Let's address each part of the problem step by step:
a. To find the tension in the control line, we can use the centripetal force formula:
Centripetal Force (Fc) = (mass * velocity^2) / radius
Fc = (0.10 kg * (6.0 m/s)^2) / 2.0 m
Fc = (0.10 kg * 36 m^2/s^2) / 2.0 m
Fc = 1.8 N
So, the tension in the control line is 1.8 Newtons.
b. The work done by the kid on the plane while it is flying around in a circle can be found using the work-energy theorem. The work done is equal to the change in kinetic energy:
Work = ΔKE = KE_final - KE_initial
Initial KE = 0.5 * mass * initial velocity^2
Initial KE = 0.5 * 0.10 kg * (6.0 m/s)^2 = 1.8 J
Final KE = 0.5 * mass * final velocity^2
Final KE = 0.5 * 0.10 kg * (6.0 m/s)^2 = 1.8 J
Work = 1.8 J - 1.8 J = 0 J
So, the work done by the kid on the plane is 0 Joules.
c. To find the work when the kid reels in the control line from a radius of 2.0 m to 1.0 m, you can integrate the centripetal force as the radius changes. The centripetal force varies with the radius. The work done is given by:
Work = ∫[initial radius (r1) to final radius (r2)] Fc(r) dr
Where Fc(r) is the centripetal force as a function of the radius.
Fc(r) = (mass * velocity^2) / r
Now, we integrate:
Work = ∫[2.0 m to 1.0 m] [(0.10 kg * (6.0 m/s)^2) / r] dr
Work = [0.10 kg * (6.0 m/s)^2] * ln(r) | from 2.0 m to 1.0 m
Work = (0.10 kg * 36 m^2/s^2) * [ln(1.0) - ln(2.0)]
Work = (0.10 kg * 36 m^2/s^2) * [0 - 0.6931]
Work ≈ -2.48 J
So, the work done by the kid when reeling in the control line is approximately -2.48 Joules.
d. The average power can be calculated using the formula:
Average Power = Work / Time
Average Power = (-2.48 J) / 30 s
Average Power ≈ -0.083 W
The negative sign indicates that the kid is doing work against the motion of the plane. So, the average power the kid must produce is approximately 0.083 Watts (or 83 mW), taking into account the negative sign.