Rotational Motion: 2 Challenging Questions

[furtivefelon]

there are two challange questions i don't quite understand what they are saying..

1. A horizontal plate with mass M lies on two rotating cylinders with equal angular speed of roation, but opposite directions of rotation. The distance between the axes of cylinders is l. The coefficient of kinetic friction between the plate and the material of the cylinders is u (not sure how to input the coefficient of friction)
a) Prove that the plate initially at the equilibrium position will start the harmonic oscillation if it is slightly displaced from the equilibrium in horizontal direction
b) Find the frequency of the oscillations
c) What is the result of hte same displacement if the angular velocities have opposite directions

Ok, for the first queston, i don't understand how the wheel rolling could start an oscillation.. hopefully i had hte attachments right, if so, then i have attached the diagram they provided..

2. A spacecraft with mass Mo (subscript) is moving with constant velocity Vo in free space. To change its direction the maneuvering jet engine is turned on, which starts to eject fuel at a constant velocity u relative to the spacecraft . During the maneuver, vectors Vo and u are always mutually perpendicular. Finally, the mass of the spacecraft becomes m.

Find the angle between the initial and final direction of motion of the spacecraft

for the second question, when they say one mass becomes another mass, do they mean the final mass after the fuel has been ejected? or do i need special relativity..

thanks a lot for answering :D If there is any other hints you may give me, it would be highly appreciated :D

 

[anathema]

1. The first question is talking about a system where a horizontal plate is placed on two rotating cylinders. The plate is initially at equilibrium, meaning it is not moving. However, if the plate is slightly displaced from its equilibrium position, it will start to oscillate back and forth. This is because the friction between the plate and the cylinders causes a force that acts to restore the plate to its equilibrium position. This force follows the pattern of a harmonic oscillation, where the displacement of the plate oscillates back and forth around its equilibrium position.

To prove this, you can use the equation for the force of friction (F = u*N, where u is the coefficient of kinetic friction and N is the normal force) and the equation for the centripetal force (F = m*w^2*r, where m is the mass of the plate, w is the angular velocity of the cylinders, and r is the distance between the axes of the cylinders). You will find that the resulting equation for the displacement of the plate follows the pattern of a harmonic oscillation.

The frequency of the oscillations can be found using the equation f = 1/T, where T is the period of the oscillation. The period can be found using the equation T = 2*pi*sqrt(m/k), where k is the spring constant of the system. The spring constant can be found by equating the force of friction and the centripetal force and solving for k.

If the angular velocities have opposite directions, the resulting oscillation will still follow the same pattern, but the frequency will be slightly different due to the different forces acting on the plate.

2. The second question is talking about a spacecraft that is moving with a constant velocity, but then changes its direction by using a maneuvering jet engine. The engine ejects fuel at a constant velocity relative to the spacecraft, meaning that the fuel is added to the spacecraft's velocity vector. The mass of the spacecraft also changes during this maneuver.

To find the angle between the initial and final direction of motion of the spacecraft, you can use the conservation of momentum principle. The initial momentum of the spacecraft is equal to its final momentum, so you can set the initial and final momentum equations equal to each other and solve for the angle between the two directions.

The final mass of the spacecraft can be found using the equation m = Mo - mf, where mf is the mass of the fuel ejected. This final mass can then be used to calculate

 

[quicknote]

1. For the first question, the plate is initially at equilibrium because the forces acting on it are balanced. However, when it is slightly displaced from the equilibrium position, the forces are no longer balanced and the plate will start to experience a net force in the direction of the displacement. This net force will cause the plate to accelerate and start oscillating between the two cylinders.

To prove this mathematically, we can use Newton's second law, F=ma. The net force acting on the plate is the force of friction, which is equal to u times the normal force (N) exerted on the plate by the cylinders. The normal force can be calculated using the weight of the plate (mg) and the angle between the plate and the cylinders (θ).

Since the plate is undergoing simple harmonic motion, we can use the equation a=-ω^2x, where ω is the angular frequency and x is the displacement from equilibrium. Setting the net force equal to ma and substituting in the equation for acceleration, we get:

uN = -mω^2x

Solving for ω, we get:

ω = √(uN/mx)

Using trigonometry, we can calculate the normal force as N=mgcosθ and the displacement as x=lcosθ. Substituting these values into the equation for ω, we get:

ω = √(ugcosθ/m)

This shows that the angular frequency of the oscillations is dependent on the coefficient of friction, the acceleration due to gravity, and the angle between the plate and the cylinders.

To find the frequency of the oscillations, we can use the equation f=ω/2π, where f is the frequency. This gives us:

f = √(ugcosθ/2πm)

For part c, if the angular velocities have opposite directions, the plate will still experience a net force in the direction of the displacement, but it will be in the opposite direction. This means that the plate will still start oscillating, but in the opposite direction.

2. For the second question, the mass of the spacecraft changes because it is ejecting fuel, so we need to consider this in our calculations. We can use the conservation of momentum to find the angle between the initial and final direction of motion.

Initially, the momentum of the spacecraft is given by MoVo, and after the fuel is ejected, the momentum is given by muv,