a train with a monochromatic headlight is approaching you at a speed of 1 18 c. if a student on the train measures the wavelength of the light to be 590 nm, what do you measure the wavelength (in nm) to be? nm

The question given is a 300 N/C uniform electric field points perpendicularly toward the left face of a large neutral conducting sheet.

The formula of the electric field. [tex]E=\frac{F}{q}[/tex]

The formula of area charge density.[tex]$$\ \sigma = \frac {q} {A} $$[/tex]

where E is the electric field.

F is the force of the electric charge.

q is the charge.

σ is the area charge density.

A is the area.

The electric field is given as E=300 N/C.

As the area is neutral and conductive, thus, there is no net charge and so σ = 0. A neutral conductor sheet doesn't have a charge on its face. Therefore the area charge density on the left and right faces is zero.

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Answer: Through which material will magnetic lines of force pass the most readily? ANSWER: iron

A), Multiply the length of the vibrating string (60.0 cm) by the ratio to find the distance x. B)No, it's impossible to play a G4 note (392 Hz) on this string without retuning, C) not possible without retuning.

Part A) To find the distance x from the bridge to play a D5 note (587 Hz), follow these steps:
1. Calculate the speed of the wave on the string using the formula: v = √(T/μ), where T is tension and μ is linear mass density.
2. Calculate the wavelength of the A4 note using the formula: λ = v/f, where f is the frequency of the A4 note (440 Hz).
3. Calculate the wavelength of the D5 note using the formula: λ = v/f, where f is the frequency of the D5 note (587 Hz).
4. Find the ratio between the A4 and D5 wavelengths: λ_A4 / λ_D5.
5. Multiply the length of the vibrating string (60.0 cm) by the ratio to find the distance x.
Part B) No, it's impossible to play a G4 note (392 Hz) on this string without retuning.
Part C) The reason why it's impossible to play a G4 note (392 Hz) without retuning is because the frequencies of the fundamental modes are fixed and cannot be changed unless the tension, mass, or length of the string is altered. To play a G4 note, the string would need to be adjusted so that its fundamental frequency is 392 Hz, which is not possible without retuning.

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If the frequency of the incoming light is decreased, the energy of the ejected electrons will decrease.

The frequency of the incoming light will affect the energy of the ejected electrons. This is because the energy of the ejected electrons is proportional to the frequency of the incoming light.

The energy of the electrons can be determined using the equation:

E = h * f,

where E is the energy, h is Planck’s constant, and f is the frequency of the incoming light. This equation shows that the energy of the electrons is directly proportional to the frequency of the incoming light.

Therefore, if the frequency of the incoming light is decreased, the energy of the ejected electrons will also decrease.

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Answer:

Explanation:

Durante as aulas, os estudantes da 3ª série deveriam escolher uma entre as três atividades físicas possíveis, sendo elas: natação, futsal e dança. Na turma, 25% escolheram dança, 15% escolheram natação, e os outros 24 estudantes escolheram futsal. Podemos afirmar que, nessa turma, existe um total de:

A) 64 alunos

B) 55 alunos

C) 48 alunos

D) 45 alunos

E) 40 alunos

Answer: B

Explanation:

The y component of the vector is given by the formula:

y component = magnitude of the vector x sin(angle between the vector and the y-axis)

In this case, the magnitude of the vector is 120 N and the angle between the vector and the y-axis is 50 degrees. So we have:

y component = 120 N x sin(50°) ≈ 92 N

Therefore, the answer is (B) 92 N.

The phases of Venus show that the solar system is in a heliocentric model instead of a geocentric model because the heliocentric model states that the Sun is at the center of the solar system, while the geocentric model states that Earth is at the center of the universe.

The phases of Venus can only be explained in the heliocentric model because the planet is orbiting the Sun.The phases of Venus are an important piece of evidence supporting the heliocentric model proposed by Nicolaus Copernicus. The geocentric model was the widely accepted model of the universe until the 16th century when Copernicus proposed the heliocentric model, which suggested that the Sun is at the center of the solar system and the Earth and other planets orbit around it.

The phases of Venus show that it orbits the Sun and not the Earth because, as it orbits the Sun, different portions of the planet's sunlit side are visible from Earth. This can only occur in a heliocentric model because Venus is between the Earth and the Sun in its orbit, which causes it to pass through phases. Therefore, the phases of Venus are not consistent with a geocentric model, which suggests that Venus orbits the Earth.

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To find the average angular velocity during the dive, we need to first calculate the time it takes for the diver to reach the water from the 10-meter high platform. Assuming zero initial vertical velocity and using the free fall equation:

h = 1/2 * g * t^2

where h = 10 meters, g = 9.81 m/s^2 (acceleration due to gravity), and t is the time in seconds.

Rearranging the equation to find t:

t^2 = 2 * h / g
t^2 = 2 * 10 / 9.81
t^2 ≈ 2.04
t ≈ √2.04 ≈ 1.43 seconds

Now that we have the time, we can calculate the average angular velocity (ω) using the formula:

ω = θ / t

where θ is the total angle in radians the diver rotates during the dive, and t is the time in seconds. The diver makes 2.5 revolutions, which is equal to 2.5 * 2π radians:

θ = 2.5 * 2π ≈ 15.71 radians

Now, we can find the average angular velocity:

ω = 15.71 radians / 1.43 seconds ≈ 10.99 radians/second

So, the average angular velocity during the dive is approximately 10.99 radians/second.

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The symbolic expression that gives the centripetal acceleration on the edge of the platform as a function of time, ac(t) is: `ac(t) = -rω²sin(ωt)`

The centripetal acceleration of an object moving in a circular path is always directed toward the center of the circle. The value of centripetal acceleration can be calculated by the formula:`ac = (v²) / r`

Here, v represents the linear velocity of the object and r is the radius of the circular path. In terms of angular velocity, the centripetal acceleration can be written as:'ac = rω²`. Therefore, the centripetal acceleration on the edge of the platform can be written as:`ac(t) = rω²sin(ωt)`

Here, ω represents the angular velocity of the platform. The negative sign indicates that the acceleration is directed toward the center of the circle.

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It would take approximately 19.2 years for the asteroid to orbit once around the sun. But that none of the answer choices match the calculated value of approximately 19.2 years.

The period (T) of an orbit of a celestial body with semimajor axis (a) around the sun can be calculated using Kepler's third law:

T² = (4π² / GM) * a³

where G is the gravitational constant and M is the mass of the sun.

Plugging in the given value for the semimajor axis (a = 4 AU), we get:

T² = (4π² / (6.674 × 10⁻¹¹ m³/(kg s²) * 1.989 × 10³⁰ kg)) * (4 AU)³

T² = 3.652 × 10¹⁶ s²

Taking the square root of both sides, we get:

T = 6.04 × 10⁸ s

We can convert this time to years by dividing by the number of seconds in a year:

T = (6.04 × 10⁸ s) / (31,536,000 s/year)

T ≈ 19.2 years

Therefore, it would take approximately 19.2 years for the asteroid to orbit once around the sun. The closest answer choice is 16 years.

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The magnitude of the upward acceleration of a load of bricks is 2.77 m/s².

Tension is the pulling force cаrried by flexible mediums like ropes, cаbles аnd string. Tension in а body due to the weight of the hаnging body is the net force аcting on the body.

The tension in the string when the body cаn be given аs,

T = m(a +g)

Here, (m) is the mаss of the body, (а) is the аccelerаtion аnd (g) is the аccelerаtion due to grаvity.

The tension due to the bricks with mаss of 15.2 kg is,

T = 15.2(a + 9.80)

The tension due to the bricks with mass of 27.2 kg is,

T = 27.2(9.80 - a)

Equate both the equation as,

15.2(a + 9.80) = 27.2(9.80 - a)

15.2a + 148.96 = 266.56 - 27.2a

42.4a = 117.6

a = 2.77 m/s²

Thus, the magnitude of the upward acceleration of a load of bricks is 2.77 m/s².

Your question is incomplete, but most probably your full question can be seen in the Attachment.

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A proton that has been accelerated from rest through a potential difference of -800 V has a speed of: 2.60 x 10⁶ m/s.

When a particle is accelerated through a potential difference, its potential energy is transformed into kinetic energy, resulting in an increase in velocity.

To calculate the velocity of a proton that has been accelerated through a potential difference of -800 V, we may use the equation: v = √(2qV/m)

where: v is the speed of the proton,

q is the charge of the proton (1.6 x 10⁻¹⁹ C),

V is the potential difference (-800 V)

m is the mass of the proton (1.67 x 10⁻²⁷ kg)

Using these values, we may calculate the velocity:

v = √(2(1.6 x 10⁻¹⁹C)(-800 V)/(1.67 x 10⁻²⁷kg))= 2.60 x 10⁶ m/s

Therefore, the velocity of a proton that has been accelerated from rest through a potential difference of -800 V is 2.60 x 10⁶ m/s.

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The amount of charge transferred is 4.5C (coulombs). This is calculated by dividing 45 J (joules) of work by 9.0 V (volts) of voltage.

We can use the equation W = qV, where W is the work done, q is the charge transferred, and V is the potential difference (voltage) across the battery.

Rearranging the equation to solve for q, we get q = W/V.

Plugging in the values given, we have:

q = 45 J / 9.0 V

q = 5.0 C

Therefore, the battery transfers 5.0 coulombs of charge from the negative to the positive terminal while doing 45 J of work.

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The given coefficient of kinetic friction is 0.55. Assuming that the acceleration is constant, so the wreck skids a distance of 0 meters.

The distance that the wreck skids while coming to a stop is calculated below.

Data Coefficient of kinetic friction = 0.55

Conversion of acceleration to m/s²0.55 coefficient of kinetic friction can be written as 0.55 times acceleration to calculate the distance that the wreck skids. We know that the acceleration due to gravity is 9.8 m/s². Hence the acceleration due to gravity can be written as follows.

a = 9.8 m/s² × 0.55a

= 5.39 m/s²

Calculation of the distance that the wreck skids is calculated by using the formula below:

d = (v² - u²)/2as = distance = initial velocity = final velocity a = acceleration

The wreck is coming to stop, so the final velocity is 0. Hence the formula can be written as:

d = (v² - u²)/2a

= (0 - u²)/2×5.39d

= -u²/10.78d

= -0.093u²

Calculation of velocity can be calculated by using the following formula below.

v² = u² + 2asv²

= u² - 2u²/10.78v²

= (8.78u²)/10.78v²

= (2u²)/2.45v

= (u²)/1.56

The final velocity is zero. Hence we can write the formula as :

0 = (u²)/1.56u² = 0

The initial velocity of the wreck is zero. Hence the wreck is moving from rest condition.

Calculation of the distance that the wreck skids is calculated by using the formula below:

d = -u²/10.78d

= 0 meters.

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When calculating the force exerted by the triceps, ignoring the mass significantly affects the torque value.

The torque is the product of the force and the distance from the force application point to the axis of rotation.

Torque= force*distance (N m)

The torque calculation for a muscle depends on the point of attachment of the muscle. Muscle mass is related to its force production capacity, and it is necessary to consider it when calculating the force applied by the triceps.

However, the force exerted by the triceps muscle would be affected by the mass of the object being lifted or moved. The force required to move an object increases with the mass of the object. Therefore, ignoring the mass of the object would result in an underestimate of the force required to move the object, and thus an underestimate of the force exerted by the triceps.

In summary, ignoring the mass of the object being lifted or moved would not significantly affect the calculated value of torque, but it would affect the calculated value of force.

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The electric motor's efficiency is 51.06%.

The majority of electric motors are made to operate between 50% and 100% of rated load. Typically, maximum efficiency is within 75% of rated load. Hence, the allowable load range for a 10-horsepower (hp) motor is between 5 and 10 hp; its peak efficiency is at 7.5 hp. Below roughly 50% load, a motor's efficiency tends to decline significantly.

To calculate the effort required to raise the object, use the formula:

Work = Force x Distance

= m x g x h (where m is the mass of the object, g is the acceleration due to gravity, and h is the height lifted)

= 34 kg x 9.81 m/s² x 11 m

= 3,769.34 J

The energy consumed by the electric motor is given as 7.4 kJ.

Therefore, the input power is:

Input power = Energy consumed / time taken

= 7.4 kJ / t

Efficiency=(Output power/Input power) x 100%

Output power = Work done/time taken

= 3,769.34 J / t

As a result, the electric motor's efficiency is:

Efficiency=(Output power/Input power)x 100%

= [(3,769.34 J / t) / (7.4 kJ / t)] x 100%

= 51.06%.

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Filling the space between a capacitor's plates with a dielectric material will increase the capacitance of the capacitor.

1. A dielectric material is inserted between the plates of the capacitor.

This material has the property of reducing the electric field between the plates.
2. The relationship between electric field (E) and voltage (V) is given by E = V/d, where d is the distance between the plates.

Since the electric field is reduced by the presence of the dielectric material, the voltage (potential difference) between the plates also decreases.

3. The relationship between capacitance (C), voltage (V), and charge (Q) is given by C = Q/V.

As the voltage decreases due to the presence of the dielectric material, the capacitance increases for a given charge on the plates.

4. The dielectric material has a property called dielectric constant (K), which is a measure of how effectively it reduces the electric field between the plates.

The capacitance of the capacitor with the dielectric material is given by C = K * C0,

where C0 is the capacitance without the dielectric material.

Since K is always greater than 1 for dielectric materials, the capacitance with the dielectric material is always higher than without it.
In conclusion, filling the empty (vacuum) space between a capacitor's plates with dielectric material will increase the capacitance of the capacitor.

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Answer:

320 km/hr = 320000 / 3600 = 88.9 m/s

(1 - v^2  / c^2) = (1 - 88.9^2 / 100^2)^1/2 = .46

Since L = L0 (1 - v^2  / c^2)^1/2

L = .46 L0 = 20.7 m

The airplane will appear to be only 20.64 m long to an observer at rest in this universe, even though its actual length is 45 m when at rest.

In this universe, the speed of light is only 100 m/s, which is much slower than in our universe, where the speed of light is approximately [tex]3 \times 10^8 m/s[/tex]. This means that the effects of special relativity will be much more noticeable in this universe.

We can use the formula for length contraction to calculate the apparent length of the airplane as seen by an observer at rest:

[tex]L' = L / \gamma[/tex]

where L is the length of the airplane at rest, L' is the apparent length of the airplane as seen by the observer, and γ is the Lorentz factor given by:

[tex]\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}[/tex]

where v is the speed of the airplane relative to the observer, and c is the speed of light in the given universe.

Converting the airplane's speed from km/h to m/s, we have:

[tex]v = (320 \ km/h) \times (1000 \ m/km) / (3600 \ s/h) = 88.89 \ m/s[/tex]

Substituting this value and c = 100 m/s into the expression for γ, we get:

[tex]\gamma = \frac{1}{\sqrt{1 - (88.89 m/s)^2 / (100 m/s)^2}} = 2.18[/tex]

Substituting this value of γ and L = 45 m into the expression for L', we get:

[tex]L' = L / \gamma = 45 \ m / 2.18 = 20.64 \ m[/tex]

Therefore, the length of the plane will appear to be 20.64 m. This significant length contraction is due to the low speed of light in this universe.

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The maximum torque produced on a current-carrying loop of wire by a magnetic field occurs when the plane of the loop is perpendicular to the direction of the magnetic field.

This can be explained using the formula for the torque on a current-carrying loop in a magnetic field,

τ = N * A * B * sin(θ)

where τ is the torque, N is the number of turns in the loop, A is the area of the loop, B is the magnetic field strength, and θ is the angle between the normal to the plane of the loop and the direction of the magnetic field. If the loop is parallel to the magnetic field, then θ = 0, and the sin(θ) term in the formula is zero. Therefore, there is no torque produced on the loop.

On the other hand, if the loop is perpendicular to the magnetic field, then θ = 90°, and the sin(θ) term in the formula is maximum, which results in the maximum torque on the loop. Therefore, to obtain the maximum torque on the loop, the plane of the loop should be perpendicular to the direction of the magnetic field.

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The length of runway needed for a light plane to take off with a constant acceleration of 3.0 m/s2 and a speed of 35 m/s is 203.7 meters.

The time taken for a light plane to reach 35 m/s from 0 m/s is given by using the formula,

v = u + at

where, v = final velocity, u = initial velocity, a = acceleration, and t = time

Here, the initial velocity of the plane is u = 0 m/s.

The final velocity of the plane is v = 35 m/s.

The acceleration of the plane is a = 3.0 m/s².

The time taken to reach the final velocity can be calculated as,

35 = 0 + (3.0)t

t = 35 / 3.0

t = 11.67 s

Therefore, the plane takes 11.67 s to reach a speed of 35 m/s for takeoff.

The distance traveled by plane during this time can be calculated as,

s = ut + 1/2 at²

s = 0 x 11.67 + 1/2 x 3.0 x (11.67)²

s = 203.7 m

Therefore, the runway should be at least 203.7 meters long if the acceleration is constant at 3.0 m/s².

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The distance traveled by the object moving with an acceleration of 6.8 m/s² is 42.35 m.

Distance is the length between two points.

To calculate the distance traveled by the object, we use the formula below.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1 and solve for s

Hence, the distance traveled by the object is 42.35 m.

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According to the laws of proportionality, if a resistor in a parallel circuit has triple the resistance of a second resistor, it will have 1/3 the current of the second resistor.

A parallel circuit is an electrical circuit in which the various components are linked together in such a way that the current can pass through multiple branches. All of the elements in a parallel circuit are linked in parallel to one another. This implies that the voltage across each element is the same, but the current through each element is different because of the different resistance values.

Total current is determined using the equation I = V/R, where V is voltage and R is resistance. The total resistance in a parallel circuit is determined using the equation 1/Rt = 1/R1 + 1/R2 + 1/R3 +…1/Rt = 1/R1 + 1/R2 + 1/R3 +… 1/Rt = 1/R1 + 1/R2 + 1/R3 +… The total resistance of a parallel circuit is always less than the resistance of the smallest resistor. According to the laws of proportionality, if a resistor in a parallel circuit has triple the resistance of a second resistor, it will have 1/3 the current of the second resistor.

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When we look into a mirror, a process called reflection occurs with the light.

What happens to the light:
1. Light source: The process begins with a light source, such as the sun or a light bulb, emitting light waves in all directions.
2. Light traveling: The light waves travel through the air and reach the mirror.
3. Mirror's surface: Mirrors have a smooth, reflective surface made of glass with a thin layer of metal, usually aluminum or silver, on the back.
4. Incident light: The light waves that strike the mirror's surface are called incident light.
5. Reflection: The mirror's smooth surface causes the incident light waves to bounce off, or reflect, at the same angle at which they arrived.

This is known as the law of reflection, which states that the angle of incidence is equal to the angle of reflection.

6. Reflected light: The light waves that bounce off the mirror are called reflected light.
7. Image formation: As the reflected light waves travel away from the mirror, they converge at a point and form an image of the object you see in the mirror.
8. Observing the image: Your eyes detect the reflected light waves, and your brain processes this information to create the perception of the image you see in the mirror.
In summary, when you look into a mirror, the light emitted from a source travels towards the mirror, strikes its reflective surface, and bounces off at the same angle, following the law of reflection.

The reflected light then forms an image, which you observe as the reflection in the mirror.

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Answer:

A

Explanation:

First, we can find the volume of the stone:

[tex]v(stone) = \frac{m(stone)}{ρ(stone)} = \frac{390}{2.7} ≈144.44 \: {cm}^{3} [/tex]

[tex]v(oil) = v(stone) = 144.44 \: {cm}^{3} [/tex]

[tex]m(oil) = v \times ρ(oil) = 144.44 \times 0.90≈130 \: g[/tex]

The car's propulsion will decrease when the static friction is reduced, as if there is an icy/slippery road. Static friction is the force that resists the movement of two surfaces in contact. When the static friction is reduced, the available friction for the car's propulsion is decreased, and the car's propulsion will be affected.

In order to understand this better, we need to understand friction. Friction is a force which acts in a direction opposite to the direction of motion and resists any change in the state of motion. Static friction is a force which acts in the direction opposite to the direction of motion and opposes any change in the state of rest of the body.

When there is a decrease in the static friction, the car will not be able to accelerate as quickly as it would on a normal road. This is because the static friction is the force which helps to accelerate the car and push it forward. On an icy/slippery road, the friction between the car and the ground will be greatly reduced and the car will be unable to move as quickly as it would on a normal road.

To summarize, the car's propulsion will decrease when the static friction is reduced, as if there is an icy/slippery road. This is because static friction is the force which helps to accelerate the car and push it forward and when there is a decrease in the static friction, the car will not be able to accelerate as quickly as it would on a normal road.

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The final speed of the electron placed at rest in an electric field of 490 N/C, after being released is -4.558 mega m/s.

Electric field = E = 490 N/C

The force acting on an electron in the electric field is:

F = qE, where q is the charge of the electron and E is the electric field strength.

q = -1.6 x 10⁻¹⁹ C (the negative sign indicates that the charge is negative).

F = qE = (-1.6 x 10⁻¹⁹ C) (490 N/C) = -7.84 x 10⁻¹⁷N.

The acceleration of the electron due to the electric field:

a = F/m = (-7.84 x 10⁻¹⁷N)/(9.11 x 10⁻³¹kg) = -8.6 x 10¹³ m/s².

According to the third law of motion, for every action, there is an equal and opposite reaction. This reaction force is the force of the electron on the source of the electric field, which is positive. Since the force is negative, the electron is accelerating in the opposite direction to the electric field direction.

The velocity can be found from the equation of motion, v = u + at

v = 0 + (-8.6 x 10¹³)(53.0 x 10⁻⁹) = 4.55 x 10⁶ m/s = 4.55 mega m/s.

The final speed of the electron is therefore -4.558 mega m/s.

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The gravitational force between two masses is inversely proportional to the square of the distance between them. This means that the two masses must be much closer together for the force to be 1.03 N. The masses 0.510 kg and 0.108 kg have to be 0.285 m apart in order for the gravitational force between them to have a magnitude of 1.03 N.

The equation for gravitational force is F=G*m1*m2/d^2, where G is the gravitational constant, m1 and m2 are the two masses, and d is the distance between them.

Assuming G=6.67*10^(-11) Nm^2/kg^2, m1=0.510 kg, and m2=0.108 kg, then d=0.285 m. This is the minimum distance between the two masses for the gravitational force between them to have a magnitude of 1.03 N.

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We need approximately 369 turns of wire to make the solenoid.

The self-inductance (L) of a solenoid is given by the formula:

L = (μ₀ * N² * A * l) / l

where:

μ₀ = permeability of free space (4π × 10^-7 H/m)

N = number of turns of wire

A = cross-sectional area of the solenoid

l = length of the solenoid

We can rearrange this formula to solve for N:

N = [tex]\sqrt{L * l) / (M_0 * A))}[/tex]

Substituting the given values, we get:

N = [tex]\sqrt{2.4(10^-^3 H * 0.045 m) / (4\pi (10^-^7 H/m * 1.80(10^-^3 m^2))}[/tex]

N ≈ 369.25

Therefore, the turn of wire that we are needed are 369.25 turns.

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The time that passes in the car is approximately 70 years. when the sister returns from the trip, she would be approximately 319.15 years old according to Earth's reference frame.

a. To calculate the time that passes in the car, we can use the concept of time dilation in special relativity. The formula for time dilation is:

[tex]\[ \Delta t' = \frac{\Delta t}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \][/tex]

where [tex]\( \Delta t' \)[/tex] is the time experienced in the moving frame (car), [tex]\( \Delta t \)[/tex] is the time experienced in the stationary frame (house), [tex]\( v \)[/tex] is the velocity of the car, and [tex]\( c \)[/tex] is the speed of light.

Given:

Velocity of the car, [tex]\( v = 89 \, \text{km/hr} = \frac{89}{3.6} \, \text{m/s} \)[/tex]

Time experienced in the stationary frame (house), [tex]\( \Delta t = 70 \, \text{years} \)[/tex]

Converting the velocity to meters per second:

[tex]\[ v = \frac{89}{3.6} = 24.72 \, \text{m/s} \][/tex]

Substituting the given values into the time dilation formula:

[tex]\[ \Delta t' = \frac{70}{\sqrt{1 - \left(\frac{24.72}{299792458}\right)^2}} \][/tex]

Simplifying the expression:

[tex]\[ \Delta t' = \frac{70}{\sqrt{1 - 8.7203 \times 10^{-17}}} \][/tex]

Since the value inside the square root is extremely close to 1, we can approximate the square root as 1:

[tex]\[ \Delta t' \approx \frac{70}{\sqrt{1}} = 70 \, \text{years} \][/tex]

Therefore, the time that passes in the car is approximately 70 years.

b. Using the same formula for time dilation, we can calculate the time that passes on the spacecraft. Given that the speed of the spacecraft is 0.95c, where c is the speed of light, we have:

[tex]\[ \Delta t' = \frac{\Delta t}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \][/tex]

Given:

The velocity of the spacecraft, [tex]\( v = 0.95c \)[/tex]

Time experienced on Earth, [tex]\( \Delta t = 70 \, \text{years} \)[/tex]

Substituting the values:

[tex]\[ \Delta t' = \frac{70}{\sqrt{1 - (0.95)^2}} \][/tex]

Simplifying the expression:

[tex]\[ \Delta t' = \frac{70}{\sqrt{1 - 0.9025}} \\\\= \frac{70}{\sqrt{0.0975}} \]\\\\\ \Delta t' = \frac{70}{0.31224} \approx 224.33 \, \text{years} \][/tex]

Therefore, during a period of 70 years on Earth, approximately 224.33 years pass on the spacecraft. The clever student will perceive that her life span has been extended because more time has passed for her relative to the observers on Earth.

c. In this scenario, the twin sister is traveling to a distant star and back at a speed of 0.99c. The time experienced in the stationary frame (Earth) is given by [tex]\( \Delta t \)[/tex], which is 45 years. The time experienced in the moving frame (spaceship) is given by [tex]\( \Delta t' \)[/tex].

Using the time dilation formula:

[tex]\[ \Delta t' = \frac{\Delta t}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \][/tex]

Given:

The velocity of the spaceship, [tex]\( v = 0.99c \)[/tex]

Time experienced on Earth, [tex]\( \Delta t = 45 \, \text{years} \)[/tex]

Substituting the values:

[tex]\[ \Delta t' = \frac{45}{\sqrt{1 - (0.99)^2}} \][/tex]

Simplifying the expression:

[tex]\[ \Delta t' = \frac{45}{\sqrt{1 - 0.9801}} \\\\= \frac{45}{\sqrt{0.0199}} \]\\\\\ \Delta t' = \frac{45}{0.141} \approx 319.15 \, \text{years} \][/tex]

Therefore, when the sister returns from the trip, she would be approximately 319.15 years old according to Earth's reference frame.

Know more about time dilation:

https://brainly.com/question/30493090

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