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

The ratio of the time period of object 1 to the time period of object 2 is
`(1)/(4)`. (T₁ : T₂) = 1 : 4

Step-by-step 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 = (2\pi)/(T)`

Where:

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

`T` - Period, measured in seconds.

The period is now cleared:

`T = (2\pi)/(\omega)`

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

`\omega = (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 = (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:

`(T_(2))/(T_(1)) = \left((v_(1))/(v_(2))\right)\cdot \left((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
`(v_(1))/(v_(2)) = 2`,
`R_(1) = 10\,m` and
`R_(2) = 20\,m`, the ratio of time periods is:

`(T_(2))/(T_(1)) = 2\cdot \left((20\,m)/(10\,m) \right)`

`(T_(2))/(T_(1)) = 4`

`(T_(1))/(T_(2)) = (1)/(4)`

The ratio of the time period of object 1 to the time period of object 2 is
`(1)/(4)`.