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A disc has mass 'M" and radius 'R'. How ...

A disc has mass 'M" and radius 'R'. How much tangential force should be applied to the rim of the disc so as to rotate with angular velocity `omega` in time 't'?

A

`(MR omega)/(4t)`

B

`(MR omega)/(2t)`

C

`(MR omega)/t`

D

`MR omega t`

Text Solution

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The correct Answer is:
To solve the problem of how much tangential force should be applied to the rim of a disc to achieve a certain angular velocity in a given time, we can follow these steps: ### Step-by-Step Solution: 1. **Identify the Given Parameters**: - Mass of the disc, \( M \) - Radius of the disc, \( R \) - Desired angular velocity, \( \omega \) - Time period to achieve this angular velocity, \( t \) 2. **Understand the Relationship Between Angular Velocity and Angular Acceleration**: - Angular acceleration \( \alpha \) can be defined as the change in angular velocity over time: \[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega - 0}{t} = \frac{\omega}{t} \] 3. **Calculate the Moment of Inertia (I) of the Disc**: - The moment of inertia \( I \) for a solid disc about its central axis is given by: \[ I = \frac{1}{2} M R^2 \] 4. **Relate Torque to Angular Acceleration**: - The torque \( \tau \) acting on the disc can be expressed as: \[ \tau = I \alpha \] - Substituting the expression for \( I \) and \( \alpha \): \[ \tau = \left(\frac{1}{2} M R^2\right) \left(\frac{\omega}{t}\right) = \frac{1}{2} M R^2 \cdot \frac{\omega}{t} \] 5. **Express Torque in Terms of Tangential Force**: - Torque can also be expressed in terms of the tangential force \( F_T \) applied at the radius \( R \): \[ \tau = F_T \cdot R \] 6. **Set the Two Expressions for Torque Equal**: - Equating the two expressions for torque: \[ F_T \cdot R = \frac{1}{2} M R^2 \cdot \frac{\omega}{t} \] 7. **Solve for Tangential Force \( F_T \)**: - Rearranging the equation to solve for \( F_T \): \[ F_T = \frac{1}{2} \frac{M R \omega}{t} \] ### Final Result: The tangential force \( F_T \) that should be applied to the rim of the disc to achieve the desired angular velocity \( \omega \) in time \( t \) is: \[ F_T = \frac{1}{2} \frac{M R \omega}{t} \]
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