# Rotational Motion of a bowling ball

• PrettyLights
In summary, the bowling ball experiences a period of sliding and rotating, during which its rotational kinetic energy increases by 5gUk/2R. After the sliding and rotating is complete, the ball enters pure rotational motion with a speed of 5Vc.
PrettyLights

## Homework Statement

A bowling ball is released with speed V and no rotational kinetic energy. After a period of sliding and rotating, the ball enters pure rotational motion. The coefficient of friction between the ball and the ground while sliding is Uk.
a. Show that the rotational acceleration of the ball during the initial period of diluting is alpha=5gUk/2R, where g is the acceleration due to gravity and r is the radius of the ball (a solid sphere).
b. Show that when sliding finishes and rolling begins, the speed of the center of mass is Vc=5V/7

## Homework Equations

Rotational Kinetic Energy= 1/2IW^2
Force due to friction= Ukmg
Kinetic Energy= 1/2mv^2

## The Attempt at a Solution

I tried this problem from the angle of energy conservation but it quickly gets complicated. I tried to work from:
Potential energy of the ball + energy lost due to friction + Rotational energy = Total Kinetic, and then subbing in I and solving for V, plugging V into w=v/R and then l x V = alpha. Any help is appreciated.

Have you tried using the torque equation?
Torque = I * alpha (Hint: use the parallel axis theorem and use the frictional force as the force producing the torque).
Conservation of energy can be used to get the final rotational speed, but
a very concise solution can be obtained using conservation of momentum.
You wrote alpha=5gUk/2R for the angular acceleration.
Are you sure the denominator should be 2 R and not 7 R.

My mistake - 2 R should be correct because the parallel axis theorem would not apply
while the ball is slipping, but it can be used when the ball reaches its final speed.

J Hann said:
the parallel axis theorem would not apply while the ball is slipping
To clarify, the difficulty with applying the parallel axis theorem while the ball is slipping is that the instantaneous centre of rotation is moving. It starts off infinitely below ground (i.e., not rotating) and finishes at ground level.
J Hann said:
a very concise solution can be obtained using conservation of momentum.
That's conservation of angular momentum, right?

Okay, so if I start with the equations for Torque this is where it takes me:

alpha=T/I = (F x R)/(2/5)MR^2 = Mguk x R/ (2/5)Mr^2 = 5guk/2R

This seems right to me because, as you said, the friction is the external force causing the torque. Thank you!

PrettyLights said:
Okay, so if I start with the equations for Torque this is where it takes me:

alpha=T/I = (F x R)/(2/5)MR^2 = Mguk x R/ (2/5)Mr^2 = 5guk/2R

This seems right to me because, as you said, the friction is the external force causing the torque. Thank you!
Looks right.

## 1. What is rotational motion?

Rotational motion refers to the movement of an object around an axis or center point. In the case of a bowling ball, it is the movement of the ball as it rolls down the lane.

## 2. How does the release of the bowling ball affect its rotational motion?

The release of the bowling ball is crucial in determining its rotational motion. The angle and speed at which the ball is released can affect its spin, direction, and speed as it travels down the lane.

## 3. What factors can affect the rotational motion of a bowling ball?

There are several factors that can affect the rotational motion of a bowling ball, including the surface of the lane, the oil pattern, the weight and shape of the ball, and the speed and angle of release.

## 4. What is the difference between angular velocity and linear velocity?

Angular velocity refers to the speed of an object's rotation, while linear velocity refers to the speed at which an object moves in a straight line. In the case of a bowling ball, its angular velocity would be its rate of spinning, while its linear velocity would be its speed down the lane.

## 5. How does friction play a role in the rotational motion of a bowling ball?

Friction between the bowling ball and the lane is important in controlling its rotational motion. The amount of friction can affect the ball's speed, direction, and rotation. This is why bowlers often use different types of balls and techniques to adjust to different lane conditions.

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