Question

Question: Consider an object on a rotating disk a distance r from its centre, held in place on the disk by static friction. Which of the following statements is not true concerning this object? (a) If the angular speed is constant, the object must have constant tangential speed. (b) If the angular speed is constant, the object is not accelerated. (c) The object has a tangential acceleration only if the disk has an angular acceleration. (d) If the disk has an angular acceleration, the object has both a centripetal acceleration and a tangential acceleration. (e) The object always has a centripetal acceleration except when the angular speed is zero.

Short answer

Answer

Among the given options, the statement, which is not true is option (b) if the angular speed is constant, the object is not accelerated.

Step-by-step solution

Defining velocity and acceleration

Velocity of an object is a vector quantity that refers to the rate at which an object changes its position. Its formula is

$$\begin{gathered} \stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}\mathbf{=}\frac{\mathbf{∆}\mathbf{x}}{\mathbf{∆}\mathbf{t}} \\ \stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}\mathbf{=}\mathbf{\text{velocity}} \\ \mathbf{∆}\mathbf{x}\mathbf{=}\mathbf{\text{displacement}} \\ \mathbf{∆}\mathit{t}\mathbf{=}\mathbf{\text{time}} \end{gathered}$$

Acceleration in simple terms is the rate of change of velocity. It’s formula is:

$\stackrel{\mathbf{\rightharpoonup }}{a}\mathbf{=}\frac{\mathbf{∆}\mathbf{x}}{\mathbf{∆}\mathbf{t}}$

It is also a vector quantity.

Detailed explanation for the object having constant angular speed but nonzero acceleration 

An object moving in a circular path undergoes a constant change In the direction of velocity. Change in the direction of velocity is acceleration. It is directed towards centre of path which is called as centripetal acceleration.

$a_c=\frac{v^2}{r}=r{\omega }^2$

Tangential velocity of an object,$v_t=r\omega$

Here ‘$\omega$’is angular velocity.

If ‘$\omega$’ is not constant, object will have both tangential and angular acceleration. So, among the given options the untrue statement is option (b). Even if ‘$\omega$’ is constant, object will still have centripetal acceleration. Thus option (b) is the correct option.