Help with rotational motion work

[Heisenburger]

Homework Statement

A billiards ball of mass M is initially motionless on a table when it is hit by a cue projecting it forward with speed V and no angular velocity. Find the speed of the ball when it eventually begins to roll. Assume the ball does not slip when it begins to roll. What proportion of the original kinetic energy is lost in the process? (The ball’s moment of inertia is 2Ma^2/5.)

Homework Equations

K=0.5mv^2
K=Iω^2
v=ωr

The Attempt at a Solution

k before = 0.5mv^2
after = Iw^2, where w=v/r

so proportion of energy lost = energy lost/original energy

=0.5mv^2-Iw^2 all over 0.5mv^2?

= 1- $\frac{2a^2}{5r^2}$

 

[Doc Al]

I assume they mean for you to find when the ball begins rolling without slipping. How can you find that final velocity of the ball's center of mass?

 

[Heisenburger]

I assume they mean for you to find when the ball begins rolling without slipping. How can you find that final velocity of the ball's center of mass?

I'm not sure, is it not the same as the start? or do i take into account energy lost before this?

 

[Doc Al]

I'm not sure, is it not the same as the start?

How can it be? What causes the ball to rotate is friction, which slows the translational speed as it increases the rotational speed. In the process, mechanical energy is lost.

or do i take into account energy lost before this?

Start by figuring out the final velocity, which will be less than the initial velocity. (There are several ways of doing this.)

 

[Nickie]

To solve this problem, we can use the conservation of energy principle, which states that energy cannot be created or destroyed, only converted from one form to another. Initially, the ball has only translational kinetic energy, given by K=0.5mv^2, since it is not rotating. When it begins to roll, it gains rotational kinetic energy, given by K=Iω^2, where I is the moment of inertia and ω is the angular velocity. We can also use the relationship v=ωr, where r is the radius of the ball, to relate the linear and angular velocities.

Using these equations, we can determine the final speed of the ball when it begins to roll. Since the ball does not slip, we can equate the linear and angular velocities at this point: v=ωr. Substituting this into the equation for rotational kinetic energy, we get K=I(v/r)^2=Iv^2/r^2. Combining this with the translational kinetic energy, we can set up an equation to solve for the final speed v:

0.5mv^2+Iv^2/r^2=Ktotal

This equation can be solved for v, giving us the final speed of the ball when it begins to roll. To determine the proportion of energy lost in the process, we can compare the initial kinetic energy (K=0.5mv^2) to the final kinetic energy (K=Iv^2/r^2). This will give us the proportion of energy lost, which is equal to 1-Iv^2/(0.5mv^2).

Substituting in the given values for the moment of inertia (I=2Ma^2/5) and the relationship v=ωr, we can simplify this equation to:

Proportion of energy lost = 1-2a^2/(5r^2)

This equation tells us that the proportion of energy lost depends on the ratio of the moment of inertia to the radius squared. This means that a larger moment of inertia or a smaller radius will result in a greater proportion of energy lost in the process. Additionally, the final speed of the ball will also depend on these factors.

In conclusion, to find the final speed of the ball when it begins to roll and the proportion of energy lost in the process, we can use the equations for kinetic energy and the conservation of energy principle. By considering the moment