A gyroscope consists of a uniform disc of mass radius M= 1 kg and radius R= 0.3 m . The disc spins with an angular speed ω= 200 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= 0.6 m from the center of the disc. What is the magnitude of the precessional angular velocity Ω (in radians/sec)?

Ω=

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what is a gyroscope?
what does the gyroscope process?

To find the magnitude of the precessional angular velocity Ω, we can use the equation:

Ω = (dω sin(θ)) / (MR)

Where:
- Ω is the precessional angular velocity
- d is the distance from the center of the disc to the pivot point (given as 0.6 m)
- ω is the angular speed of the disc (given as 200 rad⋅s^(-1))
- θ is the angle between the axle of the gyroscope and the horizontal (given as 30∘)
- M is the mass of the disc (given as 1 kg)
- R is the radius of the disc (given as 0.3 m)

Plugging in the given values, we have:

Ω = (0.6 * 200 * sin(30∘)) / (1 * 0.3)

First, we convert the angle from degrees to radians:
θ = 30∘ * (π/180) = (π/6) rad

Now we can substitute the values and calculate Ω:

Ω = (0.6 * 200 * sin(π/6)) / (1 * 0.3)
Ω = (0.6 * 200 * 0.5) / (1 * 0.3)
Ω = 60 rad/s

Therefore, the magnitude of the precessional angular velocity Ω is 60 radians/sec.