A disk with mass m = 5.8 kg and radius R = 0.47 m hangs from a rope attached to the ceiling. The disk spins on its axis at a distance r = 1.46 m from the rope and at a frequency f = 18.7 rev/s (with a direction shown by the arrow).
a. What is the magnitude of the angular momentum of the spinning disk?
b. What is the torque due to gravity on the disk?
c. What is the period of precession for this gyroscope?
d. What is the direction of the angular momentum of the spinning disk at the instant shown in the picture? (choose correct answer)
(i) up
(ii) down
(iii) left
(iv) right
e. What is the direction of the precession of the gyroscope? (choose correct answer)
(i) it does not precess
(ii) clockwise as seen from above (looking down the rope)
(iii) counterclockwise as seen from above (looking down the rope)
To answer these questions, we can use the following formulas:
1. Angular momentum (L): L = Iω, where I is the moment of inertia and ω is the angular velocity.
2. Moment of inertia (I) for a disk rotating about its central axis: I = (1/2)mr², where m is the mass of the disk and r is the radius.
3. Torque (τ): τ = Iα, where α is the angular acceleration.
4. Torque due to gravity (τ_gravity): τ_gravity = mgrsinθ, where m is the mass of the disk, g is the acceleration due to gravity, r is the distance from the axis of rotation to the center of mass of the disk, and θ is the angle between the rope and the direction of gravity.
5. Period of precession (T_p): T_p = 2π / ω_p, where ω_p is the angular velocity of precession.
Now let's solve each part of the question:
a. To find the magnitude of the angular momentum, we need to find the moment of inertia (I) and the angular velocity (ω). Given the mass (m = 5.8 kg) and radius (R = 0.47 m) of the disk, we can calculate the moment of inertia using the formula mentioned earlier: I = (1/2)mr². Plugging in the values, we get:
I = (1/2)(5.8 kg)(0.47 m)² = 0.515 kg·m²
The angular velocity (ω) is given as 18.7 rev/s. To convert this to rad/s, we multiply by 2π (since 1 rev = 2π rad):
ω = 18.7 rev/s * 2π rad/rev ≈ 117.42 rad/s
Now we can calculate the angular momentum using L = Iω:
L = (0.515 kg·m²) * (117.42 rad/s) ≈ 60.25 kg·m²/s
Therefore, the magnitude of the angular momentum of the spinning disk is approximately 60.25 kg·m²/s.
b. To find the torque due to gravity, we can use the formula τ_gravity = mgrsinθ. Since the disk hangs vertically from the rope, the angle θ between the rope and the direction of gravity is 90 degrees. The mass of the disk (m) is given as 5.8 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s². The distance from the axis of rotation to the center of mass of the disk (r) is given as 1.46 m. Plugging in the values, we get:
τ_gravity = (5.8 kg) * (9.8 m/s²) * (1.46 m) = 99.37 N·m
Therefore, the torque due to gravity on the disk is approximately 99.37 N·m.
c. To find the period of precession, we need to find the angular velocity of precession (ω_p). Using the formula T_p = 2π / ω_p, we can rearrange it to find ω_p = 2π / T_p. The period of precession (T_p) is not given directly in the problem, so we cannot calculate the exact value. However, we can say that for a gyroscope, the period of precession is usually much longer than the period of rotation. Therefore, the period of precession (T_p) should be significantly larger than the period of rotation (T = 1 / f). Given that the frequency of rotation (f) is 18.7 rev/s, we can approximate the period of rotation as:
T = 1 / f = 1 / 18.7 = 0.0533 s
Since the period of precession is much longer, we can estimate it to be around several seconds or even minutes.
d. To determine the direction of the angular momentum of the spinning disk at the instant shown in the picture, we need to refer to the arrow direction given in the problem. If the arrow points upward, then the direction of the angular momentum would be upward as well (i.e., answer (i) up).
e. To determine the direction of the precession of the gyroscope, we need to consider the rotation of the disk and the direction of the torque due to gravity. In this case, since the disk is spinning clockwise as seen from above (looking down the rope), and the torque due to gravity acts in the opposite direction (counterclockwise), the precession of the gyroscope would be counterclockwise as seen from above (looking down the rope), i.e., answer (iii) counterclockwise as seen from above (looking down the rope).
a. The magnitude of angular momentum (L) can be calculated using the equation L = Iω, where I is the moment of inertia and ω is the angular velocity. In this case, the moment of inertia of the disk can be given as I = (1/2)MR^2, where M is the mass of the disk and R is the radius. Therefore, the magnitude of angular momentum is L = (1/2)MR^2ω.
b. The torque due to gravity (τ) can be calculated using the equation τ = mgr, where m is the mass of the disk, g is the acceleration due to gravity, and r is the distance between the axis of rotation and the attachment point of the rope. Therefore, the torque due to gravity on the disk is τ = mgr.
c. The period of precession (T_p) can be calculated using the equation T_p = 2π/ω, where ω is the angular velocity. Therefore, the period of precession for this gyroscope is T_p = 2π/ω.
d. The direction of the angular momentum of the spinning disk at the instant shown in the picture can be determined by the right-hand rule. Since the disk is spinning counterclockwise as shown by the arrow, the direction of the angular momentum would be (iv) right.
e. The direction of the precession of the gyroscope can also be determined by the right-hand rule. If you curl the fingers of your right hand in the direction of the disk's spin (counter-clockwise), the thumb will point in the direction of the precession. In this case, the precession is (ii) clockwise as seen from above (looking down the rope).