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

Answer:

35.1688 seconds

Explanation:

M = Mass of sphere = 8200 kg

R = Radius of sphere = 90 cm

F = Force = 30 N

Moment of inertia is given by

$I=\dfrac{2}{5}MR^2\\\Rightarrow I=\dfrac{2}{5}\times 8200\times 0.9^2\\\Rightarrow I=2656.8\ kgm^2$

Torque is given by

$\tau=I\alpha\\\Rightarrow FR=I\alpha\\\Rightarrow \alpha=\dfrac{FR}{I}\\\Rightarrow \alpha=\dfrac{30\times 0.9}{2656.8}\\\Rightarrow \alpha=0.01016\ rad/s^2$

For one rotation

$\theta=\omega_it+\dfrac{1}{2}\alpha t^2\\\Rightarrow 2\pi=0\times t+\dfrac{1}{2}\times 0.01016\times t^2\\\Rightarrow t=\sqrt{\dfrac{2\pi2}{0.01016}}\\\Rightarrow t=35.1688\ s$

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