Question

There is a girl pushing on a large stone sphere. The sphere has a mass of 8200 kgand a radius of 90 cm and floats with nearly zero friction on a thin layer of pressurized water. Suppose that she pushes on the sphere tangent to its surface with a steady force of F = 30N and that the pressured water provides a frictionless support.

How long will it take her to rotate the sphere one time, starting from rest?

Answer 1

35.1688 seconds

Step-by-step explanation:

M = Mass of sphere = 8200 kg

R = Radius of sphere = 90 cm

F = Force = 30 N

Moment of inertia is given by

`I=(2)/(5)MR^2\\\Rightarrow I=(2)/(5)* 8200* 0.9^2\\\Rightarrow I=2656.8\ kgm^2`

Torque is given by

`\tau=I\alpha\\\Rightarrow FR=I\alpha\\\Rightarrow \alpha=(FR)/(I)\\\Rightarrow \alpha=(30* 0.9)/(2656.8)\\\Rightarrow \alpha=0.01016\ rad/s^2`

For one rotation

`\theta=\omega_it+(1)/(2)\alpha t^2\\\Rightarrow 2\pi=0* t+(1)/(2)* 0.01016* t^2\\\Rightarrow t=\sqrt{(2\pi2)/(0.01016)}\\\Rightarrow t=35.1688\ s`

The time taken to rotate the sphere from rest is 35.1688 seconds