What causes the nucleus of an isotope to be radioactive

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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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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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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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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(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.

(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

(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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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 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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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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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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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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Answer: Light is not a form of matter, so a is not a property of light.

Explanation:

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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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

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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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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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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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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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 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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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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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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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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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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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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To calculate the work done by the force when the car moves from x=0.0m to x=7.0m, we need to integrate the force over the distance traveled.W = ∫Fdx



Since the force varies with the x-coordinate of the car, we need to know the equation for the force as a function of x. Without that information, we can't calculate the work done.

To calculate the speed of the car at x=4.0m, we need to use the equations of motion. Assuming that the force is the only external force acting on the car, we can use:

F = ma

where F is the force, m is the mass of the car (2.0 kg), and a is the acceleration of the car.

Since the force varies with x, we need to know the equation for the force as a function of x. Without that information, we can't calculate the acceleration of the car, and therefore we can't calculate the speed at x=4.0m.

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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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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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(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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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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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].