a body of mass 5kg starts rotaing from rest on a circular track of radius 8m after 1mint it towards 5 rotations per30sec find the torque
To find the torque acting on the body, we need to know the moment of inertia and the angular acceleration.
1. Calculate the angular velocity:
- Convert the given rotational speed of 5 rotations per 30 seconds to radians per second.
- Each rotation is equivalent to 2π radians, so 5 rotations is 10π radians.
- The time of 30 seconds can be converted to 0.5 minutes.
- Therefore, the angular velocity (ω) is (10π radians) / (0.5 minutes).
2. Calculate the angular acceleration (α):
- Since the body is starting from rest, its initial angular velocity is zero.
- The final angular velocity can be calculated by taking the difference between the final and initial angular displacements and dividing it by the time.
- In this case, the final angular displacement is 10π radians (5 rotations), and the time is 1 minute.
- Therefore, the angular acceleration (α) is (10π radians) / (1 minute).
3. Find the moment of inertia (I):
- The moment of inertia depends on the shape and mass distribution of the object.
- Since we know the mass of the object (5 kg) and it is rotating on a circular track, we can use the formula for the moment of inertia of a ring.
- The moment of inertia of a ring with mass (m) and radius (r) is given by I = m * r^2.
- In this case, the radius is 8 m, so the moment of inertia (I) is 5 kg * (8 m)^2.
4. Calculate the torque (τ):
- The torque (τ) acting on the body can be calculated using the formula τ = I * α.
- Substitute the values we obtained: torque (τ) = (5 kg * (8 m)^2) * ((10π radians) / (1 minute)).
5. Calculate the torque:
- Perform the calculations to find the torque.
Please note that the final torque will be in the appropriate units according to the given values (e.g., Nm or kgm^2/s^2), depending on the units used in the calculation.