Maximum Angle for Non-Null Initial Velocity of Billiard Ball Hit by Cue

[TSny]

You have four conditions that must be satisfied if "null velocity" means no translational velocity and no rotational velocity:
(1) Net horizontal linear impulse = 0
(2) Net rotationall impulse = 0
(3) Net vertical impulse = 0
(4) $f \leq \mu K$

The first two already restrict $\theta$ to one value.
Then, imposing the last two conditions restricts the allowed values of $\mu$.

The fact that the problem statement makes no mention of a restriction on $\mu$ is just another oddity of the wording of the problem.

 

[anon90]

You have four conditions that must be satisfied if "null velocity" means no translational velocity and no rotational velocity:
(1) Net horizontal linear impulse = 0
(2) Net rotationall impulse = 0
(3) Net vertical impulse = 0
(4) $f \leq \mu K$

The first two already restrict $\theta$ to one value.
Then, imposing the last two conditions restricts the allowed values of $\mu$.

The fact that the problem statement makes no mention of a restriction on $\mu$ is just another oddity of the wording of the problem.

So, by your logic, I have μ \geq sqrt(2) /(2K)
But I know what (1) -> Jsinθ=K, hence μ \geq 1 /J (2)
All the values of μ satisfying (2) are good, and there's only one possible value for θ.

 

[TSny]

So, by your logic, I have μ \geq sqrt(2) /(2K)

How did you get this? Note that $\mu$ should be dimensionless.

 

[anon90]

How did you get this? Note that $\mu$ should be dimensionless.

Right.
f≤μsK=μsJsinθ

Together with θ= 45° and Jcosθ−f=0, I get μ \geq 1

 

[TSny]

OK, that's what I got, too.

 

[anon90]

OK, that's what I got, too.

So, in the end, the two conditions should be
μ≥1
θ=45°
So the trick with this kind of problems is to figure out which forces can be though as impulsive in that laps of time, and then you just draw the fbd and solve it. Am I forgetting about anything else?

 

[TSny]

I think that's essentially it.

 

[anon90]

I think that's essentially it.

I see, thank you for your help and merry Christmass.

 

[TSny]

Thanks. Merry Christmas to you, too.