A turntable turning at rotational speed 21rpm stops in 50s when turned off. The turntable's rotational inertia is 1.2×10−2kg⋅m2.
a)Determine the magnitude of the resistive torque that slows the turntable.
To determine the magnitude of the resistive torque that slows the turntable, you can use the law of rotational motion.
The law of rotational motion states that the torque applied to an object is equal to the moment of inertia times the angular acceleration. In this case, the torque applied to the turntable when it stops rotating is the resistive torque.
The equation for torque is:
τ = Iα
Where:
τ is the torque,
I is the moment of inertia,
α is the angular acceleration.
Since the turntable stops rotating, the final angular velocity is 0. Therefore, the angular acceleration can be calculated using the formula:
α = (final angular velocity - initial angular velocity) / time
First, convert the rotational speed from revolutions per minute (rpm) to radians per second (rad/s). Since 1 revolution is equal to 2π radians, the conversion factor is:
1 rpm = 2π rad/min
To find the initial angular velocity, use the formula:
initial angular velocity = (initial rotational speed) × (2π rad/1 min)
In this case, the initial rotational speed is 21 rpm, so:
initial angular velocity = (21 rpm) × (2π rad/1 min)
Next, plug the values into the formula to calculate the angular acceleration:
α = (0 rad/s - initial angular velocity) / time
Since the final angular velocity is 0 rad/s and the time taken to stop is 50 s, the equation becomes:
α = (0 rad/s - initial angular velocity) / 50 s
Substitute the known values:
α = (0 rad/s - (21 rpm) × (2π rad/1 min)) / 50 s
Now, we have the angular acceleration. To find the resistive torque, we can use the formula:
τ = Iα
Substitute the given moment of inertia (1.2 x 10^(-2) kg·m²) and the calculated angular acceleration (in rad/s²):
τ = (1.2 x 10^(-2) kg·m²) × α
Finally, calculate the magnitude of the resistive torque by taking the absolute value of τ.
Keep in mind that this formula assumes that the resistive torque is constant during the deceleration process.