Question

A bicycle with tires $68cm$ in diameter travels $9.2km$. How many revolutions do the wheels make?

Answer 1

Hint: A tire moves in circular motion as well as in a straight line. It moves by making revolutions. In one revolution, it completes one full angle. To calculate the number of revolutions, we calculate its circumference and divide the total distance covered by the circumference.
Formulas used:
$C=2\pi r$
$n=\dfrac{9.2\times {{10}^{5}}cm}{C}$

Complete step-by-step solution:
A circle covers a full angle of $2\pi $ in one revolution. Also, it will move across its circumference to complete one revolution.
The circumference of a circle is calculated as
$C=2\pi r$ - (1)
Here, $C$ is the circumference of the circle
$r$ is the radius of the circle
Given, the diameter of the tires is $68cm$, so the radius will be-
$\begin{align}
  & r=\dfrac{d}{2} \\
 & \Rightarrow r=\dfrac{68}{2}=34cm \\
\end{align}$
Substituting the value of radius in eq (1), we get,
$\begin{align}
  & C=2\pi r \\
 & \Rightarrow C=2\times 3.14\times 34 \\
 & \therefore C=213.52cm \\
\end{align}$
Therefore, the circumference of the tires is $213.52cm$. So, the tires cover $213.52cm$ in one revolution.
Number of revolutions covered in $9.2km$ is-
$n=\dfrac{9.2\times {{10}^{5}}cm}{C}$
Here, $n$ is the number of revolutions
$\begin{align}
  & \Rightarrow n=\dfrac{9.2\times {{10}^{5}}cm}{213.52} \\
 & \therefore n=4308.73 \\
\end{align}$
Therefore, the total number of revolutions made by the bicycle to cover $9.2km$ is $4308.73$.
Additional information: When an object moves in a circular motion, it covers angular displacement in radians and moves with an angular velocity which is angular displacement covered in unit time. The parameters which describe circular motion are analogous to the parameters of motion in a straight line. The centripetal force acts on an object moving with a circular motion.

Note: The number of revolutions made in one second is called the frequency and the time taken to complete one frequency is called the time period. The wire follows circular as well as translational motion. Convert the units as required. The tires cover its circumference in one revolution.