A gyroscope consists of a uniform disc of mass radius M= 2 kg and radius R= 0.2 m . The disc spins with an angular speed ω= 300 rad⋅s-1 as shown in the figure below. The gyroscope precesses, with its axle at an angle 30∘ below the horizontal (see figure). The gyroscope is pivoted about a point d= 0.4 m from the center of the disc. What is the magnitude of the precessional angular velocity Ω (in radians/sec)?
Ω=
(M*g*d)/(0.5*M*R^2*w)
To calculate the magnitude of the precessional angular velocity Ω, we can use the equation:
Ω = (Mgd)/(Iω)
First, let's calculate the angular momentum of the disc, I, using the formula:
I = (1/2)MR^2
Substituting the given values:
M = 2 kg
R = 0.2 m
I = (1/2)(2 kg)(0.2 m)^2 = 0.04 kg·m^2
Next, we need to calculate the torque, τ, acting on the gyroscope. The torque can be calculated using the equation:
τ = Mgd
Substituting the given values:
M = 2 kg
g = 9.8 m/s^2
d = 0.4 m
τ = (2 kg)(9.8 m/s^2)(0.4 m) = 7.84 N·m
Now we can substitute the values of I, ω, and τ into the equation for Ω:
Ω = (Mgd)/(Iω) = (7.84 N·m)/(0.04 kg·m^2)(300 rad/s)
Ω = (7.84 N·m)/(0.04 kg·m^2)(300 rad/s) = 65.33 rad/s
Therefore, the magnitude of the precessional angular velocity Ω is 65.33 rad/s.