Question

A particle makes 240 revolution per minutes on a circular radius of 2m find: (I) it's period (ii) it's angular velocity (III) it's linear velocity and (iv) it's acceleration

Answer 1

(i) The period is 0.25 seconds, (ii) The angular velocity is approximately 25.133 radians per second, (iii) The linear velocity of the particle is approximately 50.266 meters per second, (iv) The acceleration experimented by the particle is approximately 1263.336 meters per square second.

Step-by-step explanation:

(i) Let suppose that particle experiments an uniform circular motion. The period (
`T`) of the particle, measured in seconds, is the time needed to complete a revolution, which is determined by this:

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

Where
`\omega` is the angular speed, measured in radians per second.

Let is convert angular speed from revolutions per minute into radians per second:

`\omega = \left(240\,(rev)/(min) \right)* \left((1)/(60)\,(min)/(s) \right)* \left(2\pi\,(rad)/(rev) \right)`

`\omega \approx 25.133\,(rad)/(s)`

The period of the particle is:

`T = (2\pi)/(25.133\,(rad)/(s) )`

`T \approx 0.25\,s`

The period is 0.25 seconds.

(ii) The angular velocity is approximately 25.133 radians per second.

(iii) The linear velocity of the particle (
`v`), measured in meters per second, is found by this expression:

`v = R\cdot \omega`

Where:

`R` - Radius, measured in meters.

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

If
`R = 2\,m` and
`\omega \approx 25.133\,(rad)/(s)`, the linear velocity of the particle is:

`v = (2\,m)\cdot \left(25.133\,(rad)/(s) \right)`

`v \approx 50.266\,(m)/(s)`

The linear velocity of the particle is approximately 50.266 meters per second.

(iv) As particle rotates at constant speed, resultant acceleration (
`a`) is entirely radial and modelled by this expression:

`a = \omega^(2)\cdot R`

`a = \left(25.133\,(rad)/(s) \right)^(2)\cdot (2\,m)`

`a \approx 1263.336\,(m)/(s^(2))`

The acceleration experimented by the particle is approximately 1263.336 meters per square second.