Determining V and Angular Velocity for Billiards Ball Rolling with Slipping

In summary, the conversation is about a billiards ball with a mass of M, radius R, and moment of inertia I=2/5MR^2 that is struck in the center with an initial velocity. The ball experiences friction as it moves across the table, causing its motion to change from pure translation to rolling with slipping and eventually to rolling without slipping. The goal is to determine the speed and angular velocity of the ball while it is rolling with slipping, as well as the time when it stops slipping. The ball stops sliding when its surface speed is equal to its overall speed and it also experiences torque from the friction.
  • #1
nlightner
1,712
0
I have started this billiards, and am very confused by one thing.
The ball has a mass of M, radius R, and mement of inertia about center of mass I=2/5MR2 .
The ball is struck in the center and slides with a initial velocity. Here is what I am having trouble with, if you could give me a hint to help me get started, I would appreciate it.
--As the ball moves across th rough billiards table it has a coefficient of sliding friction, its motion gradually changes from pure translation through rolling with slipping to rolling without slipping.
I have to determine a v and an angular velocity for rolling with slipping, and a time when it stops slipping.
If you could get me started I would appreciate it.
 
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  • #2
There is friction between the ball and the table.

The fricton will slow the ball down, and it will also have torque on the ball.

The ball stops sliding when the surface speed of the ball is equal to the speed of the ball.
 
  • #3


To determine the velocity and angular velocity for rolling with slipping, we can use Newton's second law of motion and the equations of motion for rotational motion. Let's start by defining our variables:

M = mass of the ball
R = radius of the ball
I = moment of inertia about the center of mass
v = linear velocity
ω = angular velocity
μ = coefficient of sliding friction
t = time

To begin, we can use Newton's second law, F=ma, to find the net force acting on the ball. The only forces acting on the ball are the applied force from the initial strike and the frictional force from the rough table. Since the ball is moving in a straight line, we can ignore any forces acting perpendicular to the motion. Therefore, the net force in the direction of motion is given by:

F = ma = μmg

Where μ is the coefficient of sliding friction and g is the acceleration due to gravity. Now, we can use the equation for torque, τ = Iα, to find the net torque acting on the ball. The torque is due to the frictional force, and it is given by:

τ = μmgR

Where R is the radius of the ball. In order for the ball to roll with slipping, the net torque must be equal to the moment of inertia times the angular acceleration, α. Therefore, we can set these two equations equal to each other and solve for α:

μmgR = Iα

Solving for α, we get:

α = μmgR/I

Now, we can use the equations of motion for rotational motion to find the angular velocity, ω, and the linear velocity, v, at any given time t. The equations are:

ω = ω0 + αt
v = v0 + at

Where ω0 and v0 are the initial angular and linear velocities, respectively. We can use these equations to find the values of ω and v at the moment when the ball stops slipping, which occurs when the angular velocity is equal to the linear velocity divided by the radius, ω = v/R.

To summarize, to determine the v and angular velocity for rolling with slipping, you will need to use Newton's second law, the equation for torque, and the equations of motion for rotational motion. Remember to set the net torque equal to the moment of inertia times the angular acceleration, and to use the condition ω = v
 

What is the purpose of determining V and angular velocity for billiards ball rolling with slipping?

The purpose of determining V and angular velocity for billiards ball rolling with slipping is to understand the motion of the ball on the table and to make accurate predictions about its trajectory and behavior. This can help players improve their game and also aid in the design and development of billiards equipment.

What factors affect the determination of V and angular velocity for billiards ball rolling with slipping?

There are several factors that can affect the determination of V and angular velocity for billiards ball rolling with slipping, including the surface of the table, the weight and size of the ball, the initial force applied to the ball, and the presence of any external forces such as friction or spin.

How is V and angular velocity measured for billiards ball rolling with slipping?

V and angular velocity can be measured using various methods, such as high-speed cameras, motion sensors, and computer simulations. These methods can provide precise measurements of the ball's speed and rotation at different points on the table.

What is the difference between slipping and rolling motion in billiards?

Slipping motion occurs when the ball is sliding on the table without any rotation, while rolling motion involves both translational and rotational motion. In billiards, slipping usually occurs at the beginning of the shot, while rolling takes over once the ball has gained enough momentum.

How can determining V and angular velocity for billiards ball rolling with slipping impact the game?

Knowing V and angular velocity can help players make strategic decisions, such as adjusting their aim and force, to achieve the desired outcome. It also allows for a better understanding of the physics behind the game, which can lead to improvements in technique and equipment design.

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