5. A big wheel has a diameter of 5 m and a mass of 1500 kg when fully laden with people. a) Work out the moment of inertia of the big wheel. (Hint: which shape from the ones given on p114 would be most suitable? b) When the wheel is rotating at full speed, a person has a linear velocity of 3 m/s. What is the angular velocity of this person? c) What is the rotational kinetic energy at this speed? d) A motor takes 10 seconds to accelerate the wheel from rest to a linear velocity on the circumference of 3 m/s. What is the power of the motor?​

a)  I = 3.75 10⁴  kg  m², b)  w = 0.6 rad / s, c)  K = 6.75 10³ J, d) P = 6.75 10² W

Explanation:

This is a rotations exercise

a) the proper shape for a wheel is that of a rim where most of the weight is in the circumference plus the point weights of the people sitting on its periphery.

We are going to approximate the reda with a thin ring

           I = M r²

           I = 1500 5²

           I = 3.75 10⁴  kg  m²

b) angular and linear velocity are related

         v = w r

         w = v / r

         w = 3/5

         w = 0.6 rad / s

c) the expression for kinetic energy is

         K = ½ I w²

         K = ½ 3.75 10⁴   0.6²

         K = 6.75 10³ J

d) the power is

         P = W / t

to find the work we use the relationship between work and the variation of kinetic energy

         W = ΔK = K_f - K₀

the system part of rest wo = 0

         W = K_f

         W = 6.75 10³ J

we calculate

         P = 6.75 10³/10

         P = 6.75 10² W

grav force prop 1/r squared 2r all squared is 4rsquared 1/4rsquared quartered