Ap physics rolling/slipping billiard ball

In summary, A billiard ball with mass M, radius R, and moment of inertia Ic=2/5MR^2 is struck by a cue stick and initially slides with a velocity Vo. As it moves across a table with a coefficient of sliding friction U, its motion transitions from pure translation to rolling without slipping. Part a of the problem asks for an expression for the linear velocity v of the center of the ball as a function of time while it is rolling without slipping. This can be found by considering the speed of the ball's surface with respect to the center, and then combining that with the translational speed of the center with respect to the ground. This speed must be zero for rolling without slipping to occur.
  • #1
Schmoozer3348
A billiard ball has mass M, radius R and moment of inertia about the center of mass Ic=2/5MR^2. The ball is struck by a cue stick along a horiontal line through the ball's center of mass so that the ball initially slides with a velocity Vo. As the ball moves across the table (which has a coefficient of sliding friction U ), its motion gradually changes from pure translation through slipping to rolling without slipping.

a)Develop an expression for linear velocity v of the center of the ball as a function of time while it is rolling without slipping
b)develop an expression for the angular velocity w of the ball as a function of time while it is rolling without slipping.
c)determine the time at which the ball begins to roll without slipping.
d) when the ball is struck it acquires an angular momentum about a fixed point P on the surface of the table. During the subsequent motion, the angular momentum about P remains constant despite the frictional force. Explain why it is so.


I'm stuck on the first part but am confident i could figure out b,c, and d if i had a little help with a. Is it kinematics V=Vi+at?
 
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  • #2
Originally posted by Schmoozer3348

I'm stuck on the first part but am confident i could figure out b,c, and d if i had a little help with a. Is it kinematics V=Vi+at?
Pretend the ball was suspended in mid-air. If it's spinning at a certain angular speed, how fast is its surface moving (linear speed) with respect to the center? Got it? Now if I want to place this spinning ball on the ground so that the surface doesn't slide, how fast had it better be moving?
 
  • #3
This kind of thing should be in the Homework Help forums, but now that we've started...

The purpose of part a is for you to define --mathematically-- the conditions for "rolling without slipping". It's got nothing to do with kinematics (except trivially) since once the ball rolls without slipping, acceleration is zero.

Try this way of thinking. For a rotating ball, what's the speed of surface with respect to the center? Then, of course, the translational speed of the ball is the speed of the center with respect to the ground. Now combine these to find the speed of the ball surface where it touches the ground with respect to the ground. "Rolling without slipping" means that that speed (surface of ball with respect to the ground) is zero.
 
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1. What is the difference between rolling and slipping in physics?

In physics, rolling refers to the motion of an object where its center of mass moves in a circular path without any slipping or sliding. Slipping, on the other hand, is when an object's center of mass moves in a non-circular path and there is friction present between the object and the surface it is moving on.

2. How does the mass and radius of a billiard ball affect its rolling and slipping motion?

The mass and radius of a billiard ball can affect its rolling and slipping motion in several ways. A larger mass will require more force to accelerate and may result in more slipping. A larger radius can also lead to more slipping as there is a larger contact area between the ball and the surface. However, a larger radius can also help reduce slipping by providing a larger moment of inertia, making it easier for the ball to maintain its circular motion.

3. What is the coefficient of friction and how does it impact the rolling and slipping of a billiard ball?

The coefficient of friction is a measure of the amount of friction between two surfaces in contact. In the case of a billiard ball, the coefficient of friction between the ball and the table surface can affect its rolling and slipping motion. A higher coefficient of friction will result in more slipping, while a lower coefficient of friction will result in smoother rolling motion.

4. How does the angle of the table impact the rolling and slipping motion of a billiard ball?

The angle of the table can greatly impact the rolling and slipping motion of a billiard ball. When the table is tilted, the force of gravity will act on the ball at an angle, causing it to accelerate and potentially slip. The steeper the angle, the more likely the ball is to slip. However, a carefully calculated angle can also be used to control the direction and speed of the ball's motion.

5. Can the rolling and slipping of a billiard ball be accurately predicted and modeled using physics equations?

Yes, the rolling and slipping motion of a billiard ball can be accurately predicted and modeled using physics equations. These equations take into account various factors such as the mass, radius, coefficient of friction, and angle of the table to determine the ball's motion. However, in real-world scenarios, there may be other external factors that can also affect the ball's motion, making it more complicated to predict with 100% accuracy.

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