Question

Two objects ① and ② are moving along two different circular paths of radii 10 m and 20 m respectively. If the ratio of their speeds ( V₁ : V₂) = 2:1, then the ratio of their time periods (T₁ : T₂) --------- [ Use numerals to write your answer] *

Answer 1

Answer:

The ratio of the time period of object 1 to the time period of object 2 is $\frac{1}{4}$. (T₁ : T₂) = 1 : 4

Explanation:

Let suppose that both objects are moving along the circular paths at constant speed, such that period of rotation of each object is represented by the following formula:

$\omega = \frac{2\pi}{T}$

Where:

$\omega$ - Angular speed, measured in radians per second.

$T$ - Period, measured in seconds.

The period is now cleared:

$T = \frac{2\pi}{\omega}$

Angular speed ($\omega$) and linear speed ($v$) are related to each other by this formula:

$\omega = \frac{v}{R}$

Where $R$ is the radius of rotation, measured in meters.

The angular speed can be replaced and the resultant expression is obtained:

$T = \frac{2\pi\cdot R}{v}$

Which means that time period is directly proportional to linear speed and directly proportional to radius of rotation. Then, the following relationship is constructed and described below:

$\frac{T_{2}}{T_{1}} = \left(\frac{v_{1}}{v_{2}}\right)\cdot \left(\frac{R_{2}}{R_{1}} \right)$

Where:

$T_{1}$, $T_{2}$ - Time periods of objects 1 and 2, measured in seconds.

$v_{1}$, $v_{2}$ - Linear speed of objects 1 and 2, measured in meters per second.

$R_{1}$, $R_{2}$ - Radius of rotation of objects 1 and 2, measured in meters.

If $\frac{v_{1}}{v_{2}} = 2$, $R_{1} = 10\,m$ and $R_{2} = 20\,m$, the ratio of time periods is:

$\frac{T_{2}}{T_{1}} = 2\cdot \left(\frac{20\,m}{10\,m} \right)$

$\frac{T_{2}}{T_{1}} = 4$

$\frac{T_{1}}{T_{2}} = \frac{1}{4}$

The ratio of the time period of object 1 to the time period of object 2 is $\frac{1}{4}$.