Linear and Angular Momentum Problem

[xenogizmo]

Hey Everyone,
I have this physics problem, and I'm not really sure how to solve it using linear and angular momentum, I know it can be solved otherwise in different methods, but I'm interested in this one..
The question is:

"A bowling ball is given an initial speed of 5 m/s on an alley such that it slides initially without rolling. The coefficient of friction between the ball and the alley is 0.30. How far must the ball travel until pure rolling motion?"

Now I know that the friction force causes the change in linear momentum, and the change in angular momentum, so we equate them and solve using the formula of work..
But after I do that it gets really messy, and I had to add a negative sign for no reason to get the right answer..

So I would reeeally appreciate it if someone could write a full explanatory solution showing the steps, thanks a lot!

Xeno


ps. the right answer is 2.08 meters, if you guys want to check your answers

 

[ehild]

The question is:

"A bowling ball is given an initial speed of 5 m/s on an alley such that it slides initially without rolling. The coefficient of friction between the ball and the alley is 0.30. How far must the ball travel until pure rolling motion?"

Now I know that the friction force causes the change in linear momentum, and the change in angular momentum, so we equate them and solve using the formula of work..
But after I do that it gets really messy, and I had to add a negative sign for no reason to get the right answer..

Do not forget that he work of friction is negative as the force is opposite to the displacement.

ehild

 

[arildno]

1. The equations of motion:
In the horizontal direction, Newton's second law states for the acceleration of the center of mass:
$$-\mu{M}g=Ma_{C.M}$$
(The force acts in the negative horizontal direction)
The torque about the C.M is:
$$-R\vec{k}\times(-\mu{M}g\vec{i})=\frac{2}{5}MR^{2}\dot{\omega}\vec{j}$$
Where R is the radius of the sphere (it will cancel out in your problem later on).
These equations yields the expressions for the C.M velocity and the angular velocity in the period up to pure rolling.

To determine the time T when the ball starts pure rolling, we use the requirement:
$$v_{G}(T)+\omega(T)\vec{j}\times(-R\vec{k})=\vec{0}$$

Hope that helped..

 

[Cyrus]

Could you explain the left side of your equation please. $$-\mu{M}g=Ma_{g}$$
You have equal and opposite forces. But I don't see why you chose -mu*M*g. Also what is A_g on the right side. Is that the acceleration due to gravity, or the center of mass?

Also in the second equation, is R the radius?

 

[arildno]

good catch, cyrus:
I've done some editing; there is only one horizontal force acting on the system.

 

[Cyrus]

If you have time arildno, can you step through the calculations. I tried to do some work on it myself but did not get very far. It seems that you equated the torque equal to = I*omega. But I thought torque is equal to I domega/dt. So what happened to the dt at the bottom. Also where does the R cancel out? That equation leaves you with one R. Thanks for your help!

 

[arildno]

First, I wrote $$\dot{\omega}$$ rather than $$\omega$$

The "dot"-notation is a hallowed tradition reaching back to Newton for writing time-derivatives..
So, we're in agreement here!
I'll post a detailed solution a bit later.

 

[arildno]

For the solution:
We find that the velocity of the C.M must be:
$$v_{G}(T)=v_{0}-\mu{g}t, v_{0}=5m/s$$ (in the $$\vec{i}$$-direction)
The equation for the angular velocity as a function of time is:
$$\omega(t)=\frac{5\mu{gt}}{2R}$$

Now, to determine "T" in the rolling condition, we note that:
$$\omega(T)\vec{j}\times(\vec{-R}\vec{k})=-\frac{5\mu{gT}}{2}\vec{i}$$
Hence, we must have, in the $$\vec{i}$$ direction:
$$v_{0}-\mu{g}T-\frac{5\mu{g}T}{2}=0$$
Or:
$$T=\frac{2v_{0}}{7\mu{g}}$$

This T-value is then used in the distance formula:
$$r(T)=v_{0}T-\frac{\mu{g}T^{2}}{2}=\frac{12v_{0}^{2}}{49\mu{g}}$$