A disk with mass m = 5.2 kg and radius R = 0.44 m hangs from a rope attached to the ceiling. The disk spins on its axis at a distance r = 1.59 m from the rope and at a frequency f = 18.6 rev/s (with a direction shown by the arrow).

Question

1. What is the magnitude of the angular momentum of the spinning disk?
2. What is the torque due to gravity on the disk?
3. What is the period of precession for this gyroscope?

I know I'm not just supposed to flat out ask questions I'm supposed to have some work but i was hopping that someone on here could teach me step by step on how to solve it? So that I further understand it.
Thanks!

Answer 1

1. To find the magnitude of the angular momentum of the spinning disk, we first have to calculate the disk's moment of inertia, and then the angular velocity.

Moment of inertia (I) for a disk is given by the formula:

I = (1/2) * m * R^2

where m is the mass of the disk and R is its radius. Plugging in the given values, we get:

I = (1/2) * 5.2 kg * (0.44 m)^2 ≈ 0.52 kg*m^2

Now we need to calculate the angular velocity (ω) from the given frequency (f):

ω = 2 * π * f

Plugging in the given frequency, we get:

ω ≈ 2 * π * 18.6 rev/s ≈ 116.79 rad/s

Now we have both the moment of inertia and the angular velocity, so we can calculate the angular momentum (L):

L = I * ω

L ≈ 0.52 kg*m^2 * 116.79 rad/s ≈ 60.73 kg*m^2/s

So the magnitude of the angular momentum of the spinning disk is approximately 60.73 kg*m^2/s.

2. To find the torque due to gravity on the disk, we need to compute the force of gravity acting on the center of mass of the disk and the lever arm.

The force of gravity (F_g) acting on the disk is given by:

F_g = m * g

where m is the mass of the disk, g is the acceleration due to gravity (approximately 9.81 m/s^2). Plugging in the given mass, we get:

F_g ≈ 5.2 kg * 9.81 m/s^2 ≈ 51 kg*m/s^2

Now we have to find the lever arm (d) that's perpendicular to the gravitational force. The lever arm is given by:

d = r * sin(θ)

where r is the distance from the rope to the axis of the spinning disk, and θ is the angle between the vertical (gravity's direction) and the line connecting the rope to the axis. Since the disk is hanging from the rope and spinning on its axis, this angle is 90°. So sin(90°) = 1:

d = r * 1 = r = 1.59 m

Now we can calculate the torque (τ) due to gravity:

τ = F_g * d

τ ≈ 51 kg*m/s^2 * 1.59 m ≈ 81.09 kg*m^2/s^2

So the torque due to gravity on the disk is approximately 81.09 kg*m^2/s^2.

3. To find the period of precession (T) for the gyroscope, we need to use the equation:

T = (4 * π^2 * I)/(τ * ω)

Plugging in the values calculated previously for I, τ, and ω, we get:

T ≈ (4 * π^2 * 0.52 kg*m^2) / (81.09 kg*m^2/s^2 * 116.79 rad/s) ≈ 0.0107 s

So the period of precession for this gyroscope is approximately 0.0107 seconds.