Question

A chain couples and rotates two wheels in a bicycle. The radii of the bigger and the smaller wheels are 0.5m and 0.1m respectively. The bigger wheel rotates at the rate of 200 rotations per minute. The rate of rotation of the smaller wheel is
a) 1000 rpm
b) 50/3 rpm
c) 200 rpm
d) 40 rpm

Answer 1

The two wheels are coupled to a chain which rotates them. Since the two wheels are connected to a common chain both the wheels will have the same linear velocity along the edges of the two wheels., The linear velocity of the wheel is proportional to the number of rotations per minute of the wheels and its radius. Since the linear velocity of both the wheels is the same, from the expression of the linear velocities of the two wheels we can determine the rpm of the smaller wheel.

Formula used:
$v=r\omega$
$\omega =2\pi f$

Complete step-by-step answer:
The linear velocity of the wheel i.e. (v)is given by $v=r\omega$ where r is the radius of the wheel and $\omega$ is the angular velocity of the wheel. The angular velocity of the wheel with (f) rotations per minute is given by $\omega =2\pi f$ .
The linear velocity of both the wheels is the same. It is given in the question that the wheel with radius R=0.5m rotates with the 200 rotations per minute. Therefore its angular velocity is ${{\omega }_{1}} = 400\pi rad/\sec$. We are asked to find the angular the rotations per minute(f) of the wheel with radius r=0.1m. Let the angular velocity of this wheel be ${{\omega }_{2}}=2\pi f$ Since their linear velocities are the same we can write,
$\begin{aligned} & {{\omega }_{1}}R={{\omega }_{2}}r \\\ & 400\pi \times 0.5=2\pi f\times 0.1 \\\ & f=200\times 5 \\\ & f=1000rpm \\\ \end{aligned}$

So, the correct answer is “Option a”.

Note: In the above question we have said that the two wheels rotate with the same linear velocity. This can be better understood with the example. Consider two wheels of one bigger and the other smaller both connected to a chain. When the wheels move forward by the time the bigger wheel completes one complete rotation the smaller wheel will move n times its circumference with respect to the bigger wheel. But the distance travelled by both of them is the same. From this we can clearly understand why the angular velocity is different and the linear velocity of the two wheels is the same.