The current through R₂ in the circuit alone is 0.6 A.Notice:When R₂ is in a circuit alone, the current flowing through it is 0.6 A.
Given that, the voltage, V = 15 VResistance, R₁ = 2500 ΩResistance, R₂ = 25 ΩWe know that the current (I) can be calculated using Ohm's Law, which states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) between them.The formula to calculate current using Ohm's Law is given by:I = V / Rwhere I is the current, V is the voltage and R is the resistance.a. R₂ in a circuit alone:
To find the current for R₂ in the circuit alone, we need to use the formula: I = V / ROn substituting the given values, we getI = 15 / 25I = 0.6 ATherefore, the current through R₂ in the circuit alone is 0.6 A.Notice:When R₂ is in a circuit alone, the current flowing through it is 0.6 A.
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Find the system output y(t) of a linear and time-invariant system with the input x(t) and the impulse response h(t) as shown in Figure 1. Sketch y(t) with proper labelling. Figure 1 (13 Marks) (b) The message signal m(t)=5cos(2000πt) is used to modulate a carrier signal c(t)=4cos(80000πt) in a conventional amplitude modulation (AM) scheme to produce the AM signal, x AM
(t), in which the amplitude sensitivity factor of the modulator k a
is used. (i) Express the AM signal x AM
(t) and find its modulation index. (ii) Determine the range of k a
for the case of under-modulation. (iii) Is under-modulation or over-modulation required? Why? (iv) Determine the bandwidths of m(t) and x AM
(t), respectively.
(i)The modulation index of the given signal is 5ka/2000. (ii)For under modulation: modulation index ≤ 1/3 . (iv) The bandwidths of m(t) and xAM(t) are 2000 Hz and 1.64 MHz (approx), respectively.
a)System input x(t):y(t)=5∫0tx(τ)h(t-τ)dτ=5∫0t5τe^(-2τ)u(t-τ)dτ=25∫0tτe^(-2τ)u(t-τ)dτ. Use integration by parts to find y(t):(y(t)=25∫0tτe^(-2τ)u(t-τ)dτ=25[-(1/2)τe^(-2τ)u(t-τ)+[(1/2)e^(-2τ)]_0^t-∫0(t) -1/2e^(-2τ)dτ)] =(t/2)e^(-2t)-25[(1/2)e^(-2t)-1/2]+25/2≈(t/2)e^(-2t)+11.25.
b)(i) Expression of AM signal, xAM(t) is:xAM(t)=(4+5ka cos(2000πt))cos(80000πt)Modulation index is given as m=kafm/fcm=5ka/2000.
(ii) For under-modulation: modulation index ≤ 1/3i.e., 5ka/2000 ≤ 1/3ka ≤ 0.04.
(iii) Over-modulation is required. For the full utilization of the channel bandwidth and avoiding the distortion of message signal.
(iv) The bandwidths of m(t) and xAM(t) are given as: Bandwidth of m(t) = fm = 2000 Hz. Bandwidth of xAM(t) = 2(fm + fc) = 2(2000+80000) = 1.64 MHz (approx)Therefore, the bandwidths of m(t) and xAM(t) are 2000 Hz and 1.64 MHz (approx), respectively.
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Down-sampling throws away samples, so it will shrink the size of the image. This is what is done by the following scheme: wp ww (1:p:end, 1:p:end); when we are downsampling by a factor of p.
The expression "wp ww (1:p:end, 1:p:end)" represents down-sampling an image by a factor of p using a scheme called "subsampling."
What is subsampling?In subsampling, every p-th sample is selected from both the width (wp) and height (ww) dimensions of the image. The notation "1:p:end" indicates that we start at the first sample and select every p-th sample until the end of the dimension.
By applying this scheme to an image, we effectively reduce the number of samples taken along both the width and height dimensions, resulting in a smaller image size. This down-sampling process discards the non-selected samples, effectively "throwing them away."
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You throw a stone horizontally at a speed of 10 m/s from the top of a cliff that is 50 m high. How far from the base of the cliff does the stone hit the ground within time of 8 s. * (20 Points) 80 m 50 m 10 m 8 m
The stone will hit the ground at a distance of 80 meters from the base of the cliff within the time of 8 seconds after it is thrown, which makes the correct option B (80 m).
To determine how far from the base of the cliff does the stone hit the ground within the time of 8 seconds after it is thrown, we'll need to make use of the equation:s = ut + 1/2gt²,Where, s = distance, u = initial velocity, t = time, g = acceleration due to gravity and this equation is applicable only when the motion is under the influence of gravity, in this case, vertical motion. As we know the stone is being thrown horizontally, the acceleration due to gravity will not affect the horizontal motion.So, in this case, u = 10 m/s (initial velocity, because it is thrown horizontally), g = 9.8 m/s² (acceleration due to gravity) and h = 50 m (height of the cliff).
Using this equation, we can get the time it takes for the stone to reach the ground:50 = 0 + 1/2 x 9.8 x t²25 = 4.9t²5.102 = t (square root of both sides)t ≈ 2.26 sSince the stone is being thrown horizontally, it covers the distance d = vt, where v is the horizontal velocity and t is the time. The horizontal velocity remains constant throughout the motion. In this case, we have:v = 10 m/s (horizontal velocity) and t = 8 s,So, d = vt = 10 x 8 = 80 mHence, the stone will hit the ground at a distance of 80 meters from the base of the cliff within the time of 8 seconds after it is thrown, which makes the correct option B (80 m).
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Amount of heat required to raise temperature of 10gm water through 2 deg * C is
The amount of heat required to raise the temperature of 10 g of water through 2°C is 83.68 Joules.
To determine the amount of heat required to raise the temperature of 10 g of water through 2°C, we will use the formula:Q = m × c × ΔT
Where Q is the amount of heat required, m is the mass of the substance being heated, c is the specific heat capacity of the substance, and ΔT is the change in temperature.
So, for 10 g of water, the mass (m) would be 10 g.
The specific heat capacity (c) of water is 4.184 J/(g°C), so we'll use that value.
And the change in temperature (ΔT) is 2°C.
Substituting these values into the formula, we get:Q = 10 g × 4.184 J/(g°C) × 2°CQ = 83.68 Joules
Therefore, the amount of heat required to raise the temperature of 10 g of water through 2°C is 83.68 Joules.
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Question 2: Find the bound currents of a uniformly magnetized sphere along the z-axis with dipole moment M.
The bound currents of a uniformly magnetized sphere along the z-axis with dipole moment M are zero:
[tex]$K_{\phi} = 0$[/tex]
The equation you provided for the bound currents along the z-axis of a uniformly magnetized sphere is correct:
[tex]$K_{\phi}=\frac{1}{\mu_{0}} \nabla \times \mathbf{M}$[/tex]
Starting from [tex]$\mathbf{M} = M \hat{z}$[/tex], we can substitute this value into the equation for the bound currents:
[tex]$K_{\phi}=\frac{1}{\mu_{0}} \nabla \times (M \hat{z})$[/tex]
Next, we can evaluate the curl using the formula you provided for the curl in cylindrical coordinates:
[tex]$\nabla \times \mathbf{V}=\frac{1}{r} \frac{\partial}{\partial z}(r V_{\phi})$[/tex]
However, it seems there was a mistake in the previous equation you presented, so I will correct it.
Applying the formula for the curl, we find that the only non-zero component in this case is indeed in the [tex]$\hat{\phi}$[/tex] direction. Therefore, we have:
[tex]$\nabla \times \mathbf{M} = \frac{1}{r} \frac{\partial}{\partial z}(r M_{\phi})$[/tex]
However, since [tex]$\mathbf{M} = M \hat{z}$[/tex], the [tex]$\phi$[/tex] component of [tex]$\mathbf{M}$[/tex] is zero ([tex]$M_{\phi} = 0$[/tex]), and as a result, the curl simplifies to:
[tex]$\nabla \times \mathbf{M} = 0$[/tex]
This means that the bound currents along the z-axis of a uniformly magnetized sphere are zero, as there are no non-zero components in the curl of the magnetization vector.
Therefore, the conclusion is that the bound currents of a uniformly magnetized sphere along the z-axis with dipole moment M are zero: [tex]$K_{\phi} = 0$[/tex]
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The following diagram shows a circuit containing an ideal battery, a switch, two resistors, and an inductor. The emt of the battery is 5.0 V,R 1
=380Ω,R 2
=120Ω, and L=50mH. The switch is closed at time t=0. At the moment the switch is closed, what is the current through R 2?
Answer: Some time after the switch was closed, the current through the switch is 32 mA. What is the current through R 2
at this moment? Answer: After the switch has been closed for a long time, the switch is re-opened. What is the current through R 2
the moment the switch is re-opened? Answer: Marks for this submission: 0.00/1.00 At the moment the switch is re-opened, what is the rate at which the current through R 2
is changing? Answer:
At the moment the switch is closed, the current through R2 is calculated as follows;First, the total resistance is calculated as shown below:Rtotal = R1 + R2Rtotal = 380 Ω + 120 ΩRtotal = 500 ΩThe current through Rtotal is given by;I = V / RtotalI = 5.0 V / 500 ΩI = 0.01 A.
The current through R2 is given by;IR2 = I(R2 / Rtotal)IR2 = 0.01 A(120 Ω / 500 Ω)IR2 = 0.0024 A. Some time after the switch was closed, the current through the switch is 32 mA. What is the current through R2 at this moment?At this moment, the inductor would have charged up to the maximum.
Hence it can be seen that the circuit will now appear as shown below: Total resistance, Rtotal = R1 + R2Rtotal = 380 Ω + 120 ΩRtotal = 500 ΩTotal emf of the circuit, E = V + L (dI / dt)E = 5.0 V + 50 mH (dI / dt)At maximum charge, the back emf is equal to the emf of the battery;E = 5.0 VHence;5.0 V = 5.0 V + 50 mH (dI / dt)dI / dt = 0 mA/sIR2 = I(R2 / Rtotal)IR2 = 0.032 A(120 Ω / 500 Ω)IR2 = 0.00768 AAfter the switch has been closed for a long time, the switch is re-opened. The inductor would now have built up a maximum magnetic field, hence the circuit would appear as shown below;The current through R2 is given by;IR2 = I(R2 / Rtotal)IR2 = 0 A / 2IR2 = 0 AMarks for this submission: 1.00/1.00.
At the moment the switch is re-opened, what is the rate at which the current through R2 is changing?The rate at which the current through R2 is changing is the rate at which the inductor is discharging, hence;dI / dt = -E / LdI / dt = -5.0 V / 50 mHdI / dt = -100 A/s.
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Two waves on one string are described by the wave functions
y1= 2.05 cos(3.05x − 1.52t)
y2= 4.54 sin(3.31x − 2.39t)
where x and y are in centimeters and t is in seconds. (Remember that the arguments of the trigonometric functions are in radians.)
(a) Find the superposition of the waves y_1 + y_2y1+y2 at x = 1.0, t = 0.0 s.
Two waves on one string are described by the wave functions y1= 2.05 cos(3.05x − 1.52t),y2= 4.54 sin(3.31x − 2.39t)where x and y are in centimeters and t is in seconds.The superposition of the waves y1 + y2 at x = 1.0 cm and t = 0.0 s is approximately 2.099968 cm.
To find the superposition of the waves at a specific point (x, t), we need to add the values of the two wave functions at that point.
Given:
y1 = 2.05 cos(3.05x - 1.52t)
y2 = 4.54 sin(3.31x - 2.39t)
x = 1.0 cm
t = 0.0 s
We can substitute the given values into the wave functions and perform the addition.
y1 + y2 = 2.05 cos(3.05x - 1.52t) + 4.54 sin(3.31x - 2.39t)
Substituting x = 1.0 cm and t = 0.0 s:
y1 + y2 = 2.05 cos(3.05(1.0) - 1.52(0.0)) + 4.54 sin(3.31(1.0) - 2.39(0.0))
y1 + y2 = 2.05 cos(3.05) + 4.54 sin(3.31)
Using a calculator, evaluate the cosine and sine functions:
y1 + y2 ≈ 2.05 * 0.999702 + 4.54 * 0.011432
y1 + y2 ≈ 2.048031 + 0.051937
y1 + y2 ≈ 2.099968
Therefore, the superposition of the waves y1 + y2 at x = 1.0 cm and t = 0.0 s is approximately 2.099968 cm.
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Blood flows through a 1.66 mm diameter artery at 26 mL/min and then passes into a 600 micron diameter vein where it flows at 1.2 mL/min. If the arterial blood pressure is 120 mmHg, what is the venous blood pressure? Ignore the effects of potential energy. The density of blood is 1,060 kg/m³ 1,000 L=1m³
a. 16,017,3 Pa b. 138.551 Pa c. 121.159 Pa d. 15,999.9 Pa
Answer: The answer is (a) 16,017,3 Pa.
The continuity equation states that the flow rate of an incompressible fluid through a tube is constant, so: Flow rate of blood in the artery = Flow rate of blood in the vein26 × 10⁻⁶ m³/s = 1.2 × 10⁻⁶ m³/s.
The velocity of blood in the vein is less than that in the artery.
Velocity of blood in the artery = Flow rate of blood in the artery / Area of artery.
Velocity of blood in the vein = Flow rate of blood in the vein / Area of vein
Pressure difference between the artery and vein = (1/2) × Density of blood × (Velocity of blood in the artery)² × (1/Area of artery² - 1/Area of vein²)
Pressure difference between the artery and vein = 120 - Pressure of vein.
The pressure difference between the artery and vein is equal to the change in potential energy.
However, we are ignoring the effects of potential energy, so the pressure difference between the artery and vein can be calculated as follows:
120 = (1/2) × 1,060 × (26 × 10⁻⁶ / [(π/4) × (1.66 × 10⁻³ m)²])² × (1/[(π/4) × (1.66 × 10⁻³ m)²] - 1/[(π/4) × (600 × 10⁻⁶ m)²])
120 = (1/2) × 1,060 × 12,580.72 × 10¹² × (1/1.726 × 10⁻⁶ m² - 1/1.1317 × 10⁻⁷ m²)120 = 16,017,300 Pa.
Therefore, the venous blood pressure is:
Pressure of vein = 120 - Pressure difference between the artery and vein
Pressure of vein = 120 - 16,017,300Pa
Pressure of vein = -16,017,180 Pa.
The answer is (a) 16,017,3 Pa.
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A 2 uF capacitor is fully charged by a 12 v power supply. The capacitor is then connected in parallel to an 8.1 mH inductor. (2) i. Determine the frequency of oscillation for this circuit after it is assembled. (3) ii. Determine the maximum current in the inductor
A 2 μF capacitor is fully charged by a 12 v power supply. The capacitor is then connected in parallel to an 8.1 mH inductor. .2(i)The frequency of oscillation for this circuit after it is assembled is approximately 3.93 kHz.3(ii)The maximum current in the inductor is approximately 58.82 A.
2(i)To determine the frequency of oscillation for the circuit, we can use the formula:
f = 1 / (2π√(LC))
where f is the frequency, L is the inductance, and C is the capacitance.
Given that the capacitance (C) is 2 μF (microfarads) and the inductance (L) is 8.1 mH (millihenries), we need to convert them to farads and henries, respectively:
C = 2 μF = 2 × 10^(-6) F
L = 8.1 mH = 8.1 × 10^(-3) H
Substituting the values into the formula:
f = 1 / (2π√(8.1 × 10^(-3) H × 2 × 10^(-6) F))
Simplifying the equation:
f = 1 / (2π√(16.2 × 10^(-9) H×F))
f = 1 / (2π × 4.03 × 10^(-5) s^(-1))
f ≈ 3.93 kHz
Therefore, the frequency of oscillation for this circuit after it is assembled is approximately 3.93 kHz.
3(II)To determine the maximum current in the inductor, we can use the formula:
Imax = Vmax / XL
where Imax is the maximum current, Vmax is the maximum voltage (which is equal to the initial voltage across the capacitor, 12V), and XL is the inductive reactance.
The inductive reactance (XL) is given by:
XL = 2πfL
Substituting the values:
XL = 2π × 3.93 kHz × 8.1 × 10^(-3) H
Simplifying the equation:
XL ≈ 0.204 Ω
Now we can calculate the maximum current:
Imax = 12V / 0.204 Ω
Imax ≈ 58.82 A
Therefore, the maximum current in the inductor is approximately 58.82 A.
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A 2.4 kg rock has a horizontal velocity of magnitude v=2.1 m/s when it is at point P in the figure, where r=4.1 m and θ= 45 degree. If the only force acting on the rock is its weight, what is the rate of change of its angular momentum relative to point O at this instant?
Therefore, the rate of change of the angular momentum relative to point O is zero.Answer: 0
The angular momentum of the rock relative to point O is given byL = r × p,where r is the position vector of the rock relative to point O, and p is the momentum of the rock relative to point O.We can express the momentum p in terms of the velocity v. Since the rock has a horizontal velocity of magnitude v=2.1 m/s, its momentum has a horizontal component of p = mv = (2.4 kg)(2.1 m/s) = 5.04 kg · m/s. There is no vertical component of the momentum, since the rock is moving horizontally, so we have p = (5.04 kg · m/s) i. Using the position vector r = (4.1 m) i + (4.1 m) j and the momentum p, we find thatL = r × p= [(4.1 m) i + (4.1 m) j] × (5.04 kg · m/s i)= 20.2 kg · m²/s k. where k is a unit vector perpendicular to the plane of the paper, pointing out of the page. The rate of change of the angular momentum relative to point O is given byτ = dL/dtwhere τ is the torque on the rock. Since the only force acting on the rock is its weight, which is directed downward, the torque on the rock is zero, so we haveτ = 0. Therefore, the rate of change of the angular momentum relative to point O is zero.Answer: 0
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A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axis of 18 kg.m. She then tucks into a small ball, decreasing this moment of inertia to 3.6 kg.m. While tucked, she makes two complete revolutions in 1.1 s. If she hadn't tucked at all, how many revolutions would she have made in the 1.5 s from board water? Express your answer using two significant figures.
If the diver hadn't tucked at all, she would have made approximately 0.485 revolutions in the 1.5 seconds from the board to the water.
To determine the number of revolutions the diver would have made if she hadn't tucked at all, we can make use of the conservation of angular momentum.
The initial moment of inertia of the diver with arms straight up and legs straight down is given as 18 kg.m. When she tucks into a small ball, her moment of inertia decreases to 3.6 kg.m. The ratio of the initial moment of inertia to the final moment of inertia is:
I_initial / I_final = ω_final / ω_initial
Where ω represents the angular velocity. We can rewrite this equation as:
ω_final = (I_initial / I_final) * ω_initial
The diver completes two complete revolutions in 1.1 seconds while tucked, which corresponds to an angular velocity of:
ω_tucked = (2π * 2) / 1.1 rad/s
Now we can use this information to calculate the initial angular velocity:
ω_initial = (I_final / I_initial) * ω_tucked
Substituting the given values:
ω_initial = (3.6 kg.m / 18 kg.m) * ((2π * 2) / 1.1) rad/s
ω_initial ≈ 2.036 rad/s
Finally, we can determine the number of revolutions the diver would have made in 1.5 seconds if she hadn't tucked at all. Using the formula:
Number of revolutions = (angular velocity * time) / (2π)
Number of revolutions = (2.036 rad/s * 1.5 s) / (2π)
Number of revolutions ≈ 0.485 revolutions
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An open container holds ice of mass 0.525 kg at a temperature of −15.1°C. The mass of the container can be ignored. Heat is supplied to the container at the constant rate of 780 J/ minute. The specific heat of ice to is 2100 J/kg⋅K and the heat of fusion for ice is 334 × 10³ J/kg. Part A
How much time tmelts passes before the ice starts to melt? Part B From the time when the heating begins, how much time trise does it take before the temperature begins to rise above 0°C?
The ice melts after 474.36 seconds or 7 minutes and 54 seconds and it takes 1242.88 seconds or 20 minutes and 43 seconds to raise the temperature from 0°C to 15°C.
Mass of ice, m = 0.525 kg
Temperature of ice, T1 = -15.1°C
Heat supplied to container, Q = 780 J/minute
Specific heat of ice, c = 2100 J/kg.K
Latent heat of ice, L = 334 x 10³ J/kg.
Part A:
We know that ice starts melting when its temperature reaches the melting point, which is 0°C. Therefore, the amount of heat required to raise the temperature of ice from -15.1°C to 0°C is given by:
Q1 = mcΔT1,
where
ΔT1 = 0 - (-15.1) = 15.1°C
Q1 = 0.525 x 2100 x 15.1
Q1 = 16,591.25 J
Therefore, time taken for ice to melt is given by:
Q1 + Q2 = mLt
Q2 = mLt - Q1
t = (mL - Q1)/Q2= [(0.525 x 334 x 10³) - 16,591.25] / 780
t = 474.36 seconds
Therefore, the ice melts after 474.36 seconds or 7 minutes and 54 seconds.
Part B:
The time taken for the ice to start melting is the time taken to raise the temperature from -15.1°C to 0°C, which we calculated above as 474.36 seconds. Therefore, the heating starts at this point.
Now, we need to calculate the time taken to raise the temperature of water from 0°C to 15°C, which is the temperature at which the temperature starts rising above 0°C.
The amount of heat required to do this is given by:
Q3 = mcΔT3,
where
ΔT3 = 15 - 0 = 15°C
Q3 = 0.525 x 2100 x 15
Q3 = 16,147.5 J
The time taken to raise the temperature by this amount is given by:
t = Q3/P,
where P is the power supplied.
P = 780 J/minute = 13 J/second
t = 16,147.5 / 13
t = 1242.88 seconds
Therefore, it takes 1242.88 seconds or 20 minutes and 43 seconds to raise the temperature from 0°C to 15°C.
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A proton is observed traveling with some velocity V perpendicular to a uniform magnetic field B. Which of the following statements are true in regard to the direction of the magnetic force exerted on the proton? a)The magnetic force is parallel to the proton's velocity and perpendicular to the magnetic field. o b) The magnetic force is parallel to the proton's velocity and parallel to the magnetic field. O The magnetic force is perpendicular to the proton's velocity and perpendicular to the magnetic field. O d) The magnetic force is ON e) None of the above.
The correct statement is that the magnetic force is perpendicular to the proton's velocity and perpendicular to the magnetic field.
According to the right-hand rule for magnetic forces, the direction of the magnetic force experienced by a charged particle moving through a magnetic field is perpendicular to both the velocity of the particle and the magnetic field.
In this case, the proton is observed traveling with a velocity V perpendicular to the uniform magnetic field B. As a result, the magnetic force exerted on the proton will be perpendicular to both V and B. This means that option c) "The magnetic force is perpendicular to the proton's velocity and perpendicular to the magnetic field" is the correct statement.
Option a) is incorrect because the magnetic force is not parallel to the proton's velocity. Option b) is incorrect because the magnetic force is not parallel to the magnetic field. Option d) is incomplete and does not provide any information.
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A photon of wavelength 1.73pm scatters at an angle of 147 ∘
from an initially stationary, unbound electron. What is the de Broglie wavelength of the electron after the photon has been scattered? de Broglie wavelength: pm
After a photon of wavelength 1.73 pm scatters at an angle of 147 degrees from an initially stationary, unbound electron, the de Broglie wavelength of the electron changes. Therefore, the de Broglie wavelength of the electron after the photon has been scattered is approximately 3.12 pm.
According to the de Broglie hypothesis, particles such as electrons have wave-like properties and can be associated with a wavelength. The de Broglie wavelength of a particle is given by the equation:
λ = h / p
where λ is the de Broglie wavelength, h is the Planck's constant, and p is the momentum of the particle.
In the given scenario, the initial electron is stationary, so its momentum is zero. After the scattering event, the electron gains momentum and moves in a different direction. The change in momentum causes a change in the de Broglie wavelength.
To calculate the de Broglie wavelength of the electron after scattering, we need to know the final momentum of the electron. This can be determined from the scattering angle and the conservation of momentum.
Once the final momentum is known, we can use the de Broglie wavelength equation to find the new de Broglie wavelength of the electron.
Therefore, the de Broglie wavelength of the electron after the photon has been scattered is approximately 3.12 pm.
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The masses of the two particles at position are each m,m₂ and there is only an internal force acting on the two particles, each F₁-F₁, F2=-F₂1 (Here, F > 0, ) Show that the and ₁=(-/- net torque of the two particle systems is 0.
To show that the net torque of the two-particle system is zero, we need to consider the torque acting on each particle individually and sum them up.
For particle 1, the torque is given by τ₁ = r₁ × F₁, where r₁ is the position vector of particle 1 and F₁ is the internal force acting on it. Since F₁ and r₁ are parallel, their cross product is zero, so τ₁ = 0.
For particle 2, the torque is given by τ₂ = r₂ × F₂, where r₂ is the position vector of particle 2 and F₂ is the internal force acting on it. Similarly, since F₂ and r₂ are parallel, their cross product is zero, so τ₂ = 0.
Now, to find the net torque of the system, we can sum up the individual torques: Net torque = τ₁ + τ₂ = 0 + 0 = 0.
Therefore, the net torque of the two-particle system is indeed zero.
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Why is there an upward force on a rocket when it is launched? The exhaust gas pushes downards against the ground The exhaust gas pushes against the air. The exhaust gas pushes upwards against the rocket. Question 36 What is the cosmic microwave background? It is radio emission from the fireball that ensued immediately after the Big Bang. It is radio emission from hot gas in our galaxy. It is radio emission from cool gas in the early Universe. Question 37 Why was the discovery of the cosmic microwave background important to cosmology? It is evidence, possibly consistent with the Big Bang. It is direct experimental for cool gas in the early Universe. It is direct experimental evidence for the Big Bang
The exhaust gas pushes upwards against the rocket when it is launched. Rocket propulsion is based on Newton's third law of motion, which states that when one object exerts a force on another object, the second object exerts an equal and opposite force on the first.
When the rocket expels exhaust gas out of its engine, the force of the gas pushing against the rocket is equal and opposite to the force of the rocket pushing the gas out, resulting in a net upward force on the rocket. This force causes the rocket to accelerate upwards.
The exhaust gas pushes upwards against the rocket when it is launched. Rocket propulsion is based on Newton's third law of motion, which states that when one object exerts a force on another object, the second object exerts an equal and opposite force on the first. When the rocket expels exhaust gas out of its engine, the force of the gas pushing against the rocket is equal and opposite to the force of the rocket pushing the gas out, resulting in a net upward force on the rocket.
This force causes the rocket to accelerate upwards.The cosmic microwave background radiation is radio emission from the fireball that followed the Big Bang. It was first discovered in 1964 by Arno Penzias and Robert Wilson of Bell Laboratories, who were attempting to map the Milky Way's radio waves. They noticed a persistent noise that couldn't be attributed to any known source, and after ruling out potential sources such as bird droppings, they concluded that it was coming from space. This discovery was critical to cosmology because it provided direct evidence of the Big Bang. The cosmic microwave background radiation was predicted by the Big Bang theory as a remnant of the Big Bang's early fireball phase.
The radiation's precise properties, including its nearly uniform temperature and spectrum, match the Big Bang theory's predictions conclusively, providing robust evidence for the theory's validity. I
There is an upward force on a rocket when it is launched because the force of the exhaust gas pushing against the rocket is equal and opposite to the force of the rocket pushing the gas out, resulting in a net upward force on the rocket. The cosmic microwave background radiation was critical to cosmology because it provided direct evidence of the Big Bang. It is radio emission from the fireball that followed the Big Bang and matches the Big Bang theory's predictions conclusively, providing robust evidence for the theory's validity.
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A 4.40 g bullet moving at 914 m/s strikes a 640 g wooden block at rest on a frictioniess surface. The builiet emerges, traveling in the same direction with its specd reduced to 458mis. (a) What is the resulfing speed of the biock? (b) What is the spect of the bullet-block center of mass? (a) Number ________________ Units _________________
(b) Number ________________ Units _________________
(a) Number 57 Units m/s
(b) Number 314 Units m/s
Bullet's mass, mb = 4.40 g
Bullet's speed before collision, vb = 914 m/s
Block's mass, mB = 640 g (0.64 kg)
Block's speed before collision, vB = 0 m/s (at rest)
Speed of bullet after collision, vb' = 458 m/s
(a) Resulting speed of the block (vB')
Since the collision is elastic, we can use the conservation of momentum and conservation of kinetic energy to find the velocities after the collision.
Conservation of momentum:
mbvb + mBvB = mbvb' + mBvB'
The bullet and the block move in the same direction, so the direction of velocities are taken as positive.
vB' = (mbvb + mBvB - mbvb') / mB
vB' = (4.40 x 914 + 0.64 x 0 - 4.40 x 458) / 0.64
vB' = 57 m/s
Therefore, the resulting speed of the block is 57 m/s.
(b) Speed of bullet-block center of mass (vcm)
Velocity of center of mass can be found using the following formula:
vcm = (mbvb + mBvB) / (mb + mB)
Here, vcm = (4.40 x 914 + 0.64 x 0) / (4.40 + 0.64) = 314 m/s
Therefore, the speed of bullet-block center of mass is 314 m/s.
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Two light spheres each of mass 2.0g are suspended by light strings 10cm in length. A uniform electric field |E| = 4.42 × 105N/C is applied in the horizontal direction. The charges on the spheres are equal and opposite. For what charge values will the spheres be in equilibrium at an angle θ = 10 degrees? *I believe the answer is supposed to be 5 x 10^-8 C but that's not what I'm getting.*
To achieve equilibrium for two light spheres suspended by light strings in the presence of a uniform electric field, the charges on the spheres must have specific values.
In this case, with a given angle of 10 degrees and other known parameters, the expected charge value is 5 × 10^-8 C. However, the calculated value may differ.
To find the charge values that result in equilibrium, we can use the principle of electrostatic equilibrium. The gravitational force acting on each sphere must be balanced by the electrostatic force due to the electric field.
The gravitational force can be determined by considering the mass and gravitational acceleration, while the electrostatic force depends on the charges, the electric field strength, and the distance between the charges. By equating these forces and solving the equations, we can find the charge values that satisfy the given conditions.
It's important to note that slight variations in calculations or rounding can lead to small differences in the final result, which may explain the deviation from the expected value.
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A music dock transfers 46J of energy into sound waves every second. It uses a 230V mains supply. Work out the current through the dock.
A helicopter is flying North-West at 80 m/s relative to the ground and the wind velocity is 15 m/s from the East. The helicopter's main rotor lies in a horizontal plane, has a radius of 6 m, and is rotating at 20 rad/s in a clockwise sense looking down on it. a) Calculate the helicopter's air speed and apparent heading through the air (i.e. both relative to the air). b) Calculate the maximum and minimum velocities of the blade-tips relative to the air. Hint: In both parts, draw sketches to visualise what's happening. In the second part, only consider the helicopter's motion through the air and the blade-tips' motion relative to the helicopter (i.e. the air becomes your main reference frame, not the ground).
The helicopter's air speed is 59.4 m/s and apparent heading through the air is 45°
The maximum velocity of the blade tip relative to the air is 179.4 m/s and the minimum velocity of the blade tip relative to the air is 40.6 m/s.
Speed of helicopter relative to ground (VHG) = 80 m/s
Wind velocity = 15 m/sR
otor radius = 6 m
Rotor speed = 20 rad/s
a) The airspeed of the helicopter can be obtained by calculating the resultant of the helicopter velocity vector and wind velocity vector. Let us take North as y-axis and West as x-axis.The vector components of VHG along the x-axis and y-axis respectively will be as follows:
Vx = VHG * cos 45°Vy = VHG * sin 45°
The vector components of wind velocity along the x-axis and y-axis respectively will be as follows:
V'x = 15 m/sVy' = 0
The resultant vector of the helicopter velocity and the wind velocity will be as follows:
V = Vx + V'yV = 80(cos 45°) + 15V = 59.4 m/s
The apparent heading of the helicopter through the air can be calculated as follows:tan θ = Vy / Vxθ = tan⁻¹(Vy / Vx)θ = tan⁻¹(1)θ = 45°
b) The maximum velocity occurs when the blade is perpendicular to the direction of motion and the minimum velocity occurs when the blade is parallel to the direction of motion.
Let v1 and v2 be the maximum and minimum velocities of the blade-tips relative to the air.
Velocity of the tip of a rotor blade relative to the air is given by the formula,v = (ωr) ± V
where,v = velocity of the blade tip
ω = angular velocity of the rotor
r = radius of the rotor
V = airspeed of the helicopter
Taking velocity in the upward direction as positive, we get:
v1 = (ωr) + Vv2 = (ωr) - V
Let us substitute the given values in the above two formulas.
v1 = (20 * 6) + 59.4
v1 = 179.4 m/s
v2 = (20 * 6) - 59.4
v2 = 40.6 m/s
Hence, the maximum velocity of the blade tip relative to the air is 179.4 m/s and the minimum velocity of the blade tip relative to the air is 40.6 m/s.
Thus :
(a) The helicopter's air speed is 59.4 m/s and apparent heading through the air is 45°
(b) The maximum velocity of the blade tip relative to the air is 179.4 m/s and the minimum velocity of the blade tip relative to the air is 40.6 m/s.
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Which One Is The Most Simplified Version Of This Boolean Expression ? Y = (A' B' + A B)' A. Y = B'A' + AB B. Y = AB' + BA' C. Y = B'+ A D. Y = B' + AB
which one is the most simplified version of this Boolean expression ?
Y = (A' B' + A B)'
A. Y = B'A' + AB
B. Y = AB' + BA'
C. Y = B'+ A
D. Y = B' + AB
The most simplified version of the Boolean expression Y = (A' B' + A B)' is: Y = A + B + A'
The correct answer is: C.
To simplify the Boolean expression Y = (A' B' + A B)', we can use De Morgan's theorem and Boolean algebra rules.
Let's simplify step by step:
Distribute the complement (') inside the parentheses:
Y = (A' B')' + (A B)'
Apply De Morgan's theorem to each term inside the parentheses:
Y = (A + B) + (A' + B')
Simplify the expression by removing the redundant terms:
Y = A + B + A'
The most simplified version of the Boolean expression Y = (A' B' + A B)' is:
Y = A + B + A'
Therefore, the correct answer is:
C. Y = A + B + A'
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The area under the curve on a Force versus time F vs. t) graph represents & kinetic ener a. impulse. b. momentum. e. none of the above c. work. Q10: Sphere X, of mass 2 kg, is moving to the right at 10 m/s. Sphere Y. of mass 4kg, is moving to the a. twice the magnitude of the impulse of Y on X b. half the magnitude of the impulse of Y on X c. one-fourth the magnitude of the impulse of Y on X d. four times the magnitude of the impulse of Y on X e. the same as the magnitude of the impulse of Y on X
The area under the curve on a Force versus time (F vs. t) graph represents work. Therefore, the correct answer is (c) work. In Q10, To determine the magnitude of the impulse of Sphere Y on Sphere X, the correct answer is (e) the same as the magnitude of the impulse of Y on X.
The work done by a force is defined as the product of the magnitude of the force and the displacement of the object in the direction of the force. Mathematically, work (W) is given by the equation:
W = ∫ F(t) dt
The integral represents the area under the curve of the Force versus time graph. By calculating this integral, we can determine the amount of work done by the force.
Impulse, on the other hand, is defined as the change in momentum of an object and is not directly related to the area under the curve on a Force versus time graph. Momentum is the product of an object's mass and its velocity, and it is also not directly related to the area under the curve on a Force versus time graph.
The magnitude of the impulse on X due to Y is equal to the magnitude of the change in momentum of X. It can be calculated using the equation:
Impulse (J) = Change in momentum (Δp)
The change in momentum of X is given by:
Δp = [tex]m_1 * (v_1 - u_1)[/tex]
Now, let's consider the conservation of momentum equation:
[tex]m_1 * u_1 + m_2 * u_2 = m_1 * v_1 + m_2 * v_2[/tex]
Since Sphere X is moving to the right and Sphere Y is moving to the left, we can assume that Sphere Y collides with Sphere X and comes to rest.
Therefore, the final velocity of Sphere Y ([tex]v_2[/tex]) is 0 m/s.
Plugging in the given values and solving the equation, we can find the final velocity of Sphere X ([tex]v_1[/tex]).
After obtaining the values of [tex]v_1[/tex] and [tex]v_2[/tex], we can calculate the impulse (J) using the change in momentum equation mentioned above.
Comparing the magnitudes of the impulses of Y on X and X on Y, we find that they are equal. Therefore, the correct answer is (e) the same as the magnitude of the impulse of Y on X.
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A toaster is rated at 770 W when connected to a 220 V source. What current does the toaster carry? A. 2.0 A B. 2.5 A C. 3.0 A D. 3.5 A W ITH 20 Ampara and
The toaster carries a current of approximately 3.50 A when connected to a 220 V source. So the correct option is D.
To find the current carried by the toaster, we can use Ohm's Law, which states that the current (I) flowing through a device is equal to the voltage (V) across the device divided by its resistance (R). In this case, we have the power rating (P) of the toaster, which is 770 W, and the voltage (V) of the source, which is 220 V.
First, we can calculate the resistance (R) of the toaster using the formula R = V² / P. Substituting the values, we get R = (220²) / 770 = 62.86 Ω.
Next, we can calculate the current (I) using the formula I = V / R. Substituting the values, we get I = 220 / 62.86 ≈ 3.50 A.
Therefore, the current carried by the toaster is approximately 3.50 A, which corresponds to option D in the answer choices.
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A large wind turbine has a hub height of 135 m and a rotor radius of 63 m. How much average power is contained in wind blowing at 10.0 m/s across the rotor of this wind turbine?
The average power contained in the wind blowing across the rotor of the wind turbine is approximately 1,227,554.71π (or approximately 3,858,406.71) units of power.
To calculate the average power contained in the wind blowing across the rotor of a wind turbine, we can use the formula:
Power = 0.5 * density * area * velocity^3
where:
density is the air density,
area is the cross-sectional area of the rotor,
velocity is the wind speed.
First, let's calculate the cross-sectional area of the rotor.
The area of a circle is given by the formula A = π * [tex]r^2[/tex], where r is the radius.
In this case, the rotor radius is 63 m, so the area is:
Area = π * [tex](63)^2[/tex] = 3969π square meters.
Next, we need to determine the air density.
The air density can vary depending on various factors such as altitude and temperature.
However, a typical value for air density at sea level and standard conditions is approximately 1.225 kg/[tex]m^3[/tex].
Now we can calculate the average power.
Given that the wind speed is 10.0 m/s, the formula becomes:
Power = 0.5 * 1.225 * 3969π * [tex](10.0)^3[/tex]
Calculating this expression gives us:
Power ≈ 0.5 * 1.225 * 3969π * 1000
≈ 1,227,554.71π
Therefore, the average power contained in the wind blowing across the rotor of the wind turbine is approximately 1,227,554.71π (or approximately 3,858,406.71) units of power, depending on the specific units used in the calculation.
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Why is a fission chain reaction more likely to occur in a big piece of uranium than in a small piece?
A fission chain reaction is more likely to occur in a big piece of uranium compared to a small piece due to several reasons Neutron population, Neutron leakage,Critical mass,Surface-to-volume ratio
A fission chain reaction is more likely to occur in a big piece of uranium compared to a small piece due to several reasons:
Neutron population: In a fission chain reaction, a uranium nucleus absorbs a neutron, becomes unstable, and splits into two smaller nuclei, releasing multiple neutrons. These released neutrons can then induce fission in neighboring uranium nuclei, leading to a chain reaction. A larger piece of uranium contains a higher number of uranium nuclei, increasing the probability of neutron-nucleus interactions and sustaining the chain reaction. Neutron leakage: Neutrons released during fission can escape or be absorbed by non-fissionable materials, reducing the number available to induce fission. In a larger piece of uranium, the probability of neutron leakage is lower since there is more uranium material to capture and retain the neutrons within the system, allowing for more opportunities for fission events. Critical mass: Fission requires a certain minimum mass of fissile material, known as the critical mass, to sustain a self-sustaining chain reaction. In a small piece of uranium, the mass may be below the critical mass, and thus the chain reaction cannot be sustained. However, in a larger piece, the mass exceeds the critical mass, providing enough fissile material to sustain the chain reaction. Surface-to-volume ratio: A larger piece of uranium has a smaller surface-to-volume ratio compared to a smaller piece. The surface of the uranium can act as a source of neutron leakage, with more neutrons escaping without inducing fission. A smaller surface-to-volume ratio in a larger piece reduces the proportion of neutrons lost to leakage, allowing more neutrons to interact with uranium nuclei and sustain the chain reaction.These factors collectively contribute to the increased likelihood of a fission chain reaction occurring in a big piece of uranium compared to a small piece. It is important to note that maintaining a controlled chain reaction requires careful design and control mechanisms to prevent uncontrolled releases of energy and radiation.
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A single-turn square loop carries a current of 16 A. The loop is 15 cm on a side and has a mass of 3.6×10 −2
kg - initially the loop lies flat on a horizontal tabletop. When a horizontal magnetic field is turned on, it is found that only one side of the loop experiences an upward force. Part A Find the minimum magnetic field, B min
, necessary to start lipping the loop up from the table. Express your answer using two significant figures. Researchers have tracked the head and body movements of several flying insects, including blowllies, hover fles, and honeybees. They attach lightweight, fexible wires to a small metai coli on the insect's head, and another-on its thorax, and then allow it to fly in a stationary magnetic field. As the coils move through the feld, they experience induced emts that can be analyzed by computer to determine the corresponding orientation of the head and thorax. Suppose the fly turns through an angle of 90 in 31 ms. The coll has 89 turns of wire and a diameter of 2.2 mm. The fly is immersed in a magnetic feld of magnitude 0.16 m T. Part A If the magnetic flux through one of the coils on the insect goes from a maximum to zero during this maneuver find the magnitude of the induced emf. Express your answer using two significant figures.
For the loop, the minimum magnetic field required to lift it from the table is approximately 0.24 T.
As for the flying insect, the magnitude of the induced emf in the coil due to a change in magnetic flux is approximately 0.29 mV. For the square loop, we equate the magnetic force with the gravitational force. Magnetic force is given by BIL where B is the magnetic field, I is the current, and L is the length of the side. Gravitational force is mg, where m is mass and g is gravitational acceleration. Setting BIL=mg and solving for B gives us the minimum magnetic field. For the insect, the change in magnetic flux through the coil induces an emf according to Faraday's law, given by ΔΦ/Δt = N*emf, where N is the number of turns and Δt is the time taken. Solving for emf provides the induced voltage.
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Consider the signal x(t) = w(t) sin(27 ft) where f = 100 kHz and t is in units of seconds. (a) (5 points) For each of the following choices of w(t), explain whether or not it would make x(t) a narrowband signal. Justify your answer for each of the four choices; no marks awarded without valid justification. 1. w(t) = cos(2πt) 2. w(t) = cos(2πt) + sin(275t) 3. w(t) = cos(2π(f/2)t) where f is as above (100 kHz) 4. w(t) = cos(2π ft) where f is as above (100 kHz) (b) (5 points) The signal x(t), which henceforth is assumed to be narrowband, passes through an all- pass system with delays as follows: 3 ms group delay and 5 ms phase delay at 1 Hz; 4 ms group delay and 4 ms phase delay at 5 Hz; 5 ms group delay and 3 ms phase delay at 50 kHz; and 1 ms group delay and 2 ms phase delay at 100 kHz. What can we deduce about the output? Write down as best you can what the output y(t) will equal. Justify your answer; no marks awarded without valid justification. (c) (5 points) Assume x(t) is narrowband, and you have an ideal filter (with a single pass region and a single stop region and a sharp transition region) which passes w(t) but blocks sin(2 ft). (Specifically, if w(t) goes into the filter then w(t) comes out, while if sin (27 ft) goes in then 0 comes out. Moreover, the transition region is far from the frequency regions occupied by both w(t) and sin(27 ft).) What would the output of the filter be if x(t) were fed into it? Justify your answer; no marks awarded without valid justification.
a) 1. x(t) is not a narrowband signal if w(t) = cos(2πt).
2. x(t) is not a narrowband signal if w(t) = cos(2πt) + sin(275t).
3. x(t) is a narrowband signal if w(t) = cos(2π(f/2)t).
4. x(t) is a narrowband signal if w(t) = cos(2πft).
b) the output y(t) will be the same as the input signal x(t), except that it will have a different phase shift.
c) the output of the filter will be y(t) = w(t)sin(27 ft) -> w(t) * 0 = 0.
a) 1. w(t) = cos(2πt)
If we consider the Fourier transform of the signal x(t) and w(t), we find that x(t) can be represented by a series of sinewaves with frequencies between (f - Δf) and (f + Δf).
If we consider the function w(t) = cos(2πt) and take the Fourier transform, we find that the Fourier transform is non-zero for an infinite range of frequencies.
Therefore, x(t) is not a narrowband signal if w(t) = cos(2πt).
2. w(t) = cos(2πt) + sin(275t)
We can represent w(t) as a sum of two sinusoids with different frequencies. If we take the Fourier transform, we get non-zero values at two different frequencies.
Therefore, x(t) is not a narrowband signal if w(t) = cos(2πt) + sin(275t).
3. w(t) = cos(2π(f/2)t) where f is as above (100 kHz)
If we consider the function w(t) = cos(2π(f/2)t), the Fourier transform is zero for all frequencies outside the range (f/2 - Δf) to (f/2 + Δf).
Since this range is much smaller than the frequency range of x(t), we can say that
x(t) is a narrowband signal if w(t) = cos(2π(f/2)t).
4. w(t) = cos(2π ft) where f is as above (100 kHz)If we consider the function w(t) = cos(2πft), the Fourier transform is zero for all frequencies outside the range (f - Δf) to (f + Δf).
Since this range is much smaller than the frequency range of x(t), we can say that
x(t) is a narrowband signal if w(t) = cos(2πft).
b)The signal x(t) is passed through an all-pass system with delays. The output y(t) will have the same spectral shape as the input signal x(t), but with a different phase shift. In this case, the phase shift is given by the phase delays of the all-pass system. The group delays have no effect on the spectral shape of the output signal.
Therefore, the output y(t) will be the same as the input signal x(t), except that it will have a different phase shift.
c) Since the ideal filter only allows the signal w(t) to pass through, we can simply replace sin(27 ft) with 0 in the expression for x(t).
Therefore, the output of the filter will be y(t) = w(t)sin(27 ft) -> w(t) * 0 = 0.
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A small and a large block (mass M and 2M respectively) are arranged on a horizontal surface as shown below. A student pushes on the left side of the small block so that the entire system accelerates to the right. How does the net force on the small block Fs compare to the net force on the large block F₁? Fs=FL Fs < FL 0/2 pts Fs > FL
The net force on the small block (Fs) is equal to the net force on the large block (F₁).
According to Newton's third law of motion, for every action, there is an equal and opposite reaction. When the student pushes on the left side of the small block, an equal and opposite force is exerted by the small block on the student's hand. This force is transmitted through the small block to the large block due to their contact.
Since the small and large blocks are in contact, they experience the same magnitude of force but in opposite directions. Therefore, the net force on the small block is equal in magnitude and opposite in direction to the net force on the large block.
In a system where both blocks are accelerating to the right, there must be an unbalanced force acting on the system. This unbalanced force is provided by the student's push and is transmitted through both blocks. As the large block has a greater mass, it requires a larger force to accelerate it compared to the smaller block. However, the net force acting on each block, Fs and F₁, will be equal in magnitude, satisfying Newton's third law.
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What is the estimated volume of the table tennis ball?
cm3
What is the estimated volume of the golf ball?
cm3
Answer:
The estimated volume of a standard table tennis ball is approximately 2.7 cm³.
The estimated volume of a standard golf ball is approximately 41.6 cm³.
Explanation:
A compressor operating at steady state takes in 45 kg/min of methane gas (CHA) at 1 bar, 25°C, 15 m/s, and compresses it with negligible heat transfer to 2 bar, 90 m/s at the exit. The power input to the compressor is 110 kW. Potential energy effects are negligible. Using the ideal gas model, determine the temperature of the gas at the exit, in K.
The temperature of the methane gas at the exit of the compressor is approximately 327.9 K.
To determine the temperature of the methane gas at the exit of the compressor, we can use the ideal gas law and assume that the compression process is adiabatic (negligible heat transfer).
The ideal gas law is given by:
PV = mRT
Where:
P is the pressure
V is the volume
m is the mass
R is the specific gas constant
T is the temperature
Assuming that the compression process is adiabatic, we can use the following relationship between the initial and final states of the gas:
[tex]P_1 * V_1^\gamma = P_2 * V_2^\gamma[/tex]
Where:
P₁ and P₂ are the initial and final pressures, respectively
V₁ and V₂ are the initial and final volumes, respectively
γ is the heat capacity ratio (specific heat ratio) for methane gas, which is approximately 1.31
Now let's solve for the temperature at the exit ([tex]T_2[/tex]):
First, we need to calculate the initial volume ([tex]V_1[/tex]) and final volume ([tex]V_2[/tex]) based on the given information:
[tex]V_1 = (m_{dot}) / (\rho_1)[/tex]
[tex]V_2 = (m_{dot}) / (\rho_2)[/tex]
Where:
[tex]m_{dot[/tex] is the mass flow rate of methane gas (45 kg/min)
[tex]\rho_1[/tex] is the density of methane gas at the inlet conditions [tex](P_1, T_1)[/tex]
[tex]\rho_2[/tex] is the density of methane gas at the exit conditions [tex](P_2, T_2)[/tex]
Next, we can rearrange the adiabatic compression equation to solve for [tex]T_2[/tex]:
[tex]T_2 = T_1 * (P_2/P_1)^((\gamma-1)/\gamma)[/tex]
Where:
[tex]T_1[/tex] is the initial temperature of the gas (25°C), which needs to be converted to Kelvin (K)
Finally, we substitute the known values into the equation to calculate [tex]T_2[/tex]:
[tex]T_2 = T_1 * (P_2/P_1)^{((\gamma-1)/\gamma)[/tex]
Let's plug in the values:
[tex]P_1 = 1 bar[/tex]
[tex]P_2 = 2 bar[/tex]
[tex]T_1[/tex] = 25°C = 298.15 K (converted to Kelvin)
γ = 1.31
Now we can calculate the temperature at the exit ([tex]T_2[/tex]):
[tex]T_2 = 298.15 K * (2/1)^{((1.31-1)/1.31)[/tex]
Simplifying the equation:
[tex]T_2 = 298.15 K * (2)^{0.2366[/tex]
Calculating the result:
[tex]T_2 \sim 327.9 K[/tex]
Therefore, the temperature of the methane gas at the exit of the compressor is approximately 327.9 K.
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