Question

1. A kid is flying a gasoline-powered, model airplane on the end of a control line (string). The plane has a mass of 0.10 kg and

it is flying around the kid in a horizontal circle with radius of 2.0 m. The speed of the plane is 6.0 m/s.
a. What is the tension in the control line?
b. What is the work done by the kid on the plane while it is flying around in a circle?
c. Suppose the kid starts to reel in the control line while the plane is flying around him at a constant 6.0 m/s. What is
the work he will do on the plane if the final radius of revolution is 1.0 m. (hint: Integrate the equation for centripetal
force between r = 2.0 m and r = 1.0 m.)
d. What is the average power the kid must produce if he takes 30 s to reel the plane in from f= 2.0 m to r= 1.0 m?

Answer 1

a. To find the tension in the control line, we can use the centripetal force equation:

`Fc = m * v^2 / r`

Where Fc is the centripetal force, m is the mass of the plane, v is the speed of the plane, and r is the radius of the circle.

Plugging in the given values:

m = 0.10 kg

v = 6.0 m/s

r = 2.0 m

Fc = (0.10 kg) * (6.0 m/s)^2 / (2.0 m)

Fc = 1.8 N

Therefore, the tension in the control line is 1.8 N.

b. The work done by the kid on the plane can be calculated using the work-energy principle:

Work = (1/2) * m * v^2

Plugging in the given values:

m = 0.10 kg

v = 6.0 m/s

Work = (1/2) * (0.10 kg) * (6.0 m/s)^2

Work = 10.8 Joules

Therefore, the work done by the kid on the plane while it is flying around in a circle is 10.8 Joules.

c. To find the work done by the kid when reeling in the control line, we need to calculate the change in work done at different radii.

`Work = ∫(Fc \: dr)`

To find the force Fc as a function of r, we can rearrange the centripetal force equation:

Fc = m * v^2 / r

Integrating the equation from r = 2.0 m to r = 1.0 m:

Work = ∫[(m * v^2) / r] dr

Work = m * v^2 * ∫(1 / r) dr

∫(1 / r) dr = ln|r|

Work = m * v^2 * [ln|r|] from r = 2.0 m to r = 1.0 m

Plugging in the given values:

m = 0.10 kg

v = 6.0 m/s

Work = (0.10 kg) * (6.0 m/s)^2 * [ln|1.0 m| - ln|2.0 m|]

Work ≈ 4.80 Joules

Therefore, the work done by the kid on the plane when reeling in the control line is approximately 4.80 Joules.

d. Average power is calculated by dividing the work done by the time taken:

Average Power = Work / Time

Plugging in the given values:

Work = 4.80 Joules

Time = 30 s

Average Power = 4.80 Joules / 30 s

Average Power ≈ 0.16 Watts

Therefore, the average power the kid must produce when reeling the plane in from r = 2.0 m to r = 1.0 m over a 30 s period is approximately 0.16 Watts.

Correct me if I'm wrong