Module 5 -- Applying Rotational Form of Newton's Second Law

Learning goals

After working through this module, you should be able to:

• Solve problems with objects rotating around a fixed axis.
• Relate rotational quantities (θ, ω and α) to linear quantities (s, v and atan)
• Apply the constraints in rotational motion imposed by strings of constant length and that do not slip relative to the pulley (non-slip condition).

Applying Constrains When Solving Problems with the Rotational Form of Newton's Second Law

A common type of problems 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 strings passing through pulleys.

Below we discuss the constraint imposed by a string 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. Ideal string of constant length
2. Non-slip condition, the string does not slip relative to the pulley.

If the string does not slip relative to the pulley, then the points of contact between the string and pulley have the same speed as the speed of a point at the edge of the pulley. In addition, if the length is constant then all the points in the string are moving at the same speed, therefore the block will move at the speed of a point at the ede of the pulley.

This is shown in the figure above. The arc length x covered by a point at the rim of the pulley (left figure) is the same as the length of the string that is displaced down by the block (right figure). Because the string does not slip relative to the pulley then x = Rθ. Taking the derivative, dx/dt = Rdθ/dt, we show that the speed of the block is the same as the speed of a point at the rim of the pulley.

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

 v = R ω

and taking the derivative with respect to time the acceleration of the block is the tangential acceleration of a point at the rim:

 atan = Rα.

The last relationship is particularly important, since it relates the acceleration of the block to the angular acceleration of the pulley.

The following example illustrates how the constraints of constant string length and non-slipping relative to pulley are applied to problems.

For a review of the relationship between the linear quantities (ds, v, a) of a point in a rigid object and the angular quantities (dθ, ω and α is presented at the end of this module.

Illustrative Example: Down the Well

A bucket for collecting water from a well is suspended by a rope which is wound around a pulley. The empty bucket has a mass of 2.0 kg, and the pulley is essentially a uniform cylinder of mass 3.0 kg on a frictionless axle. Suppose a person drops the bucket (from rest) into the well.

Review

A summary of the relationship between the linear and the angular quantities is presented below.