What is the Velocity of a Billiard Ball after a Horizontal Impulse of 2 N*s?

[Raios168]

Homework Statement

A billiard ball is imparted a horizontal impulse of 2 N*s at a height 4 cm above the center of the ball. The mass of the ball is 0.02 kg and it has radius 5 cm. Find velocity of the center of mass right after the impulse.

Note: I simplified the question, I was actually given that a linearly increasing force was applied from 0 to 40 000 N in 0.001 seconds and then was decreased linearly from 40 000 N to 0 in 0.001 seconds. So this created a triangle when I created a Force-time graph, calculating the area of the triangle yielded 2 N*s

2. Homework Equations
L = Iw

The Attempt at a Solution

Okay so the impulse causes more rotational motion than translational meaning Rw is greater than Vcm since it is hit above 2/3 its radius.This means friction (not sure if kinetic or static) will point opposite the direction of rotation. This makes sense since it will create a torque in the opposite direction of rotation causing rotation to decrease and translation to increase (will eventually start pure rolling). Alright so this means the impulse (2) simultaneously creates angular momentum and also linear momentum. With this I got:

2 = MVcm + Iw

I also have the net force equation:

F = Ma
f = Ma --> f is friction
μMg = Ma ----> I don't know μ so this doesn't get me anywhere

Then I looked at the torque about the center of mass of the ball:

∑τ = fR ----> I once again do not know μ so not helpful

Alright so this is where I've gotten. I have no clue where to go from here, could really use some assitance. Thanks in advance.

 

[ehild]

Homework Statement

A billiard ball is imparted a horizontal impulse of 2 N*s at a height 4 cm above the center of the ball. The mass of the ball is 0.02 kg and it has radius 5 cm. Find velocity of the center of mass right after the impulse.

Note: I simplified the question, I was actually given that a linearly increasing force was applied from 0 to 40 000 N in 0.001 seconds and then was decreased linearly from 40 000 N to 0 in 0.001 seconds. So this created a triangle when I created a Force-time graph, calculating the area of the triangle yielded 2 N*s

2. Homework Equations
L = Iw

The Attempt at a Solution

Okay so the impulse causes more rotational motion than translational meaning Rw is greater than Vcm since it is hit above 2/3 its radius.This means friction (not sure if kinetic or static) will point opposite the direction of rotation. This makes sense since it will create a torque in the opposite direction of rotation causing rotation to decrease and translation to increase (will eventually start pure rolling). Alright so this means the impulse (2) simultaneously creates angular momentum and also linear momentum. With this I got:

2 = MVcm + Iw

I also have the net force equation:

F = Ma
f = Ma --> f is friction
μMg = Ma ----> I don't know μ so this doesn't get me anywhere

Then I looked at the torque about the center of mass of the ball:

∑τ = fR ----> I once again do not know μ so not helpful

Alright so this is where I've gotten. I have no clue where to go from here, could really use some assitance. Thanks in advance.

Check the value of the impulse. The force of friction is much less then the average applied force, it can be ignored.

 

[Raios168]

Check the value of the impulse. The force of friction is much less then the average applied force, it can be ignored.

Sorry I don't quite understand what you mean. Do you mean that I made an error in calculating impulse and why can I ignore friction?

If I ignored friction, what would be causing rotational momentum to decrease?

 

[ehild]

Sorry I don't quite understand what you mean. Do you mean that I made an error in calculating impulse and why can I ignore friction?

If I ignored friction, what would be causing rotational momentum to decrease?

Yes, the numerical value for the impulse is not 2 Ns.
The problem asks " Find velocity of the center of mass right after the impulse" Friction will decrease the rotational motion gradually after the applied force ceased.

 

[Raios168]

Yes, the numerical value for the impulse is not 2 Ns.
The problem asks " Find velocity of the center of mass right after the impulse" Friction will decrease the rotational motion gradually after the applied force ceased.

Uhm okay. I see my error now. I wrote down the wrong number for time. It's actually 10^-4 seconds. But that still doesn't solve my problem. Even with the correct value of impulse (which I am now pretty sure is 4 Ns), how could I find the velocity of the center of mass?

 

[ehild]

What is the relation between impulse and change of momentum of a rigid body? What forces act on the ball?

 

[Raios168]

What is the relation between impulse and change of momentum of a rigid body? What forces act on the ball?

Okay so the relation is this initial momentum plus impulse equals final momentum. In this case final momentum would be when pure rolling occurs correct?

So I'm getting:

For translational:
Mvi + μmgt = Mvf --(1)

For rotational:
Iωi - μmgRt = I(vf/R)

Dividing this by R: Iωi/R -μmgt = Ivf/R^2 ---(2)

(1) + (2) yields:

Mvi + Iωi/R = Mvf + Ivf/R^2

I now have 3 unkowns: vi, ωi, and vf. And I'm stuck. Where to go from here?

 

[ehild]

Okay so the relation is this initial momentum plus impulse equals final momentum. In this case final momentum would be when pure rolling occurs correct?

No, the problem asks " Find velocity of the center of mass right after the impulse"
Till there is force, impulse is imparted to the body. When the force becomes zero, it is the end of imparting impulse. You need the velocity of the CM at that moment.
You overcomplicate the problem.

 

[Raios168]

No, the problem asks " Find velocity of the center of mass right after the impulse"
Till there is force, impulse is imparted to the body. When the force becomes zero, it is the end of imparting impulse. You need the velocity of the CM at that moment.
You overcomplicate the problem.

Okay! I think I understand you. So, I know the change in momentum is equal to the external force which is equal to Ma. Change in momentum is equal to impulse which is 4 if I calculated it correctly. I also know the time it took for this was 0.0002 seconds. So I setup the following equation:

4/0.0002 = Ma
4/0.0004 = M(vcm/0.0004)
4 = Mvcm
4/0.02 kg = vcm
200 m/s = vcm

Could you confirm if I got the correct answer?

 

[ehild]

Okay! I think I understand you. So, I know the change in momentum is equal to the external force which is equal to Ma. Change in momentum is equal to impulse which is 4 if I calculated it correctly. I also know the time it took for this was 0.0002 seconds. So I setup the following equation:

You need only one equation: the impulse transmitted is equal to the change of momentum.

4/0.0002 = Ma

The force is given, and it is not 4/0.0002, it changes with time. The end result is correct, but the derivation is not.

 

[Raios168]

No, the problem asks " Find velocity of the center of mass right after the impulse"
Till there is force, impulse is imparted to the body. When the force becomes zero, it is the end of imparting impulse. You need the velocity of the CM at that moment.
You overcomplicate the problem.

You need only one equation: the impulse transmitted is equal to the change of momentum.The force is given, and it is not 4/0.0002, it changes with time. The end result is correct, but the derivation is not.

But isn't change in momentum divided by time equal to the net force which is equal to Ma? I did 4 (the change in momentum) divided by time (0.0002) to equal Ma.

 

[ehild]

But isn't change in momentum divided by time equal to the net force which is equal to Ma? I did 4 (the change in momentum) divided by time (0.0002) to equal Ma.

Ma=F. If the force changes with time, so does the acceleration. You have the equation ##\int{Fdt}=\Delta(mv)##. You get v directly from this equation, and it is the question, not the average acceleration during the impulse.