What have I done wrong? (torque and angular momentum)

In summary, the billiardball will roll without slipping when it is hit if the velocity of the center of mass is greater than the velocity of the cue.
  • #1
PhyIsOhSoHard
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[SOLVED] What have I done wrong? (torque and angular momentum)

Homework Statement


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A billiardball is hit from rest by the cue at height "h" from the table with a force F in the time interval Δt. The mass M and radius R of the ball is known as well as the moment of inertia which is [itex]I=\frac{2}{5}MR^2[/itex]
Find an expression for the height "h" at which the billiardball will roll without slipping when it is hit.


Homework Equations


Condition for roll with no slipping:
[itex]v_{CM}=R\omega[/itex]


The Attempt at a Solution


I start by finding an expression for [itex]v_{cm}[/itex].

Center of mass differentiated by Δt gives:
[itex]v_{CM}=\frac{Mv}{M}=v[/itex]


Newton's 2nd law:
[itex]F=M\frac{v}{Δt}[/itex]

Isolating velocity gives:
[itex]v=MFΔt[/itex]

Since the velocity is equal to the velocity of the center of mass:
[itex]v_{CM}=MFΔt[/itex]

Now I find an expression for the angular velocity.

The net torque is given by:
[itex]∑τ=Iα[/itex]

The only force is the force F from the cue which gives the torque [itex]τ=F(h-R)[/itex] where (h-R) is the perpendicular length from the force F to the center of mass of the ball.
[itex]F(h-R)=I\frac{\omega}{Δt}[/itex]

The angular velocity is:
[itex]\omega=\frac{F(h-R)Δt}{I}[/itex]

Now I insert the velocity of CM and the angular velocity into the rolling without slip equation:
[itex]MFΔt=R\frac{F(h-R)Δt}{I}[/itex]

And I end up with:
[itex]h=R(2/5M^2+1)[/itex]

But my expression for the height has the mass squared in it. What did I do wrong?
 
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  • #2
PhyIsOhSoHard said:
Newton's 2nd law:
[itex]F=M\frac{v}{Δt}[/itex]

Isolating velocity gives:
[itex]v=MFΔt[/itex]
Try that step again.
(Do you know how to do dimensional analysis? That's a very useful way to sanity-check an equation.)
 
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  • #3
haruspex said:
Try that step again.
(Do you know how to do dimensional analysis? That's a very useful way to sanity-check an equation.)

That's it! Now the answer is correct, thanks! :)
 

Related to What have I done wrong? (torque and angular momentum)

What is torque?

Torque is a measure of the force that causes an object to rotate around an axis or pivot point. It is often described as a twisting or turning force.

How is torque related to angular momentum?

Torque and angular momentum are closely related. Torque is the force that causes an object to rotate, while angular momentum is the measure of an object's tendency to keep rotating. In simple terms, torque causes angular momentum to change, and angular momentum affects the amount of torque required to change an object's rotation.

What happens if I apply too much torque?

If too much torque is applied to an object, it can cause the object to rotate faster or in a different direction than intended. This can also lead to the object becoming unstable or breaking if it is not strong enough to withstand the force.

How do I calculate torque?

To calculate torque, you need to know the force applied to the object and the distance from the force to the axis of rotation. The formula for torque is: torque = force x distance. The unit for torque is usually Newton-meters (Nm) in the metric system.

Can torque and angular momentum be negative?

Yes, torque and angular momentum can be negative. A negative torque indicates that the force is causing the object to rotate in the opposite direction of a positive torque. Similarly, negative angular momentum means the object is rotating in the opposite direction of a positive angular momentum.

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