Question

A uniform, solid metal disk of mass 6.00 kg and diameter 29.0 cm hangs in a horizontal plane, supported at its center by a vertical metal wire. You find that it requires a horizontal force of 4.28 N tangent to the rim of the disk to turn it by 3.32 ∘, thus twisting the wire. You now remove this force and release the disk from rest.

Answer 1

To calculate the angular velocity of the solid metal disk, we can use the equation for rotational motion: τ = Iα. We are given the torque applied to the disk (4.28 N) and the angle it is turned (3.32°). Using the formulas for moment of inertia and angular acceleration, we can calculate the final angular velocity.

Step-by-step explanation:

To calculate the angular velocity of the solid metal disk, we can use the equation for rotational motion:

τ = Iα

Where τ is the torque applied to the disk, I is the moment of inertia of the disk, and α is the angular acceleration of the disk.

We are given the torque applied to the disk (4.28 N) and the angle it is turned (3.32°). We can calculate the moment of inertia of the disk using the formula:

I = (1/2)mr^2

Where m is the mass of the disk and r is the radius of the disk.

Using the given diameter (29.0 cm), we can calculate the radius (14.5 cm or 0.145 m).

Plugging in the values into the formulas, we can solve for the angular acceleration:

α = τ / I

Finally, we can use the equation for angular motion to find the angular velocity:

ω = ω0 + αt

Since the disk is released from rest, the initial angular velocity (ω0) is 0. Plugging in the values, we can solve for the final angular velocity.

Answer 2

k = 21.42 Nm / rad

Step-by-step explanation:

The torque is related to the torsion constant (k) by the equation:

τ = - k*θ

We know that when we apply the force 4.28 N the system is in equilibrium with a twist angle

θ = 3.32° = 0.0579 rad.

The torque in this case is τ = R*F = 0.29m*4.28N = 1.2412 Nm.

Then

k = τ / θ

⇒ k = (1.2412 Nm) / (0.0579 rad) = 21.42 Nm / rad