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Afina-wow [57]
3 years ago
5

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 spe

eds ( V₁ : V₂) = 2:1, then the ratio of their time periods (T₁ : T₂) --------- [ Use numerals to write your answer] *
Physics
1 answer:
Leona [35]3 years ago
8 0

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}.

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