Question

Question: 48. In a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis as shown in Figure P6.48. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smooth-walled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of p = 68.0^oabove the horizontal. If the radius of the tub is r = 0.330 m, what rate of revolution is needed?

Short answer

The required rate of revolution of cylindrical tub is $0.835rev/s$.

Step-by-step solution

Centripetal force

If a particle is moving in circular path, there must be some centripetal force acting towards the centre.

The expression for the centripetal force is given by

$F_c=m\frac{v^2}{r}$

HereF_c is the centripetal force,m is the mass of the body andr is the radius of the circular path.

Step 2: Calculate the rate of revolution

The free body diagram for the cylindrical tub containing wet clothes is shown below.

The forces acting on the piece of cloth are

1. Normal force (N) exerted by the tub.

2. Gravitational force $\left(m\overrightarrow{g}\right)$exerted by the Earth.

In order for the piece of cloth to lose contact with the tub, the forces acting toward the centre must be the forces needed for uniform circular motion.

The summation of the forces in x direction is given by

$\begin{array}{c}\sum F_x=mg\sin \theta \\ =m{a}_c\\ =\frac{m{v}^2}{r}\end{array}$

Solve for v,

$v=\sqrt{rg\sin \theta }$

The rate of revolution is given by

$f=\frac{v}{2\pi r}$

Substitute $\sqrt{rg\sin \theta }$ for v in the above equation.

$\begin{array}{c}v=\frac{\sqrt{rg\sin \theta }}{2\pi r}\\ =\frac{1}{2\pi }\sqrt{\frac{g\sin \theta }{r}}\end{array}$

Substitute 9.8 m/s² forg , 68^o for θ and 0.33 m forr in the above equation.

$\begin{array}{c}f=\frac{1}{2\pi }\sqrt{\frac{9.8\sin 68°}{0.33}}\\ =0.835\text{\hspace{0.17em}rev}/\text{s}\end{array}$

Therefore, the required rate of revolution of cylindrical tub is $=0.835\text{\hspace{0.17em}rev}/\text{s}$.