Chapter 6.2, Problem 43E

a.

To determine

To find: the direction in which rollers $B$ and C turns.

Answer to Problem 43E

Direction of roller B = Clockwise

Direction of roller C = Counter-clockwise

Explanation of Solution

Given information:

Roller A turns counter-clockwise.

The side view of three rollers tangent to each other is shown below in the figure.

  

Calculation:

The direction of rotation of roller A will be opposite to that of roller B but same as that of roller C . So, if roller A turns counter-clockwise, then the roller B will turn clockwise and roller C will turn in counter-clockwise direction.

b.

To determine

To find: the angular velocity (in rpm) of rollers $B$ and C .

Answer to Problem 43E

Roller B : $180\text{ rpm}$

Roller C : $\text{75 rpm}$

Explanation of Solution

Given information:

The side view of three rollers tangent to each other is shown below in the figure.

  

Angular velocity of roller A $\left(\omega \right)=120\text{ revolutions per minute}$

Calculation:

The radius of roller A , B and C given in the figure is 3.0cm, 2.0 cm, and 4.8 cm respectively.

The linear velocity of all the rollers will be same which can be evaluated as follows:

  $\begin{array}{c}v=r\omega \\ =\left(3\right)\left(120\right)\left[\text{Put }r=3\text{ cm for roller }A\right]\\ =360\text{ rpm}\end{array}$

Thus the linear velocity of all the three rollers is 360 rpm.

The angular velocity $\left({\omega }_B\right)$ of roller B is evaluated as follows:

  $\begin{array}{c}{v}_B=r_B{\omega }_B\\ 360=\left(2\right){\omega }_B\left[\text{Put }{v}_B=360\text{ rpm and }{r}_B=\text{2 cm}\right]\\ {\omega }_B=\frac{360}{2}\\ =180\text{ rpm}\end{array}$

Thus the angular velocity of roller B is 180 rpm.

The angular velocity $\left({\omega }_C\right)$ of roller C is evaluated as follows:

  $\begin{array}{c}{v}_C=r_C{\omega }_C\\ 360=\left(4.8\right){\omega }_C\left[\text{Put }{v}_C=360\text{ rpm and }{r}_C=4.8\text{ cm}\right]\\ {\omega }_C=\frac{360}{4.8}\\ =75\text{ rpm}\end{array}$

Thus the angular velocity of roller C is 75 rpm.