A hoop of mass M and radius 1? rolls without slipping along a track which has the shape of a circle with radius 4R. It is subject to gravity. It is confined to a plane, so when the no-slip constraint is imposed there is just one degree of freedom.

Use the angle d as your coordinate. (This angle gives the location of the center of the hoop,
as measured from the center of the track circle.) Find T, U, L, and the Lagrange equation.
For small oscillations find the angular frequency of oscillation c..'. (Hint: Think carefully about
what the no-slip condition says about the rate of rotation of the hoop compared to the rate of
movement of its center, given by d. Remember you will have two terms in the kinetic energy -
which you can assume are due to the translational of its center of mass, and the rotation about its center of mass.

Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here. Thank you for using Brainmass.
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------- Diagram -----------------

Constrain of motion is
R = 4R
R α = 4R θ + C

Where C is the ...

Solution Summary

I have shown very detailed step by step solution to this classical mechanics problem using Lagrange mechanics concepts.

A hoop of radius 0.5 m and a mass of 0.2kg is released from rest and allowed to roll down an inclined plane. How fast is it moving after being dropped from a vertical distance of 3m?
a. 5.4 m/s
b. 3.8 m/s
c. 7.7 m/s
d. 2.2 m/s

Q#1
Explain (a) how it is possible for a large force to produce only a small or even zero, torque, and (b) how it is possible for a small force to produce a large torque.
q#2
A hoop, a solid cylinder, a spherical shell, and a solid sphere are placed at rest at the top of an incline. All the objects have the same ra

With reference to Figure 2, a small cylinder sits initially on top of a large cylinder of radius a, the latter being attached rigidly to a table. The smaller cylinder has mass m and radius b. A small perturbation sets the small cylinder in motion, causing it to roll down the side of the large cylinder. Assume that the coefficien

A string is wrapped several times around the rim of a small hoop with radius r and mass m . The free end of the string is held in place and the hoop is released from rest. Calculate the angular speed of the rotating hoop after it has descended a distance, h. (See attached file for diagram and figures)

1. Competitive divers pull their limbs in a curl up their bodies when they do flips. Just before entering the water, they fully extend their limbs to enter straight down. Explain the effect of both actions on their angular velocity. What is the effect of these actions on their angular momentum?
2. What is the final velocity

Question #1
A grindstone of radius 4.0m is initially spinning with an angular speed of 8.0 rad/s. The angular speed is then increased to 10 rad/s over the next 4.0 seconds. Assume that the angular acceleration is constant.
A. What is the average angular speed of the grindstone?
B. What is the magnitude of the angular acceler

A firm produces output, y, by using capital, k, and labor, l, according to the production function.
y=k^at^b
The firm can purchase all the capital and labor it wants at prices r and w, respectively.
a) Use the method of Lagrange multipliers to find the cost function c(r,w,y). Find the average and marginal cost.
b)