# How Do You Calculate the Ratio h/R for a Spinning Billiard Ball?

• AFlyingKiwi
In summary: So the total angular momentum about the z axis is mhv+Iω.In summary, a spherical billiard ball of mass m and radius R, initially at rest on a table, is given a horizontal impulse and eventually acquires a maximum speed of 9/7 times its initial speed. The problem is to find the ratio h/R, where h is the distance from the cue stick to the centerline and R is the ball's radius. This can be solved using the equations for angular momentum and momentum, and taking angular momentum about a point at the level of the table where friction has no moment. The total angular momentum consists of the rotational component and the linear component multiplied by the radius. This is valid even if there is a
AFlyingKiwi
1. Homework Statement
A spherical billiard ball of uniform density has mass m and radius R and moment of inertia about the center of mass ( ) 2 cm I = 2/ 5 mR^2 . The ball, initially at rest on a table, is given a sharp horizontal impulse by a cue stick that is held an unknown distance h above the centerline (see diagram below). The coefficient of sliding friction between the ball and the table is µk . You may ignore the friction during the impulse. The ball leaves the cue with a given speed v0 and an unknown angular velocity ω0 . Because of its initial rotation, the ball eventually acquires a maximum speed of 9 / 7 v0 . The point of the problem is to find the ratio h / R.

L=Iw
p=mv
L_i = L_f

## The Attempt at a Solution

I have the solution in this link (http://web.mit.edu/8.01t/www/materials/InClass/IC-W15D2-5.pdf) but I don't get a certain part of it. When they give us initial L, it says L_i = mv_0(R) + I_cm(w_0). Why is the mv_0(R) there? Also, why can we use conservation of angular momentum if there is a net external force?

AFlyingKiwi said:
When they give us initial L, it says L_i = mv_0(R) + I_cm(w_0). Why is the mv_0(R) there? Also, why can we use conservation of angular momentum if there is a net external force?
The answer to both questions is that they are taking angular momentum about some point at the level of the table. Friction therefore has no moment, and angular momentum is conserved. The total angular momentum consists of that due to its rotation about its centre plus the contribution from its linear motion, mvR.

haruspex said:
The answer to both questions is that they are taking angular momentum about some point at the level of the table. Friction therefore has no moment, and angular momentum is conserved. The total angular momentum consists of that due to its rotation about its centre plus the contribution from its linear motion, mvR.
So just to clarify, for any object that is rolling, it's total angular momentum is its rotational component and it's linear component times the radius?

AFlyingKiwi said:
So just to clarify, for any object that is rolling, it's total angular momentum is its rotational component and it's linear component times the radius?
It is not to do with whether it is rolling.
In general, angular momentum (and torque, and moment of a force...) is only meaningful in the context of a given axis. You can decompose the motion of a rigid body into the sum of a linear motion and a rotation about its mass centre. Given the axis, you can find the angular momentum contributions from those two motions and add them to get the total angular momentum about the axis.
E.g. consider a mass m traveling at speed v along y=h, z=0, rotating at rate ω about an axis parallel to the z axis and through its mass centre. Let the MoI about that rotation axis be I. Pick the z axis through the origin as the reference axis.
The linear motion contributes mhv, while the rotation contributes Iω.

## 1. What is the Angular Momentum Problem?

The Angular Momentum Problem is a concept in physics that describes the conservation of angular momentum in a system. It states that the total angular momentum of a closed system remains constant, unless acted upon by an external torque.

## 2. How is Angular Momentum calculated?

Angular Momentum is calculated by multiplying the moment of inertia of an object by its angular velocity. The moment of inertia is a measure of an object's resistance to changes in its rotation and is dependent on the mass and distribution of the object's mass.

## 3. What is the significance of Angular Momentum?

Angular Momentum is an important concept in physics as it helps us understand and predict the behavior of rotating objects and systems. It is also a conserved quantity, meaning that it remains constant in a closed system, allowing us to make accurate calculations and predictions.

## 4. What are some real-world examples of the Angular Momentum Problem?

Some examples of the Angular Momentum Problem in action include the rotation of planets and satellites around their axes, the spinning of a top or gyroscope, and the motion of a figure skater performing a pirouette.

## 5. How does the Angular Momentum Problem relate to other conservation laws?

The Angular Momentum Problem is closely related to other conservation laws, such as the conservation of energy and the conservation of linear momentum. These laws all describe the fundamental principles of the universe, and they work together to help us understand and predict the behavior of physical systems.

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