Angular Impulse, finding the angle of action?

[Phan]

Homework Statement

A cue hits the billiard ball at the position shown. The collision lasts 90 ms with an
average force of 420 N. If R = 7 cm, m = 300g, k = 0.1.
(ignore friction during impact)
*Didn't post the rest since they are irrelevant.

Homework Equations

J(angular) = (x*F*t)/I
J(linear) = (F*t)/m

The Attempt at a Solution

I know how to figure out everything else after the impulse force (since you just solve for time using the two angular and linear equations), but I'm still stuck on the idea of angular impulse. I know that it works like Torque (r*F) and that you have to either use the moment arm angle or sine of the smallest angle, but I'm not sure which angle that is on the diagram @_@

Is it the one that I have labelled? If so, I'm feeling stupid here, but I can't figure out how to get that angle. Any help would be greatly appreciated :)

 

[anathema]

Thank you for your post. It seems like you are working on a problem involving a billiard ball and its collision with a cue. I would like to offer some guidance on how to approach this problem.

Firstly, let's start by defining some variables and equations that will be useful in solving this problem. We have the following:

- J(angular) = angular impulse
- J(linear) = linear impulse
- x = moment arm (distance between point of impact and center of mass of the ball)
- F = average force during collision
- t = duration of collision
- I = moment of inertia of the ball
- m = mass of the ball
- R = radius of the ball
- μk = coefficient of kinetic friction

Now, let's consider the angular impulse first. As you correctly mentioned, it is similar to torque (r*F), where r is the moment arm and F is the average force during collision. However, in this case, we need to take into account the angle between the direction of the force and the moment arm. This angle can be found by using trigonometry. We can use the sine of the angle (θ) to calculate the moment arm, as shown in the equation below:

x = R*sin(θ)

Once we have the value of x, we can use the equation for angular impulse to find the value of J(angular):

J(angular) = (x*F*t)/I

Now, let's move on to the linear impulse. This can be calculated using the following equation:

J(linear) = (F*t)/m

Since we know the value of J(angular) and J(linear), we can equate them to find the duration of the collision (t):

J(angular) = J(linear)

(x*F*t)/I = (F*t)/m

Solving for t, we get:

t = (I/m)*x

Now, we have all the information we need to solve the problem. We can use the value of t to find the final velocity of the ball after the collision, using the equation for linear impulse:

J(linear) = m*(vf - vi)

Where vi is the initial velocity of the ball, which we can assume to be zero. Rearranging this equation, we get:

vf = J(linear)/m

Finally, we can use the coefficient

 

[thomate1]

Dear student,

Thank you for your question. Angular impulse is a measure of the change in angular momentum of an object during a collision or interaction. It is calculated by multiplying the force applied to the object by the time it is applied, and then dividing by the moment of inertia of the object.

In this case, the moment of inertia (I) can be calculated using the formula I = mR^2, where m is the mass of the object and R is the distance from the axis of rotation to the point of impact (in this case, the radius of the billiard ball).

To find the angle of action, you will need to use the concept of torque. Torque is the product of the force and the distance from the axis of rotation to the point where the force is applied. In this case, the force is the average force of 420 N and the distance is the radius of the billiard ball (7 cm).

To find the angle of action, you can use the formula torque = rFsinθ, where r is the distance from the axis of rotation to the point where the force is applied, F is the force, and θ is the angle between the force and the lever arm (the distance from the axis of rotation to the point where the force is applied).

In this case, r = 7 cm, F = 420 N, and you can solve for θ. Once you have the angle of action, you can use it to calculate the angular impulse using the formula J(angular) = (rFt/I)sinθ.

I hope this helps with your understanding of angular impulse and finding the angle of action. Let me know if you have any further questions.

Best,