The constant torque required to accelerate a disk (of rotational inertia I) from rest to a speed of f rotations per second in a time interval of t is:
To find the constant torque required to accelerate a disk from rest to a speed of f rotations per second in a time interval of t, we can use the rotational version of Newton's second law, which states that the torque applied to an object is equal to the moment of inertia times the angular acceleration.
The moment of inertia (I) of a disk is a property that depends on its mass distribution and shape. For a solid disk rotating about its central axis, the moment of inertia is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the disk and r is the radius of the disk.
Angular acceleration (α) can be obtained by dividing the change in angular velocity (Δω) by the time interval (t):
α = Δω / t
In this case, the change in angular velocity is given by:
Δω = (2πf) - 0
since the disk starts from rest.
So, the angular acceleration becomes:
α = (2πf) / t
Finally, we can calculate the torque (τ) required to achieve this angular acceleration:
τ = I * α
Substituting the expressions for I and α, we get:
τ = ((1/2) * m * r^2) * ((2πf) / t)
This gives you the constant torque required to accelerate a disk from rest to a speed of f rotations per second in a time interval of t.