- #1

Karol

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## Homework Statement

A truck with a cylinder of mass M and moment of inertia ##I=kMR^2## on top has initial velocity v

_{0}and decelerates with deceleration B. the cylinder is attached with a rope to a weight m. the coefficient of friction is μ.

The outer radius of the cylinder is R and the smaller is r. The cylinder moves without sliding and there isn't friction between the weight m and the truck.

What is the relation between the angular acceleration of the cylinder and the linear acceleration of the weight.

What are the forces and the moments that act on the weight and the cylinder.

What is the linear acceleration of the cylinder in the reference frame of the truck.

What is the condition on B so that the cylinder will move to the right relative to the truck.

## Homework Equations

Torque and moment of inertia: ##M=I\ddot{\theta}##

## The Attempt at a Solution

The relation between the accelerations:

$$\frac{\ddot{x}}{r}=\frac{\ddot{y}}{r}=\ddot{\theta}$$

The tension in the rope is T. the forces:

$$\left\{ \begin{array}{l} MBR+Tr=kMR^2\cdot \frac{\ddot{\theta}}{r} \\ mg-T=m\ddot{y} \end{array} \right.$$

$$\rightarrow MBR+m\left( g+\ddot{y} \right)r=kMR^2\cdot \frac{\ddot{\theta}}{r}$$

$$\ddot{y}=\frac{r\left( MBR+mgr \right)}{kMR^2+mr^2}$$

The linear acceleration of the cylinder in reference to the truck:

$$\frac{\ddot{x}}{R}=\ddot{\theta}=\frac{\ddot{y}}{r}=\frac{MBR+mgr}{kMR^2+mr^2}$$

The condition on B:

$$\left\{ \begin{array}{l} mg-T=\ddot{y}m=\ddot{\theta}mR\rightarrow T=m\left( g-R\ddot{\theta} \right) \\ Tr-MBR=kMR^2\cdot \ddot{\theta} \end{array}\right.$$

$$\ddot{\theta}=0 \rightarrow B=\frac{mr}{MR}g$$