Maximizing Distance and Angular Velocity: Solving a Billiard Ball Problem

In summary, this conversation discusses a problem involving a billiard ball being struck by a cue and determining how far it will move before it starts to roll on the table instead of slipping. The initial velocity, radius, mass, gravity, and coefficient of kinetic friction are all known. The solution involves using Newton's 2nd law to solve for the point where the ball rolls without slipping, and the condition for this is when the angular speed increases as the translational speed decreases.
  • #1
Nanoath
8
0

Homework Statement



A billiard ball is struck by a cue as is shown in the figure. The line of action of the applied impulse is horizontal and passes through the center of the ball. The initial velocity v0 of the ball after impact, its radius R, its mass m, the acceleration due to gravity g, and the coefficient of kinetic friction µk between the ball and the table are all known.
a) How far will the ball move before it ceases to slip on the table and starts to roll?
b) What will its angular velocity be at this point?

Homework Equations


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The Attempt at a Solution



I need help getting started, this problem is driving me nuts.
We are looking for distance X between the points where the ball is not rotating and where it starts to rotate due to friction force. If I'm correct, initially the angular velocity is 0, but at the point where it starts moving it becomes greater than 0. The velocity of the ball is highest at V0.
meeh... I'm not sure how to prove that with calculus -_-
 
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  • #2
Apply Newton's 2nd law to both translational and rotational motion of the ball due to the friction force. Hint: The angular speed increases as translational speed decreases. Solve for the point where it rolls without slipping. (What's the condition for rolling without slipping?)
 
  • #3
Thanks, trying it out now :D
 

Related to Maximizing Distance and Angular Velocity: Solving a Billiard Ball Problem

1. What is impulse and how is it related to momentum?

Impulse refers to the change in momentum of an object over a certain period of time. It is calculated by multiplying the force applied to an object by the time period over which the force is applied. Since momentum is the product of an object's mass and velocity, impulse is directly related to momentum as it can cause a change in an object's velocity and therefore its momentum.

2. How is impulse different from force?

Force is a push or pull that can cause an object to accelerate, while impulse is the change in momentum of an object over a period of time. Force is a vector quantity, meaning it has both magnitude and direction, while impulse is a scalar quantity, meaning it only has magnitude.

3. Can impulse be negative or positive?

Yes, impulse can be both negative and positive. A negative impulse means that the object's momentum is decreasing, while a positive impulse means that the object's momentum is increasing. This can happen when a force acts in the opposite direction of an object's motion, causing its velocity to decrease, or when a force acts in the same direction as an object's motion, causing its velocity to increase.

4. How does angular velocity differ from linear velocity?

Angular velocity refers to the rate at which an object rotates around a fixed axis, while linear velocity refers to the rate at which an object moves in a straight line. Angular velocity is measured in radians per second, while linear velocity is measured in meters per second. Additionally, angular velocity is a vector quantity as it has both magnitude and direction, while linear velocity is a scalar quantity.

5. What is the relationship between angular velocity and angular momentum?

Angular velocity and angular momentum are directly proportional to each other. Angular momentum is the product of an object's moment of inertia (which is a measure of how resistant an object is to changes in its rotational motion) and its angular velocity. This means that an increase in angular velocity will result in an increase in angular momentum, and vice versa.

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