Rotational Motion Homework: Solid Sphere Mass m, Radius r, Speed V[0]

[nybui]

Homework Statement

A solid sphere with mass "m" and radius "r", slides (without rotating) on a frictionless horizontal surface, with speed of V[0].

At time t[0] the sphere roll on a surface with friction "u", at time t[0] the ball is starts rotating as it glides aswell. At time t[1] the ball only rotates.

Homework Equations

What is the balls center mass velocity as a function of time, from time t[0] to time t[1].

The Attempt at a Solution

I have been trying to take a look at the balls inertia and torque on the ball from the friction. But the only result I got was that the ball gains speed, an that is nonsence.

Plzz... help this is a homeasignment that comes down on my final grade.

 

[Doc Al]

I have been trying to take a look at the balls inertia and torque on the ball from the friction. But the only result I got was that the ball gains speed, an that is nonsence.

You probably have the torque going in the wrong direction. In any case, consider the translational motion.

Hint: What forces act on the sphere? Note that the sphere is slipping. Apply Newton's 2nd law.

 

[m2003]

First, it is important to note that the ball's center of mass velocity will remain constant throughout this scenario, as there is no external force acting on the ball in the horizontal direction. The only change that occurs is the rotation of the ball about its center of mass.

To find the ball's angular velocity at time t[0], we can use the equation τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. Since the ball is sliding without rotating at t[0], the torque is zero, and thus the angular acceleration is also zero. This means that the ball's angular velocity at t[0] is also zero.

At time t[1], the ball starts rotating due to the friction force. To find the angular velocity at this time, we can use the equation τ = Iα once again. The torque in this case is μmgR, where μ is the coefficient of friction, m is the mass of the ball, g is the acceleration due to gravity, and R is the radius of the ball. The moment of inertia for a solid sphere is (2/5)mr^2. Plugging these values into the equation, we get:

μmgR = (2/5)mr^2 * α

Solving for α, we get:

α = (5μgR)/2r

Since the ball is only rotating at this point, its center of mass velocity remains the same. Therefore, the ball's angular velocity at t[1] can be used to find its center of mass velocity using the equation v = ωr, where ω is the angular velocity and r is the radius of the ball.

In summary, the ball's center of mass velocity remains constant from t[0] to t[1], but its angular velocity changes from 0 to (5μgR)/2r. I hope this helps with your homework assignment. Remember to always double check your calculations and equations to ensure accuracy. Good luck!