Question

Suppose a disk rotates at constant angular velocity. (a) Does a point on the rim have radial and or tangential acceleration? (b) If the disk’s angular velocity increases uniformly, does the point have radial and or tangential acceleration? (c) For which cases would the magnitude of either component of linear acceleration change?

Short answer

(a)A point on the rim has only radial acceleration.It does not have tangential acceleration.

(b)A point on the rim has tangential and radial acceleration.

(c)For case (a), none of the linear acceleration components are changing.For case (b),only the radial component of the acceleration changes.

Step-by-step solution

Understanding the rotation of a disk

The disk rotates with some angular velocity. The angular acceleration of the point on the rim is zero.When the angular velocity of the disk increases, the disk has both components of acceleration. If the angular velocity of the disk increases, the angular acceleration remains constant.

(a) Determination of the acceleration of a point on the rim when the disk rotates with constant angular velocity 

The radial acceleration or centripetal acceleration can be expressed as:

$\begin{array}{rl}{a}_r& =a_c\\ & ={\omega }^2\times r\\ & =\frac{v^2}{r}\end{array}$

Here,$\omega$is the angular velocity and$r$is the radius of the disk.

The tangential acceleration can be expressed as:

$\begin{array}{rl}{a}_t& =\frac{dv}{dt}\\ & =\frac{d\left(\omega \times r\right)}{dt}\\ & =r\times \frac{d\omega }{dt}\\ & =0\end{array}$

Here, the tangential acceleration is zero as the rate of change of angular velocity is zero.

The disk is rotating with constant angular velocity. Therefore, a point on the rim has zero tangential acceleration.

The velocity changes in the same direction in which the disk is rotating. It has only radial acceleration that is the centripetal acceleration which generally acts in a radially inward direction.

Thus, a point on the rim has only radial acceleration. It does not have tangential acceleration.

(b) Determination of the acceleration of a point on the rim when the angular velocity of the disk increases 

The angular velocity of the disk starts increasing uniformly. It has constant angular acceleration. Therefore, a point on the rim has both accelerations.

Thus, a point on the rim has tangential and radial acceleration.

(c) Determination of the cases in which the magnitude of acceleration changes  

In case (a), when the disk rotates with constant angular velocity, none of the linear acceleration components change. Both tangential and radial acceleration remain constant.

Thus, in case (a), none of the linear acceleration components change.

In case (b), when the disk rotates with increasing angular velocity, the value of radial acceleration varies. However, the other component, namely tangential acceleration, remains constant having some finite value.

Thus, in case (b), only the radial component of the acceleration changes.