Max Speed & Displacement of Particle in Rotating System

[lovelyrita24]

I have a problem involving the coriolis and gravitational forces on a rotating coordinate system:

A free particle of mass m is release from a state of rest on a rotating, sloping, rigid plane. The angular rotation rate about a vertical axis is omega and the angle formed by the plane with the horizontal is theta. Friction and centrifugal forces are negligible. What is the maximum speed acquired by the particle, and what is its maximum downhill displacement?

- I am pretty sure the answers are derived symbolically by determining when the coriolis force comes into balance with the component of the gravitational force down the slope.

Any insight would be much appreciated...

 

[Nate810]

thank you!The maximum speed acquired by the particle can be found by setting the net force on the particle equal to zero. In this case, the net force is the gravitational force down the slope minus the Coriolis force. The Coriolis force is equal to 2mωv, where v is the velocity of the particle. The gravitational force down the slope is equal to mgsinθ, where g is the gravitational acceleration and θ is the angle formed by the plane with the horizontal. Setting the net force equal to zero and solving for v gives:v = gsinθ/2ω The maximum downhill displacement can be found by integrating the velocity equation above (assuming the particle starts from rest at x = 0) to get the following expression for the position of the particle x:x = gsinθ/(4ω^2) Therefore, the maximum downhill displacement of the particle is gsinθ/(4ω^2).

 

[quicknote]

I can provide some insight and solutions to your problem involving the coriolis and gravitational forces on a rotating coordinate system. The first step in solving this problem is to understand the forces acting on the free particle in this system.

The coriolis force, caused by the rotation of the coordinate system, will act perpendicular to the velocity of the particle. This force will be in the direction of the rotation if the particle is moving towards the center of rotation, and in the opposite direction if the particle is moving away from the center of rotation.

The gravitational force, on the other hand, will act straight down the slope of the plane. In order to find the maximum speed and displacement of the particle, we need to determine when the coriolis force and the component of the gravitational force down the slope are in balance.

To find the maximum speed, we can set the coriolis force equal to the component of the gravitational force down the slope. This will give us the equation:

mωv = mg sinθ

Where m is the mass of the particle, ω is the angular rotation rate, v is the velocity of the particle, g is the acceleration due to gravity, and θ is the angle formed by the plane with the horizontal.

Solving for v, we get:

v = (g sinθ)/ω

This is the maximum speed acquired by the particle in this system.

To find the maximum downhill displacement, we can use the equation for displacement:

s = ut + 1/2at^2

Where s is the displacement, u is the initial velocity (which in this case is 0), a is the acceleration, and t is the time.

The acceleration in this case is the component of the gravitational force down the slope, which is given by:

a = g sinθ

Plugging this into the displacement equation, we get:

s = 1/2(g sinθ)t^2

To find the time, we can use the equation for velocity:

v = u + at

Where u is again 0 and a is the same as before. Solving for t, we get:

t = v/a = (g sinθ)/(ωg sinθ) = 1/ω

Plugging this into the displacement equation, we get:

s = 1/2(g sinθ)(1/ω)^2 = (g sinθ)/(2ω^2)

This is the maximum downhill