Simple yo-yo work problem

Question

I'm taking an introduction to modern physics course and would appreciate some help on this problem:

A yo-yo is attached to a string, whose end is held tightly. It is released from rest.

1. After the yo-yo has fallen a distance h, find the total kinetic energy of the yo-yo.

2. Find the speed of the yo-yo after it has fallen a distance h.

3. Find the magnitude of the tension in the string while the yo-yo is falling.

Attempt at a solution:
$$\Delta KE = KE_f-KE_i=KE_f$$ Going from a point particle system, $$\Delta KE_{trans} = W_{gravity}=mgh$$ But I am not sure this is true because the string also does work on the yo-yo, but I don't know that force.

Answer 1

Since energy is conserved, the total kinetic energy after the yo-yo has fallen a distance $h$ is simply the initial energy, $E_1=mgh$. However, gravitational potential energy is converted into two forms of kinetic energy: translational kinetic energy ($\frac{1}{2}mv^2$) and rotational kinetic energy ($\frac{1}{2}I\omega^2$). Here, $I$ is the moment of inertia of the yo-yo, and $\omega$ is the angular velocity.

Now, since the rope is always tangent to the yo-yo, the speed of the rope is simply the tangential velocity of the yo-yo, $v=\omega r$. Thus you can use conservation of energy $E_1=E_2$, where $E_2=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2$ to determine $v$.

Finally, to find the tension, apply Newton's laws $F=ma$, and $\tau=I\alpha,$ with $\alpha$ the angular acceleration. Both the linear force equation and the torque equation apply! This will allow you to determine the tension in the rope in terms of the mass, radius of the yo-yo, etc.