Question

An aeolipile is also known as a Hero engine. It is one of the earliest examples of steam power. They are made as a sphere with pipes coming out the sides as shown below. If the sphere has a radius of 45 cm and a mass of 400 g, how much force is needed from each spout to cause it to go from rest to an angular speed 5 rev/s in a span of 5 seconds? How much angular momentum does it have at t=5s?

Answer 1

final angular speed wf = 5 rev /s = 5 x 2pi rad /s = 31.42 rad/s

using wf = wi + alpha*t

31.42 = 0 + alpha*5

alpha = 6.28 rad/s^2

then using torque = I x alpha

i am considering sphere as hollow sphere if answer comes wrong then consider it as solid sphere

for hollow sphere I = 2 m r^2 / 3

for solid sphere I = 2 m r^2 / 5

everthing remains same.

using hollow sphere ,

torque = (2 m r^2 / 3) ( 6.28) = (2 x 0.400 x 0.45^2 / 3) ( 6.28) =0.339 N m

and there two forces aacting,

torque = 2(rF)

2 x 0.45 x F = 0.339

F = 0.377 N .......Ans

angular momentum = Iw = (2 m r^2 /3) (w)

= (2 x 0.400 x 0.45^2 / 3) ( 31.42) = 1.70 kg m^2 / s ......Ans