- #1

- 1,380

- 22

## Homework Statement

1) What are the COM and angular velocities and accelerations at B.

2) What is the tension T

_{d}in the wire during the descend?

3) What is the tension T

_{u}in the wire during the ascend?

4) On a time, a boy pulls the string. at what angular acceleration will the yo-yo rotate if the COM stays in place?

5) At which pace must the boy pull the string in order to keep the COM in place.

6) If the string's length is l, how long can this situation last?

## Homework Equations

Kinetic energy of a solid body: ##E_k=\frac{1}{2}I\omega^2##

Shteiner's theorem: ##I_c=I_{c.o.m.}+Mr^2##

Moment of inertia of a ball round it's center: ##I_{cen}=\frac{2}{5}mr^2##

Torque and angular acceleration: ##M=I\alpha##

In accordance with motion with constant acceleration: ##\omega^2=2\dot\omega\theta##

3. The Attempt at a Solution

3. The Attempt at a Solution

During descent and at B:

$$r\cdot mg=(I_c+mr^2)\dot\omega~~\rightarrow~~\dot\omega=\frac{mgr}{I_c+MR^2}$$

The center's acceleration: ##\dot y=\dot\omega r=\frac{mgr^2}{I_c+MR^2}##

The total angle m has traveled along the string's length l is ##\theta=\frac{l}{2\pi r}2\pi=\frac{l}{r}##

The angular velocity at B:

$$\omega_B^2=2\dot\omega_B\theta=\frac{2mlg}{I_c+mr^2}$$

The linear velocity: ##v_B=\omega_B\cdot r##

The tension T

_{d}during descent:

$$mg-T_d=m\dot y=mr\dot\omega~~\rightarrow~~T_d=mg\left( 1-\frac{mr^2}{I_c+mr^2} \right)$$

The tension T

_{u}during ascend is the same since the force-acceleration equation ##mg-T_d=m\dot y## is the same.

4) In order to keep the center in place the boy must pull with a force F that equals: ##F=mg## , so the torque round the center is:

$$F\cdot r=mg\cdot r=I_c\dot\omega~~\rightarrow~~\dot\omega=\frac{mgr}{I_c}$$

5) The boy must pull the string with acceleration of ##\dot y=\dot\omega r##

6) The string's length is l and from kinematics:

$$l=\frac{1}{2}\dot\ y t^2=\frac{1}{2}\frac{mgr^2}{I_c}t^2~~\rightarrow~~t^2=\frac{2lI_c}{mgr^2}$$