rotational motion

Question

  A uniform disc of mass m = 2 kg and radius R = 0.75 m is suspended horizontally by a wire. Refer to the figure. A force of F = 65 N is applied to the edge of the disc tangentially. As a result, the disc turns and reaches equilibrium position at an angle of θ = 1.4 rad from its initial orientation.

a) determine the value of the torsion constant of the wire, in units of Newton meters per radian.

b) determine the torque in newton meters, that would be required to rotate the disc by one complete rotation

c) if the disc is rotated as described above and then released, at what angular frequency in radians per second will it oscillate?

Answer 1

Let,

m = 2 kg be the mass of the disk,

R = 0.75 m be the radius of the disk,

F be the force applied to the edge (different for questions a and b),

θ be the angle of displacement (different for questions a, b, and c),

k be the torsion constant (to be calculated).

The term "torsion constant" can have different meanings.

Here it apparently means

torsion constant = torque/(angle of displacement)

a)

F = 65 N

θ = 1.4 rad

The torque

τ = RF

The torsion constant

k = τ/θ = RF/θ = 0.75×65/1.4 = 34.82 Nm/rad

b)

θ = 2π

τ = kθ = 34.82×2π = 218.79 Nm

c)

Let

θ(t) be the angle of twist at time t,

α(t) be the angular acceleration at time t,

τ(t) be the torque the wire applies to the disc at time t,

I = mR²/2 be the moment of inertia of the disk.

The disc follows Newton's Second Law applied to rotation:

τ(t) = Iα(t) (1)

The torque τ(t) is proportional to the angle θ(t), but in the opposite direction:

τ(t) = -kθ(t) (2)

Combining (1) and (2)

Iα(t) = -kθ(t)

I do not know what mathematical tools are at your disposal, but this is a

differential equation

d²θ/dt² = -(k/I)θ

This equation has the solution

θ = Asin(2πft + φ),

where the amplitude A and the phase φ are determined from initial conditions, and

the frequency

f = (1/2π)√(k/I) = (1/2π)√(k/(mR²/2)) = (1/2πR)√(2k/m)

Substituting actual numbers

f = (1/(2×π×0.75))√(2×34.82/2) = 1.25 rad/sec