Write an expression for the magnitude of the net torque required for the disc to

Question

Write an expression for the magnitude of the net torque required for the disc to achieve a given rotational speed when starting from rest in terms of M, R, ?, and t.

What is an expression for the force that produces this torque if it is applied to the edge of the disc and in a direction that is tangent to the disc in terms of ? and R?

If, as soon as the angular speed is ? = 23 rad/s, the force is immediately removed and a mass m = 1.7 kg is dropped at r = 0.37 m on the disc and sticks there, what would be the new rotational velocity of the disc in rad/s?

Explanation / Answer

part 1:

moment of inertia about x axis of the disk=0.5*M*R^2

as we know, torque=moemnt of inertia*angular acceleration

hence torque T=0.5*M*R^2*(final angular velocity-initial angular velocity)/time

as initial angular velocity=0 rad/sec,

T=0.5*M*R^2*w/t

part 2:

as we know, torque=force*perpendicular distance

here distance is R.

then force required=T/R

part 3:

initial moment of inertia=0.5*M*R^2=0.5*4.5*0.87^2=1.703 kg.m^2

when a mass of 1.7 kg is dropped at r=0.37 m, final moment of inertia=initial moment of inertia+1.7*0.37^2

=1.703+1.7*0.37^2

=1.9357 kg.m^2

as the external force is removed, angular momentum is conserved.

hence initial moment of inertia*initial angular speed=final moment of inertia*final angular speed

==>1.703*23=1.9357*final angular speed

==>final angular speed=1.703*23/1.9357=20.235 rad/sec

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