9 - 1) * 3 × 10^8
v_r = -7.63 m/s
Therefore, the radial-velocity component of the raindrops 51.4° north of east is -7.63 m/s.
Bearing 52.2° north of east
Substituting the values gives:
v_r = ((f - 262)/f_0 - 1) * c
v_r = ((f - 262)/2.85 × 10^9 - 1) * 3 × 10^8
v_r = 7.63 m/s
Therefore, the radial-velocity component of the raindrops 52.2° north of east is 7.63 m/s.
(b) The angular speed of their rotation (in rad/s) is given by:
Δv_r = 2 * v_r
The distance between both bearings is 52.2° - 51.4° = 0.8°
The time taken for the radar pulses to go and return is 180 ps = 180 × 10^-12 s
The distance between the station and the raindrops is given by:
d = Δv_r * t
where
Δv_r is the difference in radial velocity
t is the time taken
Substituting the values gives:
d = 2 * 7.63 * 180 × 10^-12
d = 2.7564 × 10^-10 m
The distance between the station and the vortex can be taken to be the average of the distances from the station to the raindrops
d_ave = d/2
d_ave = 1.3782 × 10^-10 m
The radius of the vortex is given by:
r = d_ave/sin(0.8°/2)
r = 9.063 × 10^-9 m
The angular speed is given by:
ω = Δv_r/r
where
Δv_r is the difference in radial velocity
r is the radius
Substituting the values gives:
ω = 2 * 7.63/9.063 × 10^-9
ω = 1.68 × 10^3 rad/s
Therefore, the angular speed of their rotation (in rad/s) is 1.68 × 10^3 rad/s.
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