A sphere of radius r and mass m has a linear velocity v0 directed to the left and no angular velocity as it is placed on a belt moving to the right with a constant velocity v1. If after sliding on the belt the sphere is to have no linear velocity relatives to the ground as it starts rolling on the belt without sliding. in terms of v1 and the coeefficient of kinetic friction mu k between the sphere and the belt. Determine- a. the required value of v0. b. the time t1 at which the sphere will start roling on the belt. and c. the distance the sphere will have moved relative to the ground at time t1 ?

To solve this problem, we'll break it down into three parts: determining the required value of v0, finding the time at which the sphere starts rolling, and calculating the distance the sphere has moved relative to the ground at that time.

a. Determining the Required Value of v0:

In order for the sphere to start rolling without sliding on the belt, the static friction force between the sphere and the belt must provide enough torque to counteract the linear velocity v0. The torque provided by static friction is given by τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

For a solid sphere, the moment of inertia I is given by I = (2/5) * m * r², where m is the mass of the sphere and r is its radius.

The torque provided by static friction is equal to the friction force multiplied by the radius of the sphere, τ = μk * m * g * r, where μk is the coefficient of kinetic friction between the sphere and the belt, and g is the acceleration due to gravity.

Setting the torque provided by static friction equal to the torque needed to counteract the linear velocity, we have:

μk * m * g * r = (2/5) * m * r² * α

Since the sphere is not slipping, the linear acceleration of the sphere is equal to the angular acceleration multiplied by the radius, a = α * r.

Substituting this into the above equation, we get:

μk * m * g * r = (2/5) * m * r² * (a/r)

Simplifying, we find:

μk * g = (2/5) * a

Since the acceleration a is equal to the difference in velocities v1 and v0 divided by the time t1, a = (v1 - v0) / t1.

Substituting this into the equation, we have:

μk * g = (2/5) * (v1 - v0) / t1

Solving for v0, we get:

v0 = v1 - (5 * μk * g * r * t1) / 2

b. Finding the Time t1 at Which the Sphere Starts Rolling:

To find the time at which the sphere starts rolling, we need to consider the motion of the sphere on the belt. Initially, while sliding on the belt, the sphere has a linear velocity v0 directed to the left. As the sphere slides, it experiences a kinetic friction force opposing its motion, caused by the friction between the sphere and the belt. This friction force slows down the sphere until the static friction force becomes strong enough to provide a clockwise torque, causing the sphere to start rolling.

The equation of motion for the sphere while sliding is given by:

m * a = μk * m * g

where a is the acceleration of the sphere while sliding and is equal to (v1 - v0) / t1.

Simplifying, we find:

(v1 - v0) / t1 = μk * g

Solving for t1, we get:

t1 = (v1 - v0) / (μk * g)

c. Calculating the Distance the Sphere Has Moved Relative to the Ground at Time t1:

To find the distance the sphere has moved relative to the ground at time t1, we need to consider the initial sliding motion and the subsequent rolling motion.

During the sliding motion, the distance traveled by the sphere relative to the ground is given by d1 = v0 * t1.

During the rolling motion, the sphere moves in a circular path with a radius equal to its own radius r. The distance traveled by the sphere relative to the ground during the rolling motion is given by d2 = 2 * π * r, which is the circumference of the circular path.

The total distance traveled by the sphere relative to the ground at time t1 is therefore:

d = d1 + d2
= v0 * t1 + 2 * π * r

Substituting the value of t1 from part b, we get:

d = v0 * ( (v1 - v0) / (μk * g) ) + 2 * π * r

This equation gives us the distance the sphere has moved relative to the ground at time t1.

In summary:
a. The required value of v0 is v0 = v1 - (5 * μk * g * r * t1) / 2.
b. The time t1 at which the sphere starts rolling is t1 = (v1 - v0) / (μk * g).
c. The distance the sphere will have moved relative to the ground at time t1 is d = v0 * ( (v1 - v0) / (μk * g) ) + 2 * π * r.