A record player turn table isof mass 2kg and 34cm diameter . From an initial speed of 3.5rad/sec, it makes 3 complete revolutions before it comes to rest when it is switched off. Calculate
1. Angulr momentum
2. The time taken for it to come to rest after it is switched off.
3. Moment of inertia of the turn table about it's axis.
4. The torque, acting on the turn table, after it is switched off
To solve this problem, we need to consider the conservation of angular momentum and the equations of rotational motion.
1. Angular momentum is given by the equation:
Angular momentum = moment of inertia * angular velocity
The moment of inertia of a solid disc about its axis is given by:
moment of inertia = (1/2) * mass * radius^2
Thus, the moment of inertia of the turntable is:
moment of inertia = (1/2) * 2 kg * (0.17 m)^2 = 0.0578 kg·m^2
The angular momentum can be calculated using the equation:
angular momentum = moment of inertia * angular velocity
angular momentum = 0.0578 kg·m^2 * 3.5 rad/s = 0.202 kg·m^2/s
2. To calculate the time taken for the turntable to come to rest, we can use the equation of rotational motion:
final angular velocity = initial angular velocity - (torque / moment of inertia) * time
When the turntable comes to rest, the final angular velocity is 0 rad/s. Rearranging the equation, we get:
time = (initial angular velocity) / ((torque / moment of inertia))
Since the torque acting on the turntable is unknown, we need to find it first.
3. The moment of inertia of the turntable about its axis is already calculated in part 1:
moment of inertia = 0.0578 kg·m^2
4. To find the torque acting on the turntable, we can use the equation:
torque = moment of inertia * angular acceleration
Since the turntable comes to rest, the final angular velocity is 0 rad/s. The initial angular velocity is 3.5 rad/s. Thus, the angular acceleration can be calculated using the equation:
angular acceleration = (final angular velocity - initial angular velocity) / time
Since the final angular velocity is 0 rad/s, the angular acceleration becomes:
angular acceleration = (0 - 3.5 rad/s) / time = -3.5 rad/s^2
Substituting this value into the torque equation, we get:
torque = moment of inertia * angular acceleration
torque = 0.0578 kg·m^2 * (-3.5 rad/s^2)
torque = -0.202 kg·m^2/s^2
Now that we have the torque, we can calculate the time taken for the turntable to come to rest. Substituting the values into the equation obtained in part 2:
time = (initial angular velocity) / ((torque / moment of inertia))
time = 3.5 rad/s / ((-0.202 kg·m^2/s^2) / (0.0578 kg·m^2))
time = 10.87 s
Therefore, the answers are:
1. Angular momentum = 0.202 kg·m^2/s
2. The time taken for it to come to rest after it is switched off is 10.87 seconds.
3. Moment of inertia of the turntable about its axis = 0.0578 kg·m^2
4. The torque, acting on the turntable after it is switched off, is -0.202 kg·m^2/s^2.