A disk is rotating at 50rps (revolution per second) about its fixed axis (delta) with respect to which its moment of inertia is J= 2×10^-2 kg.m^2
A braking couple of moment -0.5N.m is applied to the disk (the sense of rotation of the disk is taken as positive ). The disk will stop after a duration t equals to what?
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To find the time duration (t) it takes for the disk to stop rotating, we can use the concept of angular acceleration and torque on the disk.
1. Angular acceleration (α):
Angular acceleration is defined as the rate of change of angular velocity. It can be calculated using the equation:
α = torque / moment of inertia
In this case, the torque acting on the disk is -0.5 N.m (negative because it opposes the rotation) and the moment of inertia (J) is given as 2×10^-2 kg.m^2. Plugging in the values, we have:
α = -0.5 N.m / (2×10^-2 kg.m^2)
2. Initial angular velocity (ω0):
The initial angular velocity of the disk is given as 50 rps (revolutions per second), which can be converted to radians per second using the equation:
ω0 = 2πf0
Where f0 is the initial frequency given in revolutions per second.
ω0 = 2π × 50
3. Final angular velocity (ω):
The final angular velocity is zero, as the disk stops rotating.
4. Time duration (t):
We can use the following equation to find the time duration (t) it takes for the angular velocity to decrease from ω0 to zero:
ω = ω0 + αt
Since ω is zero, we can solve for t:
0 = ω0 + αt
Substituting the values, we have:
0 = 2π × 50 + (-0.5 N.m / (2×10^-2 kg.m^2)) × t
Solving this equation will give us the time duration (t) it takes for the disk to stop rotating.