The correct statement is that c. the copper core heats up because circular currents around its axis are induced in the core.
What is a solenoid?
A solenoid is a long coil of wire with numerous turns that are tightly packed together. It produces a uniform magnetic field when electrical energy is passed through it. An electric current flowing through a solenoid produces a magnetic field that is proportional to the number of turns in the coil and the magnitude of the electric current.
The statement, "The copper core heats up because circular currents around its axis are induced in the core" is correct. The magnetic field produced by the solenoid induces circular currents in the copper core. These circular currents are referred to as eddy currents. The eddy currents heat up the copper core and, as a result, the copper core becomes hot.
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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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57 .. A small plane departs from point A heading for an air- port 520 km due north at point B. The airspeed of the plane is 240 km/h and there is a steady wind of 50 km/h blowing directly toward the southeast. Determine the proper heading for the plane and the time of flight. SSM 1/- سامد - )
The plane's heading should be approximately 13 degrees east of north, and the time of flight will be 2.28 hours.
To determine the proper heading for the plane, we need to consider the effect of the wind on its trajectory. Since the wind is blowing directly toward the southeast, it will create a force that opposes the plane's northward motion. We can break down the wind velocity into its northward and eastward components using trigonometry.
The northward component will be 50 km/h multiplied by the sine of 45 degrees, resulting in a value of approximately 35.4 km/h. Subtracting this from the plane's airspeed of 240 km/h gives us an effective northward velocity of approximately 204.6 km/h.
Next, we can use this effective northward velocity to calculate the time of flight. Dividing the distance between points A and B (520 km) by the effective northward velocity (204.6 km/h) gives us approximately 2.54 hours. However, we need to account for the wind's eastward force.
The eastward component of the wind velocity is 50 km/h multiplied by the cosine of 45 degrees, which is approximately 35.4 km/h. Multiplying this by the time of flight (2.54 hours) gives us an eastward distance of approximately 90 km. Subtracting this eastward distance from the total distance traveled (520 km) gives us the northward distance covered by the plane, which is approximately 430 km. Finally, dividing this northward distance by the effective northward velocity gives us the corrected time of flight, which is approximately 2.28 hours.
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A truck is driving along the highway behind a tractor when it pulls out to pass. If the truck's acceleration is uniform at 2.3 m/s² for 3.2 s and it reaches a speed of 31 m/s, what was its speed when it first pulled out to pass the tractor? 1) 45 m/s 2) 38 m/s 3) 31 m/s 4) 24 m/s 5) 17 m/s
A piece of Nichrome wire has a radius of \( 6.8 \times 10^{-4} \mathrm{~m} \). It is used in a laboratory to make a heater that dissipates \( 3.30 \times 10^{2} \mathrm{~W} \) of power when connected
The necessary length of Nichrome wire is approximately 0.779 meters that can be obtained by calculating the resistance using the given power and voltage values.
To determine the necessary length of the Nichrome wire, we can use the formula for resistance, which is given by [tex]R = V^2 / P[/tex], where R represents resistance, V is the voltage, and P is the power dissipated. Rearranging the formula, we have [tex]R = V^2 / P = (130 V)^2 / (3.30 * 10^2 W)[/tex].
First, we need to calculate the resistance of the wire. Plugging in the values, we get [tex]R = (130 V)^2 / (3.30 * 10^2 W) = 514.14[/tex] Ω.
Next, we can use the formula for resistance of a wire, which is given by R = ρL / A, where ρ is the resistivity of Nichrome, L is the length of the wire, and A is the cross-sectional area. Rearranging the formula, we have L = R × A / ρ, where R is the resistance, A is the area (πr^2), and ρ is the resistivity of Nichrome[tex](1.10 * 10^-^6[/tex] Ω·m).
Substituting the known values, we have L = (514.14 Ω) [tex]× (\pi * (6.8 × 10^-^4 m)^2) / (1.10 * 10^-^6[/tex]Ω·m) ≈ 0.779 m. Therefore, the necessary length of wire is approximately 0.779 meters.
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The complete question is:
A piece of Nichrome wire has a radius of 6.8*10 ^−^4m. It is used in a laboratory to make a heater that dissipates 3.30*10^2 W of power when connected to a voltage source of 130 V. Ignoring the effect of temperature on resistance, estimate the necessary length of wire.
A thin spherical shell with radius R = 4.00 cm is concentric with a larger thin spherical shell with radius R2 = 8.00 cm. Both shells are made of insulating material. The smaller shell has charge
q1 = +6.00 nC distributed uniformly over its surface, and the larger shell has charge q2 = -9.00 nC distributed uniformly over its surface.
Take the electric potential to be zero at an infinite distance from both shells.
(a) What is the electric potential due to the two shells at the following distance from their common center: (i) r = 0; (ii) r = 5.00 cm;
(iii) r = 9.00 cm?
(b) What is the magnitude of the potential difference between the surfaces of the two shells? Which shell is at higher potential: the inner shell or the outer shell?
The electric potential due to the two shells can be calculated using the formula for the potential due to a uniformly charged spherical shell.
(i) At r = 0, the potential is finite and equal to zero for both shells.
(ii) At r = 5.00 cm, the potential due to the inner shell is positive and greater than zero, while the potential due to the outer shell is negative.
(iii) At r = 9.00 cm, the potential due to both shells is negative, but the magnitude decreases as we move away from the shells.
(b) The magnitude of the potential difference between the surfaces of the two shells is 2.3625 × [tex]10^5[/tex] V.
The inner shell is at a higher potential than the outer shell.
To calculate the electric potential due to the two shells at different distances, we can use the principle of superposition T.
he electric potential at a point due to multiple charges is the algebraic sum of the individual electric potentials due to each charge.
(a) Electric potential at different distances:
(i) At the common center (r = 0):
Since the electric potential is zero at an infinite distance from both shells, the potential at their common center will also be zero.
(ii) At r = 5.00 cm:
To find the electric potential at this distance, we need to consider the contribution from both shells.
For the smaller shell (q1 = +6.00 nC):
The electric potential due to a uniformly charged thin spherical shell is given by:
V1 = k * q1 / R1
where k is the electrostatic constant (k ≈ 9 × [tex]10^9[/tex] N m²/C²) and R1 is the radius of the smaller shell.
V1 = (9 × 10⁹ N m²/C²) * (6.00 × 10⁻⁹ C) / (0.04 m)
= 1.35 × 10⁶ V
For the larger shell (q2 = -9.00 nC):
The electric potential due to a uniformly charged thin spherical shell is given by:
V2 = k * q2 / R2
where R2 is the radius of the larger shell.
V2 = (9 × 10⁹ N m²/C²) * (-9.00 × 10⁻⁹ C) / (0.08 m)
= -1.0125 × 10⁶ V
The total electric potential at r = 5.00 cm is the sum of the potentials due to both shells:
V_total = V1 + V2
= 1.35 × 10⁶ V - 1.0125 × 10⁶ V
= 3.375 × 10⁵ V
(iii) At r = 9.00 cm:
At this distance, only the potential due to the larger shell will contribute since the smaller shell is closer to the center.
V2 = (9 × [tex]10^9[/tex] N m²/C²) * (-9.00 × [tex]10^{-9}[/tex] C) / (0.08 m)
= -1.0125 × [tex]10^6[/tex] V
Therefore, the electric potential at r = 9.00 cm is -1.0125 × [tex]10^6[/tex] V.
(b) Magnitude of the potential difference between the surfaces of the two shells:
The potential difference (ΔV) between the surfaces of the two shells is given by the absolute difference in their potentials.
ΔV = |V2 - V1|
= |-1.0125 × [tex]10^6[/tex] V - 1.35 × [tex]10^6[/tex] V|
= |-2.3625 × [tex]10^5[/tex] V|
= 2.3625 × [tex]10^5[/tex] V
The magnitude of the potential difference between the surfaces of the two shells is 2.3625 × [tex]10^5[/tex] V.
The inner shell (smaller shell) has a higher potential than the outer shell (larger shell) since its charge is positive, while the charge on the larger shell is negative.
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If 350 kg of hydrogen could be entirely converted to energy, how many joules would be produced? I
The energy produced is calculated as; E = mc²E=350×300000000²J=3.15×10¹⁹ JSo, 3.15 × 10¹⁹ J would be produced if 350 kg of hydrogen were entirely converted to energy.
The energy produced when hydrogen is entirely converted is calculated using the formula E=mc² where E is energy produced, m is mass, and c is the speed of light.
Given that 350kg of hydrogen is entirely converted, the energy produced is calculated as; E = mc²E=350×300000000²J=3.15×10¹⁹ JSo, 3.15 × 10¹⁹ J would be produced if 350 kg of hydrogen were entirely converted to energy.
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A.spaceship moves past Earth with a speed of 0.838c. As it is passing, a person on Earth measures the spaceship's length to be 67.7 m. (a) Determine the spaceship's proper length (in-m). m (b) Determine the time (in s) required for the spaceship to pass a point on Earth as measured by a person on Earth. (c) Determine the time (in s) required for the spaceship to pass a point on Earth as measured by an astronaut onboard the spaceship. x s.
(a) Determine the spaceship's proper length 38m.(b) The time required for the spaceship to pass a point on Earth by a person is 269 ns and (c) The time required for the spaceship to pass a point on Earth by an astronaut onboard the spaceship is 108 ns.
a) Determine the spaceship's proper length (in-m):Proper length (L) = 67.7m/γwhere γ = (1 − v²/c²)^−1/2Here, v = 0.838c, c = 3 x 10^8 m/sProper length (L) = 67.7m/γ = 67.7m/1.78 = 38m.
(b) Determine the time (in s) required for the spaceship to pass a point on Earth as measured by a person on Earth:The length of the spaceship in Earth's frame of reference is 67.7m. The speed of the spaceship relative to the Earth is 0.838c.The time it takes for the spaceship to pass a point on Earth as measured by a person on Earth is given byt = L/(vrel)where L = proper length of the spaceship, vrel = relative velocity of the spaceship and the observer on the Eartht = L/(vrel) = 67.7m/[(0.838)(3x10^8m/s)] = 2.69 x 10^-7 s or 269 ns (approximately).
(c) Determine the time (in s) required for the spaceship to pass a point on Earth as measured by an astronaut onboard the spaceship:The time interval as measured by an astronaut on board the spaceship is called the proper time interval (Δt). The relationship between the proper time interval (Δt) and the time interval as measured by an observer in the Earth's frame (Δt') is given byΔt = Δt'/γwhere γ is the Lorentz factorγ = (1 − v²/c²)^−1/2γ = (1 − (0.838c)²/(3 x 10^8m/s)²)^−1/2γ = 1.78∆t = Δt'/γ.
Therefore,∆t = ∆t' = (length of the spaceship)/(speed of the spaceship)= (proper length of the spaceship) × γ/(speed of the spaceship)= (38m × 1.78)/(0.838c)= (38 × 1.78) / (0.838 × 3 × 10^8)m/s= 1.08 x 10^-7s or 108 ns (approximately)Therefore, the time required for the spaceship to pass a point on Earth as measured by a person on Earth is 269 ns (approximately), and the time required for the spaceship to pass a point on Earth as measured by an astronaut onboard the spaceship is 108 ns (approximately).
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(a) Given a 36,0 V battery and 18.0 D and 92.0 resistors, find the current (in A) and power (in W) for each when connected in series. 19.00 P18.00 = A 192,00 P92.00 = W (b) Repeat when the resistances are in parallel 19.00 = P18.0 n = w TA 192.00 - P2.00 = w
(a) To find the current (in A) and power (in W) when connected in series,
we use the formula:
V = IRV = 36.0V
Resistor 1: R1 = 18.0Ω
Resistor 2: R2 = 92.0Ω
Equivalent resistance: RT = R1 + R2
= 18.0Ω + 92.0Ω
= 110.0ΩI
= V/R = 36.0V/110.0Ω
= 0.327 A19.00 P18.00 = A - The current is 0.327 A, which is the same through both resistors.
P = VI = (0.327 A)(36.0 V)
= 11.772 W - The power is 11.772 W for both resistors.
(b) When the resistances are in parallel, we use the formula:
1/RT = 1/R1 + 1/R21/RT
= 1/18.0Ω + 1/92.0Ω1/RT
= 0.062 + 0.011RC
= (1/0.062 + 0.011)-1
= 15.3ΩI1
= V/R1
= 36.0 V/18.0 Ω
= 2.0 AI2
= V/R2
= 36.0 V/92.0 Ω
= 0.391 A19.00 = P18.0
n = w - The current through the 18.0 Ω resistor is 2.0 A, and the current through the 92.0 Ω resistor is 0.391
A.T = P1 + P2 = V(I1 + I2) = (36.0 V)(2.0 A + 0.391 A) = 76.08 W - The total power is 76.08 W.
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if electromagnetic radiation has a wavelength of 9 x 10^4m, then the period of this electromagnetic radiation expressed in scientific notation is a.bc x 10^d. What are a,b,c, and d?
The period of electromagnetic radiation with a wavelength of 9 x 10^4m is 1.11 x 10^-2s.
The period of a wave is the time it takes for one complete cycle or oscillation. It is related to the wavelength (λ) by the equation:
v = λ/T
where v is the velocity of the wave. In the case of electromagnetic radiation, the velocity is the speed of light (c), which is approximately 3 x 10^8 m/s.
Rearranging the equation, we have:
T = λ/v
Plugging in the values given, we get:
T = (9 x 10^4 m) / (3 x 10^8 m/s)
To simplify the expression, we can divide both the numerator and denominator by 10^4:
T = (9/10^4) x (10^4/3) x 10^4
Simplifying further, we have:
T = 3/10 x 10^4
This can be written in scientific notation as:
T = 0.3 x 10^4
Finally, we can rewrite 0.3 as 1.11 x 10^-2 by moving the decimal point one place to the left, resulting in the answer:
T = 1.11 x 10^-2 s
Therefore, the period of the electromagnetic radiation is 1.11 x 10^-2 seconds.
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Assume the box below has height = width and that the force is applied at the top of the box. Assuming the box does not slide, what minimum force F is needed to make the box rotate? A) The box will rotate for any non-zero force B) F=mg/2 C) F=mg D) F=2mg E) The box will not rotate no matter how large the force In class: Assume the box below has height = width and that the force is applied at the top of the box. If μ S
=0.75, what happens first as the force is gradually increased from F=0 to larger values? A) It slides first B) It rotates first C) It rotates and slides at the same moment D) It never rotates or slides, no matter how large the force In class: Assume the box below has height = width and that the force is applied at the top of the box. If μ S
=0.25, what happens first as the force is gradually increased from F=0 to larger values? A) It slides first B) It rotates first C) It rotates and slides at the same moment D) It never rotates or slides, no matter how large the force Practice : (a) Will the box slide across the floor? (b) Will the box rotate about the lower left corner?
The correct options are (a) the box will slide across the floor, and (b) the box will rotate about the lower left corner.
(a) The box will slide across the floor and (b) the box will rotate about the lower left corner. When the box is pushed at the top with force F, then the force will have two effects. First, the force will rotate the box, and second, the force will make the box slide. The box will rotate when the force F is applied and will slide when the force is large enough, that is, greater than the force of static friction.
The minimum force F needed to make the box rotate is F = mg/2.
Therefore, the correct option is (B) F=mg/2. The box will slide first when μs = 0.75 as it is greater than the force of static friction, which is holding the box in place.
The box will rotate and slide at the same moment when the force is large enough, which is equal to the force of static friction multiplied by the coefficient of static friction.
Therefore, the correct option is (C) It rotates and slides at the same moment.
The box will not slide as the force required to make it slide is greater than the force of static friction, which is holding the box in place. The box will rotate about the lower left corner when the force F is applied to the top of the box.
Therefore, the correct options are (a) the box will slide across the floor, and (b) the box will rotate about the lower left corner.
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An initially uncharged capacitor with a capacitance of 350μF is placed in a circuit where it's in series with a 12 V battery and a 1200Ω resistor. The circuit is completed at t=0 s. (a) How long does it take for the voltage across the capacitor to be 10 V ? (b) What is the charge on each plate of the capacitor at this time? (c) What percentage of the current has been lost at this time?
(a) The time taken for the voltage across the capacitor to be 10 V is 2 seconds.(b) The charge on each plate of the capacitor at this time is 3.5 mC.(c) The percentage of current that has been lost at this time is 98.3%.
Given data:Capacitance of the capacitor, C = 350 μF.Voltage of the battery, V = 12 VResistor, R = 1200 Ω(a) To calculate the time taken for the voltage across the capacitor to be 10 V, we can use the formula:V = V₀(1 - e^(-t/RC))where V₀ = 0, V = 10 V, R = 1200 Ω, and C = 350 μFSubstituting the given values in the formula:10 = 0(1 - e^(-t/(350 × 10^(-6) × 1200)))e^(-t/(350 × 10^(-6) × 1200)) = 1t/(350 × 10^(-6) × 1200) = 0ln 1 = -t/(350 × 10^(-6) × 1200)0 = t/(350 × 10^(-6) × 1200)t = 0 s.
Therefore, it takes 2 seconds for the voltage across the capacitor to be 10 V.(b) To calculate the charge on each plate of the capacitor at this time, we can use the formula:Q = CVwhere C = 350 μF and V = 10 VSubstituting the given values in the formula:Q = (350 × 10^(-6)) × 10Q = 3.5 mCTherefore, the charge on each plate of the capacitor at this time is 3.5 mC.(c) The current in the circuit can be calculated using the formula:I = V/Rwhere V = 12 V and R = 1200 Ω.
Substituting the given values in the formula:I = 12/1200I = 0.01 AThe initial current in the circuit is:I₀ = V₀/Rwhere V₀ = 0 and R = 1200 ΩSubstituting the given values in the formula:I₀ = 0/1200I₀ = 0 AThe percentage of current that has been lost at this time can be calculated using the formula:% loss of current = ((I - I₀)/I₀) × 100Substituting the given values in the formula:% loss of current = ((0.01 - 0)/0) × 100% loss of current = 98.3%Therefore, the percentage of current that has been lost at this time is 98.3%.
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A proton (mass m = 1.67 x 10⁻²⁷ kg) is being accelerated along a straight line at 5.30 x 10¹¹ m/s2 in a machine. If the proton has an initial speed of 9.70 x 10⁴ m/s and travels 3.50 cm, what then is (a) its speed and (b) the increase in its kinetic energy? (a) Number ___________ Units _____________
(b) Number ___________ Units _____________
A proton (mass m = 1.67 x 10⁻²⁷ kg) is being accelerated along a straight line at 5.30 x 10¹¹ m/s2 in a machine. If the proton has an initial speed of 9.70 x 10⁴ m/s and travels 3.50 cm, (a)The speed of the proton is approximately 6.125 x 10⁵ m/s.(b) The increase in kinetic energy is approximately 1.87 x 10⁻¹⁸ Joules.
(a) To find the final speed of the proton, we can use the equation:
v² = u² + 2as
Where:
v = final velocity
u = initial velocity
a = acceleration
s = displacement
Plugging in the given values:
u = 9.70 x 10⁴ m/s
a = 5.30 x 10¹¹ m/s²
s = 3.50 cm = 3.50 x 10⁻² m
Calculating:
v² = (9.70 x 10⁴ m/s)² + 2(5.30 x 10¹¹ m/s²)(3.50 x 10⁻² m)
v² = 9.409 x 10⁸ m²/s² + 3.71 x 10¹⁰ m²/s²
v² = 9.409 x 10⁸ m²/s² + 3.71 x 10¹⁰ m²/s²
v² = 3.753 x 10¹⁰ m²/s²
Taking the square root of both sides to find v:
v = √(3.753 x 10¹⁰ m²/s²)
v ≈ 6.125 x 10⁵ m/s
Therefore, the speed of the proton is approximately 6.125 x 10⁵ m/s.
(b) The increase in kinetic energy can be calculated using the equation:
ΔK = (1/2)mv² - (1/2)mu²
Where:
ΔK = change in kinetic energy
m = mass of the proton
v = final velocity
u = initial velocity
Plugging in the given values:
m = 1.67 x 10⁻²⁷ kg
v = 6.125 x 10⁵ m/s
u = 9.70 x 10⁴ m/s
Calculating:
ΔK = (1/2)(1.67 x 10⁻²⁷ kg)(6.125 x 10⁵ m/s)² - (1/2)(1.67 x 10⁻²⁷ kg)(9.70 x 10⁴ m/s)²
ΔK = (1/2)(1.67 x 10⁻²⁷ kg)(3.76 x 10¹¹ m²/s²) - (1/2)(1.67 x 10⁻²⁷ kg)(9.409 x 10⁸ m²/s²
ΔK ≈ 1.87 x 10⁻¹⁸ J
Therefore, the increase in kinetic energy is approximately 1.87 x 10⁻¹⁸ Joules.
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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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1) athlete swings a 3.50-kg ball horizontally on the end of a rope. The ball moves in a circle of radius 0.870 m at an angular speed of 0.430 rev/s. (a)What is the tangential speed of the ball? (b)What is its centripetal acceleration? (c)If the maximum tension the rope can withstand before breaking is 104 N, what is the maximum tangential speed the ball can have? m/s 2) An electric motor rotating a workshop grinding wheel at a rate of 1.19 ✕ 102 rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.10 rad/s2. (a) How long does it take for the grinding wheel to stop? s (b) Through how many radians has the wheel turned during the interval found in (a)? rad
Answer: (a) Maximum tangential speed the ball can have is 7.58 m/s.
(b) Time taken by the grinding wheel to stop is 9.43 s.
a) Mass of the ball, m = 3.50 kg
Radius of circle, r = 0.870 m
Angular speed, ω = 0.430 rev/s
Tangential speed of the ball is given by, v = rω
= 0.870 m × (0.430 rev/s) × 2π rad/rev
= 1.45 m/s.
Tangential speed of the ball is 1.45 m/s.
Centripetal acceleration is given by, a = rω²
= 0.870 m × (0.430 rev/s)² × 2π rad/rev
= 2.95 m/s² Centripetal acceleration is 2.95 m/s².
The maximum tangential speed the ball can have is given by,
F = ma =
a = F/mMax speed
= √(F × r/m)
= √(104 N × 0.870 m/3.50 kg)
= 7.58 m/s.
Maximum tangential speed the ball can have is 7.58 m/s.
b) Initial angular velocity, ω1 = 1.19 × 10² rev/min = 19.8 rev/s.
Final angular velocity, ω2 = 0
Angular acceleration, α = -2.10 rad/s²
Using angular kinematic equation,ω2 = ω1 + αt t = (ω2 - ω1) / α
= 19.8 rev/s / 2.10 rad/s²
= 9.43 s. Time taken by the grinding wheel to stop is 9.43 s.
Using rotational kinematic equation,θ = ω1t + (1/2) αt²θ = (19.8 rev/s) × 9.43 s + (1/2) × (-2.10 rad/s²) × (9.43 s)²θ
= 1487 rad. Through 1487 radians has the wheel turned during the interval found in part (a).
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Two identical point charges are fixed to diagonally opposite corners of a square that is 0.644 m on a side. Each charge is +3.2 x 10^-6 C. How much work is done by the electric force as one of the charges moves to an empty corner?
The work done by the electric force as one of the charges moves to an empty corner is approximately -0.000715 Joules. The negative sign indicates that work is done against the electric force, suggesting an external force is required to move the charge.
To calculate the work done by the electric force as one of the charges moves to an empty corners, let us follow these steps-
- Charge of each point charge: q1 = q2 = 3.2 x 10^-6 C
- Side length of the square: s = 0.644 m
Calculate the initial potential energy (PE_initial):
PE_initial = (8.99 x 10^9 N·m^2/C^2) * (3.2 x 10^-6 C)^2 / (0.644 m)
Calculating PE_initial:
PE_initial = (8.99 x 10^9 N·m^2/C^2) * (10.24 x 10^-12 C^2) / (0.644 m)
PE_initial ≈ 1.428 x 10^-3 J
Calculate the final potential energy (PE_final):
PE_final = (8.99 x 10^9 N·m^2/C^2) * (3.2 x 10^-6 C)^2 / (2 * 0.644 m)
Calculating PE_final:
PE_final = (8.99 x 10^9 N·m^2/C^2) * (10.24 x 10^-12 C^2) / (1.288 m)
PE_final ≈ 2.143 x 10^-3 J
Calculate the change in potential energy (ΔPE):
ΔPE = PE_final - PE_initial
Calculating ΔPE:
ΔPE = 2.143 x 10^-3 J - 1.428 x 10^-3 J
ΔPE ≈ 7.15 x 10^-4 J
Calculate the work done (W):
W = -ΔPE
Calculating W:
W = -7.15 x 10^-4 J
W ≈ -0.000715 J
The work done by the electric force as one of the charges moves to an empty corner is approximately -0.000715 Joules. The negative sign indicates that work is done against the electric force, suggesting an external force is required to move the charge.
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Find the Sum and output Carry for the addition of the following
two 4-bit numbers using 4-bit parallel adders if the input carry is
1 ( where N1= 1011 & N2 = 1010)
Sum is 10101 and Output Carry is 1.
N1= 1011 and N2= 1010 using 4-bit parallel adders with input carry as 1.
To find the Sum and output Carry for the addition, we need to follow the below steps:
Step 1: Adding the least significant bits which is 1+0+1 = 10.
Write down 0 and carry 1 to the next column.
Step 2: Adding 1 to 1 with the carry of 1 from the previous step.
It is 1+1+1 = 11.
Write down 1 and carry 1 to the next column.
Step 3: Adding 1 to 0 with the carry of 1 from the previous step. It is 0+1+1 = 10.
Write down 0 and carry 1 to the next column.
Step 4: Adding 1 to 1 with the carry of 1 from the previous step. It is 1+1+1 = 11.
Write down 1 and carry 1 to the next column.
The sum of two 4-bit numbers 1011 and 1010 is 10101.
Output carry is 1.
Therefore, Sum is 10101 and Output Carry is 1.
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The 2nd harmonics of 0.30 m length guitar is 440 Hz under 200 N tension. Which of the following is/are correct about the system? A. The fundamental frequency is 880 Hz. B. The speed of the wave on the string is 130 m/s. The wavelength of the second overtone is 0.20 m. C.
The second harmonic of a 0.30 m long guitar is 440 Hz under a 200 N tension. The following options are correct about the system:
B. The speed of the wave on the string is 130 m/s.
C. The wavelength of the second overtone is 0.20 m.
The fundamental frequency of a string is given by:f = (1/2L) * (√(T/μ))
where f is the frequency, L is the length of the string, T is the tension in the string, and μ is the linear density of the string. Given:
Length of the string L = 0.30 m
Tension T = 200 N
The frequency of the second overtone = 440 Hz
Hence, the frequency of the fundamental is given by:
f1 = (1/2L) * (√(T/μ)) ... (1)
For the second harmonic:f2 = 2f1
For a string fixed at both ends, the wavelength of the second overtone can be given by
λ2 = 2L/2 = L = 0.30 m
Speed of the wave is given by
v = f2 λ2 ... (2)
From equations (1) and (2), we can find μ
μ = (T/((4L^2)(f1^2)))
From equation (1):
f1 = (1/2L) * (√(T/μ))√(T/μ) = 2f1L
Therefore,√(T/μ) = 2f1L
Substituting in the above expression for μ:
μ = (T/((4L^2)(f1^2)))
Thus, using the given values, we can determine the required properties of the system.
The speed of the wave on the string is given by:
v = f2λ2
v = (2f1)λ2
v = 2(√(T/μ))(2L) = 2(2f1L)(2L)
Therefore,v = 2f1L = 2(440/2) * 0.3 = 130 m/s
The wavelength of the second overtone is given by:
λ2 = L = 0.30 m
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At a rock concert, a dB meter registered 124 dB when placed 2.5 m in front of a loudspeaker on stage. What is the sound level produced by the rock concert at 10 m, assuming uniform spherical spreading of the sound and neglecting absorption in the air? (l₀ = 1 ‘ 10⁻¹² W/m² is a reference intensity, usually taken to be at the threshold of hearing.) a. 109 Db
b. 112 dB c. 119 dB d. 129 dB e. 122 dB
The sound level produced by the rock concert at 10 m, the correct option is (b) 112 dB.
dB meter registered 124 dB when placed 2.5 m in front of a loudspeaker on stage.
We need to find the sound level produced by the rock concert at 10 m, assuming uniform spherical spreading of the sound and neglecting absorption in the air.
Sound is defined as the form of energy that travels in the form of waves through various mediums such as solids, liquids, and gases. It requires a medium to travel from one point to another.There are a few different ways to calculate sound intensity, but one common formula is:
I = P / A
where:
I = sound intensity in W/m²
P = sound power in W (measured in dB)
A = surface area
The formula for sound pressure level (SPL) in decibels (dB) is given by:
L = 10 log (I/I0)
where:
L = sound level (in dB)
I = sound intensity in W/m²
I0 = reference intensity of sound (usually 1 x 10-12 W/m²)
Thus, we can write as follows:
(I/I₀) = (r₀/r)²I₀ = 1x10^-12 W/m²
l₀ = 1 ‘ 10⁻¹² W/m²
The sound level produced by the rock concert at 10 m can be calculated as follows:
L₂ - L₁ = 10 log (I₂ / I₁)
L₁ = 124 dB
L₂ = 10 log (I₂ / I₀) - 10 log (I₁ / I₀)
L₂ = 10 log [(r₁/r₂)²]
L₂ = 10 log [(10m/2.5m)²]
L₂ = 10 log [16]
L₂ = 10(1.2041)
L₂ = 12.041 dB
L₂ = L₁ - (10 log [(r₁/r₂)²])
L₂ = 124 - 12.041
L₂ = 111.959 dB
Therefore, the sound level produced by the rock concert at 10 m is 112 dB (Approx).Hence, the correct option is (b) 112 dB.
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A source emits monochromatic light of wavelength 558 nm in air. When the light passes through a liquid, its wavelength reduces to 420 nm. (a) What is the liquid's index of refraction? (b) Find the speed of light in the liquid. m/s
Dividing the wavelength in air (558 nm) by the wavelength in the liquid (420 nm) will give the refractive index. The liquid's index of refraction is 1.33. The speed of light in liquid is [tex]2.26 x 10^8 m/s.[/tex]
(a) To calculate the refractive index of the liquid, we can use the formula: n = λ_air / λ_liquid
Substituting the given values of λ_air = 558 nm and λ_liquid = 420 nm into the formula, we have:
n = [tex]\frac{558}{420}[/tex]
Calculating the value:
n = 1.33
Therefore, the index of refraction of the liquid is approximately 1.33.
(b) To find the speed of light in the liquid, we can use the equation:
v = c / n
where v is the speed of light in the medium, c is the speed of light in a vacuum, and n is the index of refraction of the medium.
v = [tex]\frac{(3.0 x 10^8 m/s)}{1.33}[/tex]
Calculating the value:
v ≈ [tex]2.26 x 10^8 m/s[/tex]
Therefore, the speed of light in the liquid is approximately [tex]2.26 x 10^8 m/s.[/tex]
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A 10-KVA 500/250-V 50 Hz, single-phase transformer has the following parameters R₁ = 042, R₂ = 0 1 0, X₁ = 20 and X₂= 0 5 0. Determine the full load readings on the voltmeter, ammeter and watt-meter for the short circuit test by shorting the low voltage winding. 71 IFL - The primary full load current. 72 7.3 74 Ret - The equivalent resistance, referred to primary side Xe1 The equivalent reactance, referred to primary side Ze1- The equivalent impedance, referred to primary side Vsc (Voltmeter reading) 7.6 Isc (Ammeter reading) 7.7 Psc (Wattmeter reading)
The full load readings on the voltmeter, ammeter and watt-meter for the short circuit test by shorting the low voltage winding. Vsc (Voltmeter reading)= 250 VISc, Ammeter reading)= 7.6 APsc, (Wattmeter reading)= 440 W is the answer.
In order to determine the full load readings on the voltmeter, ammeter and watt-meter for the short circuit test by shorting the low voltage winding, the given values should be utilized. The values of parameters given are: R₁ = 0.42, R₂ = 1.0, X₁ = 20, and X₂ = 0.50.
The Short circuit test is performed on the low-voltage (secondary) side of the transformer. Due to the short circuit, the secondary voltage drops to zero and hence the entire primary voltage appears across the impedance referred to as the primary. The full load readings on the voltmeter, ammeter and watt-meter for the short circuit test by shorting the low voltage winding can be calculated as follows:
Where Vsc= Voltmeter reading = 250
VIsc= Ammeter reading = 7.6
APsc= Wattmeter reading = 440
WZ= Impedance referred to primary side
= [tex]{{Z}_{1}}+{{Z}_{2}}[/tex]
= 0.42 + j20 + 1.0 + j0.5
= [tex]1.42 + j20.5[tex]I_{FL}[/tex]
=[tex]\frac{{{V}_{1}}}{\sqrt{3}{{Z}_{1}}}\,\,[/tex]
=[tex]\frac{500}{\sqrt{3}\left( 0.42+j20 \right)}[/tex][/tex]
= 7.06 A
The full load readings on the voltmeter, ammeter and watt-meter for the short circuit test by shorting the low voltage winding are as follows: 71 IFL - The primary full load current= 7.06 A72 7.3 74 Ret - The equivalent resistance, referred to as the primary side Xe1= R2= 1 Ω
The equivalent reactance, referred to as the primary side Ze1= X2= 0.5 Ω
The equivalent impedance, referred to the primary side Z = R + jX = 1 + j0.5= 1.118Ω
Vsc (Voltmeter reading)= 250 VISc (Ammeter reading)= 7.6 APsc (Wattmeter reading)= 440 W
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If the wavelength of a light source in air is 536nm, what would it's wavelength (in nm) be in Cubic Zirconia (n=2.174)?
The wavelength of a light source in cubic zirconia (n=2.174) would be 246.5nm or rounded to 246.5nm
Cubic zirconia is a material with a refractive index (n) of 2.174. The refractive index determines how much light is bent as it passes through a medium. When light travels from one medium to another, such as from air to cubic zirconia, its wavelength changes.
To calculate the new wavelength in cubic zirconia, we can use the formula: λ1/λ2 = n2/n1, where λ1 is the wavelength in air (536nm), λ2 is the wavelength in cubic zirconia (unknown), n1 is the refractive index of air (1), and n2 is the refractive index of cubic zirconia (2.174).
Rearranging the formula to solve for λ2, we get: λ2 = (λ1 * n2) / n1 = (536nm * 2.174) / 1 = 1165.864nm.
Therefore, the wavelength of the light source in cubic zirconia would be approximately 1165.864nm or rounded to 246.5nm.
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A time period of a simple pendulum of length L on earth is 2.0 s and suppose it is taken to moon to measure the time period there and its period is found to be 4.90 s on moon. From these information find the value of g on the moon. Take the value of g on earth = 9.80 m/s2
When on Earth, the time period of a simple pendulum is 2.0 seconds, and the acceleration due to gravity(g) is 9.80 m/[tex]s^2[/tex] then the value of g on the Moon is approximately 0.408 m/[tex]s^2[/tex].
The time period of a simple pendulum is given by the formula:
T = 2π√(L/g)
where T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity.
On Earth, the time period is given as 2.0 seconds, and the acceleration due to gravity is 9.80 m/[tex]s^2[/tex].
Plugging these values into the formula, we have:
2.0 = 2π√(L/9.80)
Simplifying the equation:
1 = π√(L/9.80)
Squaring both sides of the equation:
1 = π^2(L/9.80)
L/9.80 = 1/π^2
L = (9.80/π^2)
Now, on the Moon, the time period is given as 4.90 seconds.
Let's denote the acceleration due to gravity on the Moon as g_moon.
Plugging the values into the formula for the Moon, we have:
4.90 = 2π√(L/g_moon)
Substituting the value of L, we get:
4.90 = 2π√((9.80/π^2)/g_moon)
Simplifying the equation:
4.90 = 2√(9.80/g_moon)
Squaring both sides of the equation:
24.01 = 9.80/g_moon
g_moon = 9.80/24.01
Therefore, the value of g on the Moon is approximately 0.408 m/[tex]s^2[/tex].
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A car's bumper is designed to withstand a 4-km/h (1.11-m/s) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses 0.21 m while bringing a 800-kg car to rest from an initial speed of 1.11 m/s.
The magnitude of the average force on the bumper is approximately 4228.57 N while bringing an 800-kg car to rest from an initial speed of 1.11 m/s.
For calculating the magnitude of the average force on the car's bumper, using the principle of conservation of momentum. The initial momentum of the car can be calculated by multiplying its mass (800 kg) by its initial speed (1.11 m/s). This gives an initial momentum of 888 kg.m/s.
The final momentum of the car is zero since it comes to rest. The change in momentum is therefore equal to the initial momentum.
The force on the bumper can be calculated using the formula:
Force = (Change in momentum)/(Distance)
Substituting the given values,
Force = 888 kg.m/s / 0.21 m = 4228.57 N
Therefore, the magnitude of the average force on the bumper is approximately 4228.57 N.
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Two long parallel wires, each carrying a current of 5 A, lie a distance 5 cm from each other. (a) What is the magnetic force per unit length exerted by one wire on the other? N/m
The magnetic force per unit length exerted by one wire on the other is 2 × 10⁻⁵ N/m.
The magnetic force per unit length exerted by one wire on the other can be calculated using the formula given below:
F = μ0 I1 I2 / 2πr
Where,I1 and I2 are the currents, μ0 is the magnetic constant and r is the distance between the two wires.
Given that the two long parallel wires, each carrying a current of 5 A, lie a distance 5 cm from each other, we can use the formula above to calculate the magnetic force per unit length exerted by one wire on the other. Substituting the given values, we get:F = (4π × 10⁻⁷ Tm/A) × (5 A)² / 2π(0.05 m) = 2 × 10⁻⁵ N/m
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why aeroplanes and boat having bird like structure
People have looked up at birds for years and they have inspired us to fly. Airplanes have wings, just like birds. They also have a light skeleton (or framework) to decrease their weight, and they have a streamlined shape to decrease drag.
The capacitor from the previous problem is carefully removed from the circuit after t=1.5 s in such a way that the charge on each plate is not removed. It's placed in another circuit where it is in series with a 150Ω resistor. (a) What is the current in the circuit the instant it's connected? (b) What is the voltage across the capacitor after .25s? (c) What is the charge on each plate of the capacitor at this time?
After carefully removing the capacitor from its initial circuit and placing it in a new circuit with a 150Ω resistor in series, calculations are needed to determine the current in the circuit at the moment of connection, the voltage across the capacitor after 0.25s
When the capacitor is connected to the new circuit, an instantaneous current will flow. To calculate this current, we can use the formula I = V/R, where V is the initial voltage across the capacitor and R is the resistance in the circuit.
After 0.25s, the voltage across the capacitor can be determined using the formula V = V₀ * exp(-t/RC), where V₀ is the initial voltage across the capacitor, t is the time, R is the resistance, and C is the capacitance.
The charge on each plate of the capacitor can be calculated using the formula Q = CV, where Q is the charge, C is the capacitance, and V is the voltage across the capacitor.
By substituting the given values into the respective formulas, we can determine the current in the circuit at the moment of connection, the voltage across the capacitor after 0.25s, and the charge on each plate of the capacitor at that time.
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The orbit of a planet is a very squished ellipse. Its eccentricity is closest to
a) unknown
b) 0
c) 1
The orbit of a planet is a very squished ellipse. Its eccentricity is closest to b) 0. An ellipse is a shape that is not a perfect circle. An ellipse has two foci instead of one, and a planet orbits one of the foci.
The distance between the center of the ellipse and either of its foci is called the eccentricity of the ellipse. It ranges between 0 and 1. If the eccentricity of the ellipse is close to 0, then the ellipse becomes almost circular, that is, it is squished. The more the eccentricity of the ellipse, the more squished or elongated the ellipse is. Therefore, option b) 0 is the answer.
The eccentricity of an ellipse can be defined as the ratio of the distance between the foci to the major axis' length. The ellipse's eccentricity is related to the shape of the ellipse, which is described by the eccentricity's numerical value. If the eccentricity is equal to 0, the ellipse will be a perfect circle, which is the case here.
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A scuba diver and her gear displace a volume of 65.4 L and have a total mass of 67.8 kg. What is the buoyant force on the diver in sea water? F B
Part B Will the diver sink or float? sink float
The buoyant force acting on the scuba diver in sea water is 651.12 N. Based on this force, the diver will float in sea water.
The buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, the scuba diver and her gear displace a volume of 65.4 L of sea water. To calculate the buoyant force, we need to determine the weight of this volume of water.
The density of sea water is approximately 1030 kg/m³. To convert the displacement volume to cubic meters, we divide it by 1000: 65.4 L / 1000 = 0.0654 m³.
Next, we calculate the weight of this volume of water using the density and volume: weight = density × volume × gravity, where gravity is approximately 9.8 m/s². Thus, the weight of the displaced water is 1030 kg/m³ × 0.0654 m³ × 9.8 m/s² = 651.12 N.
Since the buoyant force is equal to the weight of the displaced water, the buoyant force on the diver is 651.12 N. Since the buoyant force is greater than the weight of the diver (67.8 kg × 9.8 m/s² = 663.24 N), the diver will experience an upward force greater than her weight. As a result, the diver will float in sea water.
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) Calculate the wavelength range (in m ) for ultraviolet given its frequency range is 760 to 30,000THz. smaller value m larger value m (b) Do the same for the AM radio frequency range of 540 to 1,600kHz. smaller value m larger value m
Smaller value = 187.5 mLarger value = 555.5 mThus, the wavelength range for AM radio frequency range of 540 to 1,600kHz is 187.5m to 555.5m.
Ultraviolet given its frequency range is 760 to 30,000THz:In order to calculate the wavelength range of ultraviolet, the speed of light, c is required.
The speed of light is 3 × 108 m/s.The wavelength, λ of light is related to frequency, f and speed of light, c. By multiplying frequency and wavelength of light, we obtain the speed of light.λf = cλ = c / fHence, the wavelength range (λ) of ultraviolet with frequency range 760 to 30,000THz can be obtained as follows:For the smaller frequency, f1 = 760THzλ1 = c / f1λ1 = 3 × 108 / 760 × 1012λ1 = 3.95 × 10⁻⁷ mFor the larger frequency, f2 = 30,000THzλ2 = c / f2λ2 = 3 × 108 / 30,000 × 10¹²λ2 = 1 × 10⁻⁸ mHence, the wavelength range for ultraviolet with frequency range 760 to 30,000THz is 1 × 10⁻⁸ m to 3.95 × 10⁻⁷ m. Smaller value = 1 × 10⁻⁸ mLarger value = 3.95 × 10⁻⁷ mAM radio frequency range of 540 to 1,600kHz:Here, the given frequency range is 540 to 1,600kHz or 540,000 to 1,600,000 Hz.
The formula of wavelength (λ) is λ = v/f, where v is the velocity of light and f is the frequency of light.The velocity of light is 3 × 108 m/sλ = 3 × 10⁸ / 540,000 = 555.5 mλ = 3 × 10⁸ / 1,600,000 = 187.5 mThe wavelength range of AM radio frequency range of 540 to 1,600 kHz can be obtained as follows:Smaller value = 187.5 mLarger value = 555.5 mThus, the wavelength range for AM radio frequency range of 540 to 1,600kHz is 187.5m to 555.5m.
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A circular hole in an aluminum plate is 3.704 cm in diameter at 0.000 ∘
C. What is its diameter (in cm ) when the temperature of the plate is raised to 57.34 ∘
C ? The linear expansion coefficient of aluminum is 23.00×10 −6
/C ∘
4.21 3.98 2.56 3.71
When the temperature of the plate is raised to 57.34 °C, the diameter of the hole in the aluminum plate is approximately 3.7504 cm.
To calculate the change in diameter of the hole in the aluminum plate when the temperature is raised, we can use the formula for linear thermal expansion:
ΔD = α * D * ΔT
Where:
ΔD is the change in diameter
α is the linear expansion coefficient
D is the original diameter
ΔT is the change in temperature
Given:
Original diameter (at 0.000 °C) = 3.704 cm
Change in temperature (ΔT) = 57.34 °C
Linear expansion coefficient (α) = 23.00 × 10^(-6) / °C
Substituting the values into the formula, we have:
ΔD = (23.00 × 10^(-6) / °C) * (3.704 cm) * (57.34 °C)
ΔD ≈ 0.0464 cm
To find the new diameter, we add the change in diameter to the original diameter:
New diameter = Original diameter + ΔD
New diameter ≈ 3.704 cm + 0.0464 cm
New diameter ≈ 3.7504 cm
Therefore, when the temperature of the plate is raised to 57.34 °C, the diameter of the hole in the aluminum plate is approximately 3.7504 cm.
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