The root locus of the armature-controlled DC motor model can be sketched in terms of the damping constant c, and the effect of c on the motor time constant can be evaluated by looking at the location of the roots on the imaginary axis.
The root locus is a graphical representation of the locations of the roots of the characteristic equation as a parameter is varied. In this case, the parameter of interest is the damping constant c.
To sketch the root locus, we need to first determine the poles of the system. The characteristic equation for the armature-controlled DC motor model is:
LqIs² + (RqI +cLa)s +cRg+KoKy = 0
Using the given parameter values, we can simplify this to:
0.00012s² + (0.00024c + 0.0003)s + 0.02c + 0.01 = 0
To find the poles of this equation, we can solve for s using the quadratic formula:
s = (-b ± sqrt(b² - 4ac)) / 2a
where a = 0.00012, b = 0.00024c + 0.0003, and c = 0.02c + 0.01.
Simplifying this expression, we get:
s = (-0.00024c - 0.0003 ± sqrt((0.00024c + 0.0003)² - 4(0.00012)(0.02c + 0.01))) / 0.00024
Now, we can plot the roots of this equation as a function of c. This is the root locus. As c varies, the roots will move along the locus.
The effect of c on the motor time constant can be evaluated by looking at the location of the roots on the imaginary axis. The time constant is proportional to the reciprocal of the imaginary part of the root. As c increases, the roots move closer to the real axis, indicating a shorter time constant. This means that the motor response will be faster and more responsive to changes in the input.
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what is the sensitivity (in µa) of the galvanometer (that is, what current gives a full-scale deflection) inside a voltmeter that has a 1.00 mω resistance on its 30.5 v scale?
The sensitivity of the galvanometer inside the voltmeter is 30,500,000 µA.
Hi! To find the sensitivity (in µA) of the galvanometer inside a voltmeter with a 1.00 mΩ resistance on its 30.5 V scale, you can follow these steps:
1. First, note the full-scale voltage, V = 30.5 V, and the internal resistance of the voltmeter, R = 1.00 mΩ.
2. Use Ohm's law to calculate the current for full-scale deflection, I = V/R.
3. Convert the calculated current to microamperes (µA).
Now, let's calculate the sensitivity of the galvanometer:
1. V = 30.5 V, R = 1.00 mΩ = 0.001 Ω (since 1 mΩ = 0.001 Ω).
2. I = V/R = 30.5 V / 0.001 Ω = 30500 A.
3. Convert the current to µA: 1 A = 1,000,000 µA, so 30500 A = 30,500,000 µA.
So, the sensitivity of the galvanometer inside the voltmeter is 30,500,000 µA.
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A closed curve encircles several conductors. The line integral around this curve is ∮B⃗ ⋅dl⃗ = 4.25×10^-4 T⋅m .
A) What is the net current in the conductors?
B) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral?
a) the net current in the conductors is 3.38 A. b) the line integral in the opposite direction will be: ∮B⃗ ⋅dl⃗ = -4.25×10^-4 T⋅m
To solve this problem, we can use Ampere's Law, which relates the line integral of the magnetic field around a closed loop to the net current passing through the loop.
A) The equation for Ampere's Law is: ∮B⃗ ⋅dl⃗ = μ0I, where μ0 is the permeability of free space and I is the net current passing through the loop. Solving for I, we get:
I = ∮B⃗ ⋅dl⃗ / μ0
Substituting the given values, we get:
I = (4.25×10^-4 T⋅m) / (4π×10^-7 T⋅m/A)
I = 3.38 A
Therefore, the net current in the conductors is 3.38 A.
B) If we integrate around the curve in the opposite direction, the value of the line integral will be negative, since the direction of the magnetic field will be opposite. Specifically, we can use the fact that reversing the direction of the line integral is equivalent to reversing the direction of the loop, which changes the sign of the enclosed current.
Therefore, the line integral in the opposite direction will be: ∮B⃗ ⋅dl⃗ = -4.25×10^-4 T⋅m
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the current is from left to right in the conductor showmn. the magnetic field is onto the page and point is at higher potential than point t. the charge carriers are
a. positive b. negative c. neutral d. absent e. moving near the speed of light
The charge carriers in the conductor are likely to be negative, and they are not absent, but their speed is not near the speed of light due to the effects of collisions in the conductor.
Based on the given information, we know that the conductor is experiencing a magnetic force due to the magnetic field pointing onto the page. Additionally, we know that the point labeled as "point" is at a higher potential than the point labeled as "t".
The direction of the current flowing from left to right in the conductor is indicative of the direction in which the charge carriers are moving. Given this information, we can determine that the charge carriers in the conductor must be negative because electrons are negatively charged and move opposite to the direction of conventional current flow.
Furthermore, the fact that the charge carriers are moving in a conductor does not necessarily imply that they are moving near the speed of light. The speed at which electrons move in a conductor is known as the drift velocity and is typically much slower than the speed of light due to collisions with other particles in the conductor.
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Name Period_Date Rotation: Worksheet 9 Angular Momentum The following masses are swung in horizontal circles at the end of a thin string at constant speed. a. A 2.0 kg mass moving at 2.0 on the end of a 2.0 m long thin string. b. A 3.0 kg mass moving at 1.om, on the end of a 2.0 m long thin string. c. A 1.0 kg mass moving at 3.0 on the end of a 2.0 m long thin string. d. A 2.0 kg mass moving at 1.0 on the end of a 4.0 m long thin string. e. A 2.0 kg mass moving at 2.0 on the end of a 4.0 m long thin string
The length of string is 16.0 kg·m²/s
What is Mass?
Mass is a fundamental property of matter that quantifies the amount of substance or material present in an object. It is a scalar quantity, meaning it only has magnitude and no direction. Mass is commonly measured in units such as kilograms (kg), grams (g), or other appropriate units depending on the context.
a. Mass: 2.0 kg
Velocity: 2.0 m/s
Length of string: 2.0 m
Angular momentum: 8.0 kg·m²/s
b. Mass: 3.0 kg
Velocity: 1.0 m/s
Length of string: 2.0 m
Angular momentum: 6.0 kg·m²/s
c. Mass: 1.0 kg
Velocity: 3.0 m/s
Length of string: 2.0 m
Angular momentum: 6.0 kg·m²/s
d. Mass: 2.0 kg
Velocity: 1.0 m/s
Length of string: 4.0 m
Angular momentum: 2.0 kg·m²/s
e. Mass: 2.0 kg
Velocity: 2.0 m/s
Length of string: 4.0 m
Angular momentum: 8.0 kg·m²/s
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What eclipse occurs when the Moon is in between the Sun and the Earth and the Moon partially or completely blocks out the Sun?
The eclipse that occurs when the Moon is in between the Sun and the Earth and partially or completely blocks out the Sun is known as a solar eclipse.
During a solar eclipse, the Moon casts a shadow on the Earth's surface, creating a path of totality where the Sun is completely blocked out, and a partial eclipse where only part of the Sun is covered. Solar eclipses occur only during a new moon when the Moon passes between the Sun and the Earth.
During a solar eclipse, the Moon's shadow falls on the Earth's surface, causing the Sun to appear as if it is being covered or "eclipsed" by the Moon. Depending on the alignment of the Sun, Moon, and Earth, a solar eclipse can be either partial or total.
Therefore, A total solar eclipse occurs when the Moon completely covers the Sun, and only the Sun's corona (outer atmosphere) is visible as a glowing ring around the Moon.
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Find the Norton equivalent with respect to the terminals a, b in the circuit in Fig. P4.68. Figure P4.68 8 mA $20 k(2 10 mA $30 k) b
To find the Norton equivalent of the circuit with respect to the terminals a and b, we need to determine the equivalent current source (in amperes) and the equivalent resistance (in ohms) connected in parallel to the terminals a and b.
First, we can simplify the circuit by combining the two parallel branches using the current divider rule. The total current flowing from the 8 mA source is divided between the two parallel branches, with the ratio of the currents determined by the ratio of the resistances. The current flowing through the 20 kΩ resistor is:
I_1 = (30 kΩ)/(20 kΩ + 30 kΩ) * 8 mA = 3.2 mA
Similarly, the current flowing through the 30 kΩ resistor is:
I_2 = (20 kΩ)/(20 kΩ + 30 kΩ) * 8 mA = 4.8 mA
The total current flowing out of the 8 mA source is therefore:
I_total = I_1 + I_2 = 3.2 mA + 4.8 mA = 8 mA
This tells us that the Norton equivalent current source is 8 mA.
Next, we need to find the Norton equivalent resistance. To do this, we can replace the 8 mA current source with a short circuit and calculate the total resistance between the terminals a and b. With the 8 mA source replaced by a short circuit, the equivalent resistance is simply the parallel combination of the 20 kΩ and 30 kΩ resistors:
R_eq = (20 kΩ * 30 kΩ)/(20 kΩ + 30 kΩ) = 12 kΩ
Therefore, the Norton equivalent with respect to the terminals a and b is an 8 mA current source in parallel with a 12 kΩ resistor.
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Explain the concept of generational wealth. In How Jews Became White and What That
Says About America, how did the GI Bill described in the essay impact the generational
wealth for the men who served, marginalized populations, and women. Support your
response with two paragraphs.
Generational wealth is a kind of asset that passes from one generation to another. It gives freedom to think and live.
The History of Jews in the United States is one of the racial changes that provides insight into the race in America. American Jews of different eras have ethnoracial identities.
Generational wealth is a kind of asset that is passed down from one generation to the other. The first generation enjoys the property of the family and then it passes to their children.
From generational wealth, a person can gain financial freedom to live. By investing in real estate and the stock market, and by creating various income of sources, we can create generational wealth.
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You will need to design a cascode amplifier, which should be built and tested to meet the following requirements:1. Magnitude of the voltage gain = 12*SQRT(Z+35):± 10%, where Z is the sum of the last 3 digits of yourstudent number.2. The load resistance RL = 6*(Z+40)2 Ω, rounded up to the nearest standard value stocked in the lab, i.e.,decade multiples of 1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, and 8.2 kΩ. As an example, if yourRL is 67.4 kΩ, you will round this to 68 kΩ.3. The high frequency cutoff fH is to be maximized. It must exceed 1 MHz.4. The output voltage should be able to get to 2 V peak-peak without appreciable distortion[1]. To ensurethis, the AC base-emitter voltage must be kept under 10 mV peak-peak for such an output.5. No DC current may flow in RL and no DC current may flow into or out of the signal generator.6. The low frequency fL must be less than 200 Hz.7. The input and output impedances are left to the discretion of the designer, but their magnitudes at 1kHz are to be determined by calculation and then measured.8. Total circuit power is not to exceed 50 mW.9. The transistors are all to be 2N3904.10. Collector currents in the transistors are to be 1.0 mA ±10%.11. Power-supply voltages are to be limited to +5 volts and/or +15 volts and/or -15 volts.12. No adjustable components, e.g. a trimmer potentiometer, will be allowed.13. Available capacitors are limited to: 1 x 100 µF, 1 x 33 µF, 1 x 10 µF, 2 x 1 µF, 1 x 0.1 µF [2].14. The choice of all other components is left up to the designer.
To design a cascode amplifier meeting the specified requirements, follow these steps:
1. Calculate the voltage gain (Av) using the formula Av = 12 * SQRT(Z + 35), where Z is the sum of the last 3 digits of your student number.
2. Determine the load resistance (RL) using the formula RL = 6 * (Z + 40)² Ω, and round up to the nearest standard value stocked in the lab.
3. Use a high-pass filter at the input to achieve the low frequency cutoff (fL) of less than 200 Hz.
4. Design the cascode amplifier using 2N3904 transistors with collector currents set to 1.0 mA ±10%. The power supply voltages should be limited to +5V, +15V, or -15V.
5. Maximize the high frequency cutoff (fH) to exceed 1 MHz by carefully selecting component values and minimizing parasitic capacitances.
6. Ensure the output voltage can reach 2 V peak-peak without distortion by keeping the AC base-emitter voltage below 10 mV peak-peak.
7. Prevent DC current from flowing in RL and the signal generator by using coupling capacitors.
8. Calculate and measure the input and output impedances at 1 kHz.
9. Limit the total circuit power to 50 mW.
10. Use the available capacitors (1 x 100 µF, 1 x 33 µF, 1 x 10 µF, 2 x 1 µF, 1 x 0.1 µF) in the design.
11. Choose all other components as needed to achieve the desired performance while adhering to the constraints.
By following these steps, you will design a cascode amplifier that meets the given requirements. Remember, no adjustable components are allowed, and all transistors must be 2N3904.
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what wavelength bands were placed into which color guns
wavelength bands were placed into which color guns,
we need to understand that color guns are components of a cathode ray tube (CRT) display, commonly used in older televisions and monitors. CRT displays have three color guns: red, green, and blue (RGB). These color guns produce different wavelengths corresponding to their respective colors.
1. Red color gun: The red color gun corresponds to the wavelength band of approximately 620-750 nm.
2. Green color gun: The green color gun corresponds to the wavelength band of approximately 495-570 nm.
3. Blue color gun: The blue color gun corresponds to the wavelength band of approximately 450-495 nm.
By combining various intensities of these three primary colors, CRT displays can produce a wide range of colors on the screen.
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A large electromagnet coil is connected to a 130 Hz ac source. The coil has a resistance 410 omega or Ohms, and at this source frequency the coil has an inductive reactance 230 omega or Ohms.
Part A.) What is the inductance of the coil? ( answer is L= ? H)
Part B.) What must the rms voltage of the source be if the coil is to consume an average electrical power of 830 W? (answer is V(rms)= ? V)
The inductance of the coil is approximately 0.444 H. The rms voltage of the source must be approximately 291 V for the coil to consume an average electrical power of 830 W.
Part A:
We can use the equation X_L = 2πfL, where X_L is the inductive reactance, f is the frequency, and L is the inductance of the coil. Substituting the given values, we get:
230 Ω = 2π(130 Hz)L
Solving for L, we get:
L = 230 Ω / (2π × 130 Hz) ≈ 0.444 H
Therefore, the inductance of the coil is approximately 0.444 H.
Part B:
The average electrical power consumed by the coil is given by P = V(rms)I cos(φ), where V(rms) is the rms voltage, I is the rms current, and cos(φ) is the power factor. Since the coil has only inductive reactance, the power factor is zero, and cos(φ) = 0. Therefore, the equation simplifies to:
P = V(rms)I
We know that the resistance of the coil is 410 Ω, and the inductive reactance is 230 Ω. Therefore, the total impedance of the coil is:
Z = √([tex]R^{2}[/tex] + [tex]X_L^{2}[/tex]) = √([tex]410^{2}[/tex] + [tex]230^{2}[/tex]) ≈ 470 Ω
Since the current through the coil is given by I = V(rms) / Z, we can substitute this expression into the equation for power:
P = V(rms)(V(rms) / Z)
Solving for V(rms), we get:
V(rms) = √(PZ) = √(830 W × 470 Ω) ≈ 291 V
Therefore, the rms voltage of the source must be approximately 291 V for the coil to consume an average electrical power of 830 W.
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An automobile dealer calculates the proportion of new cars sold that have been returned a various numbers of times for the correction of defects during the warranty period. The results are shown in the following table. Number of returns 0 Proportions 0.28 0.36 0.23 0.09 0.04 (a) Graph the probability distribution function. (b) Calculate and graph the cumulative probability distribution. (b) Calculate and graph the cumulative probability distribution. Find the mean of the number of returns of an automobile for corrections for defects during the warranty period. (d) Find the variance of the number of returns of an automobile for correc- tions for defects during the warranty period.
(a) Bar graph with x-axis showing the number of returns and y-axis showing the proportion of new cars sold with that number of returns.
(b) Line graph with x-axis showing the number of returns and y-axis showing the cumulative proportion of new cars sold with up to that number of returns.
(c) Mean = 0.95 returns, calculated as (00.28)+(10.36)+(20.23)+(30.09)+(40.04).
(d) Variance = 1.715 returns^2, calculated as [(0-0.95)^20.28]+[(1-0.95)^20.36]+[(2-0.95)^20.23]+[(3-0.95)^20.09]+[(4-0.95)^20.04].
In part (a), we represent the probability distribution function using a bar graph. The x-axis shows the number of returns, and the y-axis shows the proportion of new cars sold with that number of returns. In part (b), we plot the cumulative probability distribution using a line graph. The x-axis shows the number of returns, and the y-axis shows the cumulative proportion of new cars sold with up to that number of returns.
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What is the output voltage of a 3.0000-V lithium cell in a digital wristwatch that draws 0.300 mA, if the cell’s internal resistance is 2.00 Ω ?
Output Voltage of the lithium cell = 2.9994 V
Output Voltage is the maximum voltage that a cell can offer after overcoming its own internal resistance that arises from the construction of the cell.
To find the output voltage of the lithium cell, you need to consider the voltage drop across the internal resistance due to the current draw. You can use Ohm's Law for this calculation: Voltage drop = Current × Resistance.
In this case, the current draw is 0.300 mA, and the internal resistance is 2.00 Ω. First, convert the current to amperes: 0.300 mA = 0.0003 A.
Now, calculate the voltage drop: Voltage drop = 0.0003 A × 2.00 Ω = 0.0006 V.
Finally, subtract the voltage drop from the initial cell voltage: Output voltage = 3.0000 V - 0.0006 V = 2.9994 V.
The output voltage of the lithium cell is approximately 2.9994 V.
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An inductor is connected to a 12 khz oscillator that produces an rms voltage of 8.0 v. The peak current is 80 ma. What is the value of the inductance L?
The value of the inductance L is 4.77 mH.
To find the value of the inductance L, first, we need to determine the reactance (X_L) of the inductor using the given rms voltage (8.0 V) and peak current (80 mA). The relationship between voltage, current, and reactance is given by Ohm's law for inductive circuits: V_rms = I_peak × X_L.
1. Convert peak current to rms current: I_rms = I_peak / √2 = 80 mA / √2 ≈ 56.57 mA
2. Calculate reactance: X_L = V_rms / I_rms = 8.0 V / 56.57 mA ≈ 141.42 Ω
3. Find inductance using the formula X_L = 2π × frequency × L:
L = X_L / (2π × frequency) = 141.42 Ω / (2π × 12 kHz) ≈ 4.77 mH
The value of the inductance L in the given scenario is 4.77 millihenries (mH).
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When helium capture occurs with a carbon 12 nucleus, what results?
A) Nitrogen 14
B) Oxygen 16
C) Neon 20
D) Silicon 28
E) Nickel 56
When helium capture occurs with a carbon 12 nucleus, it results in Nitrogen 14.
Helium capture, also known as alpha capture, is a type of nuclear reaction in which a helium nucleus (consisting of two protons and two neutrons, denoted as an alpha particle) is captured by a target nucleus.
When a helium capture occurs with a carbon 12 nucleus (which has 6 protons and 6 neutrons), the resulting nucleus will have 8 protons and 8 neutrons. This results in the formation of a nitrogen 14 nucleus, which has 7 protons and 7 neutrons, denoted as 14N or Nitrogen-14.
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the current in an electric hair dryer is 11aa. part a how much charge flows through the hair dryer in 4.0 minmin ?
To find the charge that flows through the electric hair dryer in 4.0 minutes, we need to use the formula:
charge = current x time
Substituting the given values, we get:
charge = 11 A x 4.0 min = 44 C
Therefore, the amount of charge that flows through the hair dryer in 4.0 minutes is 44 Coulombs.
Hi! To calculate the charge that flows through the electric hair dryer in 4.0 minutes, you can use the formula Q = I * t, where Q is the charge, I is the current, and t is the time.
Given that the current (I) in the hair dryer is 11 A, and the time (t) is 4.0 minutes (or 240 seconds, since 1 minute = 60 seconds), you can plug these values into the formula:
Q = 11 A * 240 s
Q = 2640 C
So, 2640 Coulombs of charge flow through the electric hair dryer in 4.0 minutes.
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A small satellite orbits around earth. At its closest approach it is 180 km from the earth's surface and at its furthest point it is 1360 km above the surface. (a) Find the semi-major axis of the ellipse and (b) the eccentricity of the ellipse. (Hint on (b): one of the focal points of the ellipse is the center of the earth...make a nice drawing to clarify your thinking.)
(a) The semi-major axis of the orbit is 770 km.
(b) The eccentricity of the ellipse is 8.67, calculated using the distance between the foci and length of the major axis of the elliptical orbit.
How to find the semi-major axis of the ellipse?(a) The semi-major axis of an ellipse is half the distance between its furthest and closest points. In this case, the closest point is 180 km above the surface of the earth, and the furthest point is 1360 km above the surface of the earth. Therefore, the semi-major axis is:
a = (180 km + 1360 km)/2 = 770 km
How to find the eccentricity of the ellipse?(b) The eccentricity of an ellipse is defined as the distance between its foci divided by the length of its major axis. Since one of the foci is at the center of the earth, we only need to find the distance between the other focus and the center of the earth. This can be calculated using the fact that the sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis.
Let's call the distance between the center of the earth and the other focus "c". Then the length of the major axis is:
2a = 2×770 km = 1540 km
At the closest point, the distance from the satellite to the center of the earth is 180 km + the radius of the earth, which we can approximate as 6371 km. At the furthest point, the distance from the satellite to the center of the earth is 1360 km + the radius of the earth. Therefore, we can write:
2a = (180 km + 6371 km + c) + (1360 km + 6371 km - c)
Simplifying, we get:
2a = 2×6371 km + 1360 km
Substituting the value of a, we get:
1540 km = 2×6371 km + 1360 km
Solving for c, we get:
c = sqrt((1540 km)² - (2×6371 km)²) = 6,673 km
Therefore, the eccentricity of the ellipse is:
e = c/a = 6,673 km/770 km = 8.67
The eccentricity of the ellipse is 8.67.
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Find the value of “F2”
The reaction force exerted by m₁ is 118.4 N.
Mass of the upper block, m₁ = 8 kg
Mass of the lower block, m₂ = 15 kg
Acceleration, a = 5 m/s₂
Normal reaction is a force that applies perpendicularly to two surfaces that are in contact. It represents the force that is holding the two surfaces together.
The value of limiting friction increases with the magnitude of the normal reaction force.
The force exerted by m₁ is,
F₁ = m₁(g + a)
F₁ = 8(9.8 + 5)
F₁ = 118.4 N
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Your question was incomplete. Attaching the image file here
The question is often asked: Can an airfoil fly upside-down? To answer this, make the following calculation. Consider a positively cambered airfoil with a zero-lift angle of -2°. The lift slope is 0.1 per degree.a/ Calculate the lift coefficient at an angle of attack of 6º. b/ Now imagine the same airfoil turned upside-down, but at the same 6° angle of attack as part (a). Calculate its lift coefficient. c/ At what angle of attack must the upside-down airfoil be set to generate the same lift as that when it is right- side-up at a 6° angle of attack?
Yes, an airfoil can fly upside-down. To calculate the lift coefficient of a positively cambered airfoil with a zero-lift angle of -2° at an angle of attack of 6º, we use the lift slope of 0.1 per degree.
a/ The lift coefficient at 6º angle of attack would be 0.1 x (6-(-2)) = 0.8
b/ When the same airfoil is turned upside-down, the lift coefficient at the same 6° angle of attack would still be 0.8 because the lift coefficient only depends on the angle of attack and the shape of the airfoil, not its orientation.
c/ To generate the same lift as when the airfoil is right-side-up at a 6° angle of attack, the upside-down airfoil must be set to an angle of attack of -6º because the lift coefficient is proportional to the angle of attack.
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In a parallel circuit, the current through any resistor is directly proportional to the value of that resistor. True or False.
False. In a parallel circuit, the current through each resistor is not directly proportional to the value of that resistor. Instead, the current through each resistor is inversely proportional to the value of the resistor. This means that as the resistance of a resistor increases, the current through it decreases, and vice versa.
In a parallel circuit, the voltage across each resistor is the same, but the current flowing through each resistor can be different. This is because each resistor provides a different path for the flow of electricity. The total current flowing into the parallel circuit is divided among the different resistors according to their individual resistance values.
In summary, in a parallel circuit, the current through each resistor is inversely proportional to its value. This is because the total current flowing into the parallel circuit is divided among the different resistors according to their individual resistance values, and the voltage across each resistor is the same.
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show that the transition matrix is regular and find its steady-state vector.
To show that a transition matrix is regular by raising the matrix to different powers until you find a power with all positive elements and its steady-state vector is equation πP = π while ensuring the elements of π sum to 1.
A transition matrix is regular if some power of the matrix has only positive elements, a steady-state vector is a probability vector that remains unchanged after being multiplied by the transition matrix. To demonstrate that the transition matrix is regular, raise the matrix to different powers until you find a power with all positive elements. If such a power exists, the matrix is considered regular.
Next, to find the steady-state vector, solve the following equation: πP = π, where π is the steady-state vector, and P is the transition matrix. Additionally, ensure the elements of π sum to 1, representing the total probability, you can solve this system of linear equations using methods like Gaussian elimination, matrix inversion, or iterative techniques. In summary, to show that a transition matrix is regular, find a power of the matrix with all positive elements, then, to find the steady-state vector, solve the equation πP = π while ensuring the elements of π sum to 1.
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Johnny, of mass 65 kg, and Lucy, of mass 45 kg, are facing each other on roller blades. The coefficient of kinetic friction between the roller blades and concrete surface is 0.20. When Johnny pushes Lucy from rest he applies a force for 1.0 s. Lucy then slows down to a stop in another 8.0 s. Calculate:
a. The applied force exerted by Johnny on Lucy.
b. How long it takes Johnny to come to rest.
I tried calculated the force exerted but I would need acceleration which I don't have...any tips on how to solve this one??? help is appreciated!!
Answer:
John applied a force of approximately [tex]795\; {\rm N}[/tex] (on average, rounded) on Lucy.
John slows down to a stop after approximately another [tex]5.37\; {\rm s}[/tex].
(Assuming that [tex]g = 9.81\; {\rm N\cdot kg^{-1}}[/tex].)
Explanation:
Assuming that the surface is level. The normal force on Johnny will be equal to the weight of Johnny: [tex]N(\text{John}) = m(\text{John})\, g[/tex]. Similarly, the normal force on Lucy will be equal to weight [tex]N(\text{Lucy}) = m(\text{Lucy})\, g[/tex].
Multiply normal force by the coefficient of kinetic friction to find the friction on each person:
[tex]f(\text{John}) = \mu_{k}\, N(\text{John}) = \mu_{k}\, m(\text{John})\, g[/tex].
[tex]f(\text{Lucy}) = \mu_{k}\, N(\text{Lucy}) = \mu_{k}\, m(\text{Lucy})\, g[/tex].
Again, because the surface is level, the net force on each person after the first [tex]1.0\; {\rm s}[/tex] will be equal to the friction. Divide that the net force on each person by the mass of that person to find acceleration:
[tex]\displaystyle a(\text{John}) = \frac{\mu_{k}\, m(\text{John})\, g}{m(\text{John})} = \mu_{k}\, g[/tex].
[tex]\displaystyle a(\text{Lucy}) = \frac{\mu_{k}\, m(\text{Lucy})\, g}{m(\text{Lucy})} = \mu_{k}\, g[/tex].
(Note that the magnitude of acceleration is independent of mass and is the same for both John and Lucy.)
[tex]a = \mu_{k}\, g= (0.2)\, (9.81\; {\rm m\cdot s^{-2}}) = 1.962\; {\rm m\cdot s^{-2}}[/tex].
In other words, after the first [tex]1\; {\rm s}[/tex], both John and Lucy will slow down at a rate of [tex]1.962\; {\rm m\cdot s^{-2}}[/tex].
To find the speed of Lucy immediately after the first [tex]1.0\: {\rm s}[/tex], multiply this acceleration by the time [tex]t = 8.0\; {\rm s}[/tex] it took for Lucy to slow down to [tex]0\; {\rm m\cdot s^{-1}}[/tex]:
[tex]\begin{aligned}& (8.0\; {\rm s})\, (1.962\; {\rm m\cdot s^{-2}}) \\ =\; & (8.0)\, (1.962)\; {\rm m\cdot s^{-1}} \\ =\; & 15.696\; {\rm m\cdot s^{-1}}\end{aligned}[/tex].
Thus, in the first [tex]1.0\; {\rm s}[/tex], Lucy accelerated (from [tex]0\; {\rm m\cdot s^{-1}}[/tex]) to [tex]15.696\; {\rm m\cdot s^{-1}}[/tex].
The average acceleration of Lucy in the first [tex]1.0\; {\rm s}[/tex] would be [tex](15.696) / (1) = 15.696\; {\rm m\cdot s^{-2}}[/tex]. Multiply this average acceleration by the mass of Lucy to find the average net force on Lucy during that [tex]1.0\; {\rm s}[/tex]:
[tex]\begin{aligned}F_{\text{net}}(\text{Lucy}) &= m(\text{Lucy})\, a \\ &= (45)\, (15.696)\; {\rm N} \\ &= 706.320\; {\rm N}\end{aligned}[/tex].
This net force on Lucy during that [tex]1.0\; {\rm s}[/tex] is the combined result of both the push from Johnny and friction:
[tex]F_{\text{net}}(\text{Lucy}) = F(\text{push}) - f(\text{Lucy})[/tex].
Since [tex]f(\text{Lucy}) = \mu_{k}\, N(\text{Lucy}) = \mu_{k}\, m(\text{Lucy})\, g[/tex]:
[tex]\begin{aligned}F(\text{push}) &= F_{\text{net}}(\text{Lucy}) + f(\text{Lucy}) \\ &= F_{\text{net}}(\text{Lucy}) + \mu_{k}\, m(\text{Lucy})\, g \\ &= (706.320) \; {\rm N}+ (0.2)\, (45)\, (9.81)\; {\rm N} \\ &= 706.320\; {\rm N} + 88.290\; {\rm N} \\ &=794.610\; {\rm N}\end{aligned}[/tex].
In other words, Johnny would have applied a force of [tex]794.610\; {\rm N}[/tex] on Lucy.
By Newton's Laws of Motion, when Johnny exerts this force on Lucy in that [tex]1.0\; {\rm s}[/tex], Lucy would exert a reaction force on Johnny of the same magnitude: [tex]794.610\; {\rm N}[/tex].
Similar to Lucy, the net force on Johnny during that [tex]1.0\; {\rm s}[/tex] will be the combined effect of the push [tex]F(\text{push})[/tex] and friction [tex]f(\text{John}) = \mu_{k}\, m(\text{John})\, g[/tex]:
[tex]\begin{aligned}F_{\text{net}}(\text{John}) &= F(\text{push}) - f(\text{John}) \\ &= F(\text{push}) - \mu_{k}\, m(\text{John})\, g\\ &= 794.610\; {\rm N} - (0.2)\, (65)\, (9.81)\; {\rm N} \\ &= 667.080\; {\rm N}\end{aligned}[/tex].
Divide net force by mass to find acceleration:
[tex]\begin{aligned}\frac{667.080\; {\rm N}}{65\; {\rm kg}} \approx 10.2628\; {\rm m\cdot s^{-2}}\end{aligned}[/tex].
In other words, Johnny accelerated at a rate of approximately [tex]10.5406\; {\rm m\cdot s^{-2}}[/tex] during that [tex]1.0\; {\rm s}[/tex]. Assuming that Johnny was initially not moving, the velocity of Johnny right after that [tex]1.0\; {\rm s}\![/tex] would be:
[tex](0\; {\rm m\cdot s^{-1}}) + (10.2628\; {\rm m\cdot s^{-2}})\, (1.0\; {\rm s}) = 10.2628\; {\rm m\cdot s^{-1}}[/tex].
After the first [tex]1.0\; {\rm s}[/tex], the acceleration of both John and Lucy (as a result of friction) would both be equal to [tex]a = \mu_{k}\, g= (0.2)\, (9.81\; {\rm m\cdot s^{-2}}) = 1.962\; {\rm m\cdot s^{-2}}[/tex]. Divide initial velocity of Johnny by this acceleration to find the time it took for Johnny to slow down to a stop:
[tex]\displaystyle \frac{10.2628\; {\rm {m\cdot s^{-1}}}}{1.962\; {\rm m\cdot s^{-2}}} \approx 5.23\; {\rm s}[/tex].
Answer:
John applied a force of approximately [tex]795\; {\rm N}[/tex] (on average, rounded) on Lucy.
John slows down to a stop after approximately another [tex]5.37\; {\rm s}[/tex].
(Assuming that [tex]g = 9.81\; {\rm N\cdot kg^{-1}}[/tex].)
Explanation:
Assuming that the surface is level. The normal force on Johnny will be equal to the weight of Johnny: [tex]N(\text{John}) = m(\text{John})\, g[/tex]. Similarly, the normal force on Lucy will be equal to weight [tex]N(\text{Lucy}) = m(\text{Lucy})\, g[/tex].
Multiply normal force by the coefficient of kinetic friction to find the friction on each person:
[tex]f(\text{John}) = \mu_{k}\, N(\text{John}) = \mu_{k}\, m(\text{John})\, g[/tex].
[tex]f(\text{Lucy}) = \mu_{k}\, N(\text{Lucy}) = \mu_{k}\, m(\text{Lucy})\, g[/tex].
Again, because the surface is level, the net force on each person after the first [tex]1.0\; {\rm s}[/tex] will be equal to the friction. Divide that the net force on each person by the mass of that person to find acceleration:
[tex]\displaystyle a(\text{John}) = \frac{\mu_{k}\, m(\text{John})\, g}{m(\text{John})} = \mu_{k}\, g[/tex].
[tex]\displaystyle a(\text{Lucy}) = \frac{\mu_{k}\, m(\text{Lucy})\, g}{m(\text{Lucy})} = \mu_{k}\, g[/tex].
(Note that the magnitude of acceleration is independent of mass and is the same for both John and Lucy.)
[tex]a = \mu_{k}\, g= (0.2)\, (9.81\; {\rm m\cdot s^{-2}}) = 1.962\; {\rm m\cdot s^{-2}}[/tex].
In other words, after the first [tex]1\; {\rm s}[/tex], both John and Lucy will slow down at a rate of [tex]1.962\; {\rm m\cdot s^{-2}}[/tex].
To find the speed of Lucy immediately after the first [tex]1.0\: {\rm s}[/tex], multiply this acceleration by the time [tex]t = 8.0\; {\rm s}[/tex] it took for Lucy to slow down to [tex]0\; {\rm m\cdot s^{-1}}[/tex]:
[tex]\begin{aligned}& (8.0\; {\rm s})\, (1.962\; {\rm m\cdot s^{-2}}) \\ =\; & (8.0)\, (1.962)\; {\rm m\cdot s^{-1}} \\ =\; & 15.696\; {\rm m\cdot s^{-1}}\end{aligned}[/tex].
Thus, in the first [tex]1.0\; {\rm s}[/tex], Lucy accelerated (from [tex]0\; {\rm m\cdot s^{-1}}[/tex]) to [tex]15.696\; {\rm m\cdot s^{-1}}[/tex].
The average acceleration of Lucy in the first [tex]1.0\; {\rm s}[/tex] would be [tex](15.696) / (1) = 15.696\; {\rm m\cdot s^{-2}}[/tex]. Multiply this average acceleration by the mass of Lucy to find the average net force on Lucy during that [tex]1.0\; {\rm s}[/tex]:
[tex]\begin{aligned}F_{\text{net}}(\text{Lucy}) &= m(\text{Lucy})\, a \\ &= (45)\, (15.696)\; {\rm N} \\ &= 706.320\; {\rm N}\end{aligned}[/tex].
This net force on Lucy during that [tex]1.0\; {\rm s}[/tex] is the combined result of both the push from Johnny and friction:
[tex]F_{\text{net}}(\text{Lucy}) = F(\text{push}) - f(\text{Lucy})[/tex].
Since [tex]f(\text{Lucy}) = \mu_{k}\, N(\text{Lucy}) = \mu_{k}\, m(\text{Lucy})\, g[/tex]:
[tex]\begin{aligned}F(\text{push}) &= F_{\text{net}}(\text{Lucy}) + f(\text{Lucy}) \\ &= F_{\text{net}}(\text{Lucy}) + \mu_{k}\, m(\text{Lucy})\, g \\ &= (706.320) \; {\rm N}+ (0.2)\, (45)\, (9.81)\; {\rm N} \\ &= 706.320\; {\rm N} + 88.290\; {\rm N} \\ &=794.610\; {\rm N}\end{aligned}[/tex].
In other words, Johnny would have applied a force of [tex]794.610\; {\rm N}[/tex] on Lucy.
By Newton's Laws of Motion, when Johnny exerts this force on Lucy in that [tex]1.0\; {\rm s}[/tex], Lucy would exert a reaction force on Johnny of the same magnitude: [tex]794.610\; {\rm N}[/tex].
Similar to Lucy, the net force on Johnny during that [tex]1.0\; {\rm s}[/tex] will be the combined effect of the push [tex]F(\text{push})[/tex] and friction [tex]f(\text{John}) = \mu_{k}\, m(\text{John})\, g[/tex]:
[tex]\begin{aligned}F_{\text{net}}(\text{John}) &= F(\text{push}) - f(\text{John}) \\ &= F(\text{push}) - \mu_{k}\, m(\text{John})\, g\\ &= 794.610\; {\rm N} - (0.2)\, (65)\, (9.81)\; {\rm N} \\ &= 667.080\; {\rm N}\end{aligned}[/tex].
Divide net force by mass to find acceleration:
[tex]\begin{aligned}\frac{667.080\; {\rm N}}{65\; {\rm kg}} \approx 10.2628\; {\rm m\cdot s^{-2}}\end{aligned}[/tex].
In other words, Johnny accelerated at a rate of approximately [tex]10.5406\; {\rm m\cdot s^{-2}}[/tex] during that [tex]1.0\; {\rm s}[/tex]. Assuming that Johnny was initially not moving, the velocity of Johnny right after that [tex]1.0\; {\rm s}\![/tex] would be:
[tex](0\; {\rm m\cdot s^{-1}}) + (10.2628\; {\rm m\cdot s^{-2}})\, (1.0\; {\rm s}) = 10.2628\; {\rm m\cdot s^{-1}}[/tex].
After the first [tex]1.0\; {\rm s}[/tex], the acceleration of both John and Lucy (as a result of friction) would both be equal to [tex]a = \mu_{k}\, g= (0.2)\, (9.81\; {\rm m\cdot s^{-2}}) = 1.962\; {\rm m\cdot s^{-2}}[/tex]. Divide initial velocity of Johnny by this acceleration to find the time it took for Johnny to slow down to a stop:
[tex]\displaystyle \frac{10.2628\; {\rm {m\cdot s^{-1}}}}{1.962\; {\rm m\cdot s^{-2}}} \approx 5.23\; {\rm s}[/tex].
A student wants to determine the angular speed w of a rotating object. The period T is 0.50 s +/- 5%. The angular speed ω is
ω = 2π/T
What is the percentage uncertainly of ω? A. 0.2%
B. 2.5% C. 5% D. 10%
The percentage uncertainty of the angular speed= 5%
To determine the percentage uncertainty of the angular speed ω, we can use the following formula:
Percentage Uncertainty of ω = (Percentage Uncertainty of T) * (Constant Value)
In this case, the percentage uncertainty of T is 5%, and the constant value is 2π (from the formula ω = 2π/T). Therefore:
Percentage Uncertainty of ω = (5%) * (2π)
However, since we're only interested in the percentage uncertainty, we don't need to multiply by the constant value (2π). So, the percentage uncertainty of ω is the same as the percentage uncertainty of T:
Percentage Uncertainty of ω = 5%
So, the correct answer is C i.e.5%
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how would you shape a given length of wire to give it the greatest self-inductance? the least?
A straight wire has much lower inductance than a coiled wire, as the magnetic fields generated by the current are less likely to interact with each other and create self-inductance.
To shape a given length of wire to give it the greatest self-inductance, you would want to create a tightly wound coil, such as a solenoid.
The self-inductance of a solenoid is proportional to the square of the number of turns, so the more turns you can create with the given length of wire, the greater the inductance will be.
Additionally, keeping the turns close together will help maximize the inductance.
On the other hand, to achieve the least self-inductance, you would want to keep the wire as straight as possible
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A straight wire has much lower inductance than a coiled wire, as the magnetic fields generated by the current are less likely to interact with each other and create self-inductance.
To shape a given length of wire to give it the greatest self-inductance, you would want to create a tightly wound coil, such as a solenoid.
The self-inductance of a solenoid is proportional to the square of the number of turns, so the more turns you can create with the given length of wire, the greater the inductance will be.
Additionally, keeping the turns close together will help maximize the inductance.
On the other hand, to achieve the least self-inductance, you would want to keep the wire as straight as possible
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A 1.60 m tall person lifts a 1.75 kg book off the ground so it is 2.00 m above the ground.What is the potential energy of the book relative to the ground?What is the potential energy of the book relative to the top of the person's head?How is the work done by the person related to the answers in parts A and B?W=Uground−UheadW=UgroundW=Uground+UheadW=Uhead−UgroundW=Uhead
The potential energy is PE_head ≈ 6.86 J (Joules) and the work done by the person is approximately 27.43 Joules.
To find the potential energy of the book relative to the ground, we can use the formula for gravitational potential energy,
PE = m * g * h
where PE is potential energy, m is the mass of the book (1.75 kg), g is the acceleration due to gravity (9.81 m/s^2), and h is the height above the ground (2.00 m).
PE_ground = 1.75 kg * 9.81 m/s^2 * 2.00 m
PE_ground ≈ 34.29 J (Joules)
To find the potential energy of the book relative to the top of the person's head, we need to determine the height above the person's head,
height_above_head = 2.00 m - 1.60 m = 0.40 m
PE_head = 1.75 kg * 9.81 m/s^2 * 0.40 m
PE_head ≈ 6.86 J (Joules)
To relate the work done by the person to the potential energies, we can use the following equation,
W = PE_ground - PE_head
where W is the work done by the person.
W = 34.29 J - 6.86 J
W ≈ 27.43 J
The work done by the person is approximately 27.43 Joules.
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A typical power for a laser used in physics labs is 0.75 mW. This laser would produce a beam that is about 1 mm in diameter I found the average intensity: to be 955 W/m2 I found the average energy density of this beam: 3.183*10^-6 J/m^3 I need help with this question. Lets say the laser beam is reflected completely off a mirror. What is the maximum force the beam can exert on the mirror? I did 955 x 2x (4pie x 10^-7)/(3*10^8) but says my answer is wrong, is this the right equation???
The maximum force of a typical power for a laser used in physics labs is 0.75 mW and would produce a beam that is about 1 mm in diameter can exert on the mirror is 5 x 10⁻⁹ N.
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. Power is a scalar quantity.
To find the maximum force the laser beam can exert on the mirror when it is reflected completely, you should use the following formula:
Force (F) = (2 × Power (P)) / Speed of light (c)
where Power (P) = 0.75 mW (convert to Watts: 0.75 x 10⁻³ W), and Speed of light (c) = 3 x 10⁸ m/s.
F = (2 × (0.75 x 10⁻³ W)) / (3 x 10⁸ m/s)
Thus, the maximum force of the beam can exert on the mirror is approximately 5 x 10⁻⁹ N.
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Which of the following is not a property of light?
a. light is a form of matter less dense than air
b. light travels in straight lines
c. light has different colors
d. light has different intensities, and cam be bright or dim
Answer: Light is not a form of matter, so a is not a property of light.
Explanation:
for a particular process at 300 k, δg δ g = -10.0 kj and δh δ h = -7.0 kj. if the process is carried out reversibly, the amount of useful work (in kj) that .
The amount of useful work that can be obtained for this particular process at 300 K is -3.0 kJ.
For a reversible process, the amount of useful work (in kJ) that can be obtained is given by the equation:
w = -δg = δh - Tδs
where δs is the change in entropy of the system.
Since the process is carried out reversibly, δs can be calculated using the equation:
δs = δqrev / T
where δqrev is the heat absorbed by the system during a reversible process.
Since δh = -7.0 kJ and δg = -10.0 kJ, we know that the process is exothermic (δh < 0) and spontaneous (δg < 0). Therefore, δqrev must be negative, indicating that heat is released from the system.
We can calculate δqrev using the equation:
δqrev = -Tδs = -T(δh / T) = -δh = 7.0 kJ
Substituting this value into the equation for work, we get:
w = -δg = δh - Tδs = -10.0 kJ - (-7.0 kJ) = -3.0 kJ
Therefore, the amount of useful work for this particular process at 300 K is -10.0 kJ.
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a 65-kg skydiver jumps out of an airplane and falls 310 m, reaching a maximum speed of 53 m/s before opening her parachute.
The equation for the final velocity of an object undergoing [tex]t ≈ 5.41 s[/tex]
Why will be reaching a maximum speed of 53 m/s?
We can use the equations of motion to solve this problem. The key is to break the problem into two parts: the freefall part before the parachute is opened, and the part where the parachute is used to slow down the skydiver.
First, we can use the equation for the final velocity of an object undergoing constant acceleration to find the time it takes for the skydiver to reach her maximum speed:
[tex]v_f = v_i + at[/tex]
where v_f is the final velocity, v_i is the initial velocity (which is 0 in this case), a is the acceleration (which is due to gravity, -9.8 m/s^2), and t is the time.
Rearranging the equation to solve for t, we get:
[tex]t = v_f / a[/tex]
Substituting the values given, we get:
[tex]t = 53 m/s / 9.8 m/s^2[/tex]
Simplifying, we get:
[tex]t ≈ 5.41 s[/tex]
This tells us that it takes 5.41 seconds for the skydiver to reach her maximum speed of 53 m/s.
Next, we can use the equation for the distance traveled by an object undergoing constant acceleration to find the distance the skydiver falls during this time:
[tex]d = v_i t + (1/2)[/tex][tex]at^2[/tex]
where d is the distance, v_i is the initial velocity (which is 0 in this case), a is the acceleration (which is due to gravity, -9.8 m/s^2), and t is the time.
Substituting the values given, we get:
[tex]d = 0 + (1/2)(-9.8 m/s^2)(5.41 s)^2[/tex]
Simplifying, we get:
d ≈ 147.9 m
This tells us that the skydiver falls about 147.9 meters during the freefall part of the jump.
Now we can calculate the velocity of the skydiver just before opening her parachute using the equation:
[tex]v_f^2 = v_i^2 + 2ad[/tex]
where v_f is the final velocity (which is 0 in this case), v_i is the initial velocity (which is 53 m/s), a is the acceleration (which is due to gravity, -9.8 m/s^2), and d is the distance traveled during the deceleration phase (which is 310 m - 147.9 m = 162.1 m).
Substituting the values given, we get:
[tex]0 = (53 m/s)^2 + 2(-9.8 m/s^2)(162.1 m - 147.9 m)[/tex]
Simplifying, we get:
[tex]0 = 2809 - 6176[/tex]
This is not possible, since it means that the final velocity is imaginary. This suggests that we made an error in our calculations. Checking our work, we find that the error is in the equation we used to calculate the distance traveled during the freefall part of the jump. We used the wrong value for the acceleration due to gravity - it should be positive, not negative:
[tex]d = v_i t + (1/2)at^2[/tex]
Substituting the correct value for a, we get:
[tex]d = 0 + (1/2)(9.8 m/s^2)(5.41 s)^2[/tex]
Simplifying, we get:
[tex]d ≈ 147.9 m[/tex]
This is the same value we found earlier, but with the correct sign for the acceleration. Now we can redo our calculations for the deceleration phase:
[tex]v_f^2 = v_i^2 + 2ad[/tex]
Substituting the values given, we
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if a ball with a weight of 30 n hangs from the end of a 1.5-m horizontal pole, what torque is produced?
The torque produced by a weight of 30 N at the end of a 1.5 m horizontal pole is 45 Nm.
To calculate the torque produced, you can use the following formula:
Torque (τ) = Force (F) × Distance (d)
In this case, the weight of the ball (30 N) is the force, and the length of the horizontal pole (1.5 m) is the distance.
1: Identify the force and distance.
Force (F) = 30 N
Distance (d) = 1.5 m
Step 2: Use the formula to calculate the torque.
Torque (τ) = Force (F) × Distance (d)
Step 3: Plug in the values and calculate the torque.
τ = 30 N × 1.5 m
Step 4: Calculate the result.
τ = 45 Nm
The torque produced when a 30 N ball hangs from the end of a 1.5-m horizontal pole is 45 Nm.
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A capacitor has parallel plates of area 12 cm2 separated by 6.0mm . The space between the plates is filled with polystyrene (K=2.6, Em=2.0x107V/m).Find the permittivity of polystyrene. Express your answer using two significant figures
A capacitor has parallel plates of area 12 cm2 separated by 6.0mm. The space between the plates is filled with polystyrene the permittivity of polystyrene is 3.0x10^-11 F/m.
The permittivity of polystyrene can be found using the formula:
ε = (C/d) / ε0A
Where C is the capacitance of the capacitor, d is the distance between the plates, A is the area of the plates, and ε0 is the permittivity of free space.
First, we need to find the capacitance of the capacitor:
C = ε0KA/d
Substituting the given values:
C = (8.85x10^-12 F/m)(2.6)(0.0012 m^2) / 0.006 m
C = 3.06x10^-11 F
Now we can substitute this value along with the other given values into the formula for permittivity:
ε = (C/d) / ε0A
ε = (3.06x10^-11 F) / (0.006 m)(8.85x10^-12 F/m) / (0.00012 m^2)
ε = 2.96x10^-11 F/m
Rounding to two significant figures, the permittivity of polystyrene is 3.0x10^-11 F/m.
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