Question

A 8.0-cm radius disk with a rotational inertia of 0.12 kg ·m2 is free to rotate on a horizontal

axis. A string is fastened to the surface of the disk and a 10-kgmass hangs from the other end.
The mass is raised by using a crank to apply a 9.0-N·mtorque to the disk. The acceleration of
the mass is:
A. 0.50m/s2
B. 1.7m/s2
C. 6.2m/s2
D. 12m/s2
E. 20m/s2

Answer 1

The acceleration of the mass is found to be 0.8 m/s^2 after calculating the net force on the mass using torque, the rotational inertia of the disk, and the gravitational force.

Step-by-step explanation:

To find the acceleration of the hanging mass, we first need to use the torque to find the angular acceleration of the disk, since torque (τ) is equal to the product of the rotational inertia (I) and the angular acceleration (α), τ = Iα. The torque given is 9.0 N·m, and the rotational inertia is 0.12 kg·m2.

To find α, we rearrange the equation to α = τ / I, which gives us:

α = 9.0 N·m / 0.12 kg·m2 = 75 rad/s2

Using the relationship between angular acceleration and linear acceleration (a = αr, where r is the radius), and knowing the radius (r = 0.08 m) we can calculate:

a = αr = 75 rad/s2 × 0.08 m = 6 m/s2

Since the only force acting on the mass besides the torque is gravity, we must calculate its effect. The force due to gravity (Fg) is:

Fg = m·g = 10 kg × 9.8 m/s2 = 98 N

The net force (Fnet) on the mass is the difference between the upward torque-induced force and gravity:

Fnet = Iα/r - Fg = (0.12 kg·m2 × 75 rad/s2) / 0.08 m - 98 N = 90 N - 98 N = -8 N

Finally, the linear acceleration (a) is:

a = Fnet/m = -8 N / 10 kg = -0.8 m/s2

Since the negative sign indicates the direction opposite to the positive torque, the actual acceleration of the mass is:

a = 0.8 m/s2

Answer 2

The acceleration of the mass cannot be determined with the information provided.

Step-by-step explanation:

The acceleration of the mass can be calculated using the formula:

a = ∂l / I

Where a is the acceleration, ∂l is the change in angular momentum, and I is the rotational inertia.

Given that the torque is 9.0 N·m and the rotational inertia is 0.12 kg·m², we can calculate the change in angular momentum:

∂l = torque * time

= 9.0 N·m * time

Now, we need to determine the time it takes for the mass to be raised. Since the question does not provide this information, we cannot determine the exact acceleration. Therefore, we cannot determine the answer from the options given.