8.01SC | Fall 2016 | Undergraduate

# Classical Mechanics

## Week 10: Rotational Motion

« Previous | Next »

### Week 10 Problem Set

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

A uniform massive pulley of radius $$R$$ and moment of inertia about its center of mass $$I_{A}$$ is suspended from a ceiling. An inextensible string of negligible mass is wrapped around the pulley and attached on one end to an object of mass $$m_1$$ and on the other end to an object of mass $$m_2$$.

Assume $$m_2> m_1$$. Find the acceleration of the objects.

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

### Constraints for Rotational Motion about a Fixed Axis

A common type of problem in rotational dynamics involves objects which rotational motion is constrained by the linear motion of other objects. A typical example is when different objects are connected by ropes or ropes passing through pulleys.

Below we discuss the constraint imposed by a rope wrapped around a massive pulley of radius $$R$$ and connected to a hanging block. The resulting rotation of the pulley is related to the translation of the block because we will assume:

1. Non-slip condition: the rope does not slip relative to the pulley.
2. Ideal Rope: massless rope with constant length.

Consider the three points A, B and C in the rope. At time $$t$$, left figure, point A is in contact with the pulley and at an angle $$\theta$$ with respect to the horizontal, point B is the point where the rope detaches from the pulley, and point C is the point in the rope in contact with the block. The length of the portion of the rope between points A and B is the arlclength $$s$$, where $$s = R\theta$$.

If the rope does not slip relative to the pulley, point A must have the same velocity as a point at the rim of the pulley. At time $$t+\Delta t$$ , left figure, the pulley has rotated an angle $$\theta$$ and point A has moved an arc of length $$s$$ at the same rate as a point at the rim. The velocity of point A is has a component tangent to the circle given by $$\displaystyle v_A = \frac{ds}{dt}= R\frac{d\theta}{dt}$$.

At the same time, because the rope has a constant length (ideal rope), when the pulley has rotated an angle $$\theta$$ the string has unwrapped a length $$s$$ (right figure). Consequently, points B and C and all the points in the hanging rope have moved down a length $$s$$. As a result, the velocity of any point in the hanging rope is the same as the one of the block equal to $$\displaystyle \frac{ds}{dt}$$.

In this problem, assumptions 1 and 2, imply that the component of the velocity of the block is:

$$\displaystyle v = \frac{ds}{dt}=R\frac{d\theta}{dt}$$

After taking the derivative with respect to time, the acceleration of the block is equal the tangential component of the acceleration of a point at the rim of the pulley:

$$\displaystyle a = R \alpha$$

« Previous | Next »

« Previous | Next »

« Previous | Next »

« Previous | Next »

Problem Set 10 contains the following problems:

1. Moment of Inertia: Disc and Washer
2. Compound Pulley
3. Suspended Rod
4. Person Standing on a Hill
5. A Cylinder Rolling in a V-Groove
6. A Massive Pulley and a Block on an Incline

« Previous | Next »

« Previous | Next »

A pulley of mass $$\displaystyle m_ p$$, radius $$\displaystyle R$$, and moment of inertia about its center of mass $$\displaystyle I_{cm}$$, is attached to the edge of a table. An inextensible string of negligible mass is wrapped around the pulley and attached on one end to block 1 that hangs over the edge of the table. The other end of the string is attached to block 2 which slides along a table. The coefficient of sliding friction between the table and the block 2 is $$\displaystyle \mu _ k$$. Block 1 has mass $$\displaystyle m_1$$ and block 2 has mass $$\displaystyle m_2$$, with $$\displaystyle m_1>\mu _ k m_2$$. At time $$\displaystyle t=0$$, the blocks are released from rest. At time $$\displaystyle t=t_1$$, block 1 hits the ground. Let $$\displaystyle g$$ denote the gravitational constant.

Find the direction and magnitude of the acceleration of the block 1 hanging over the edge of the table. The positive direction is chosen to be downward.

« Previous | Next »

« Previous | Next »

« Previous | Next »

Fall 2016
Lecture Videos
Problem Sets
Online Textbook