Energy - An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length l)

An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length l), whose other end is pivoted from the ceiling to let it swing freely in a vertical plane, as shown in Figure 4.26. The pendulum's position can be specified by its angle σ from the equilibrium position. (It could equally be specified by its distance s from equilibrium-indeed s=lσ-but the angle is a little more convenient.) (a)Prove that the pendulum's potential energy (measured from the equilibrium level) is

U(σ)=mgL_little(l-cosσ)

Write down the total energy E as a function of σ_dot. (b) Show that by differentiation your expression for E with respect to t you can get the equation of motion for σ and that the equation of motion is just the familiar Ґ=Ialpha (Where Ґ is the torque, I is the moment of inertia, and alpha is the angular acceleration σ_double-dot). (c) Assuming that the angle σ remains small throughout the motion, solve for σ(t) and show that the motion is periodic with period
τ_0=2π√(L_little/g)

I need help with this problem. Outline of solution please

Solution Summary

An interesting one-dimensional system is the simple pendulum, consisting of a point mass m, fixed to the end of a massless rod (length l).

A torture technique used by inquisition included a pendulum with attached blade. The length and mass of the wooden rod are L=5m and M=10kg, and the steel disk blade has m=30kg and radius R=30cm. While the pendulum swings, the whole system moves vertically down at a rate of 1mm/min. Considering that the pendulum has a small amp

A double pendulum consists of a pendulum of mass m2 hanging from a pendulum of mass m1. The motion of both parts of the double pendulum is constrained to the x-y plane. Both strings are "unstretchable" and having length I2 and I1, respectively.
a) How many degrees of freedom does this system have?
Using the variable theta1

Consider a simple plane pendulum consisting of a mass m connected to a string of length L.
After the pendulum is set in motion , the length of the string is shortened at a constant rate:
dL/dt = -k
The suspension point remains fixed.
Compute the following:
a) The Lagrangian and Hamiltonian functions
b) Compare

A pendulum is modeled by a mass m, attached to one end of a light rod of lenght l, whose other end is attached to an axis. Write down the equation of motion (Newton's Law) modeling the pendulum. Rewrite the model as a system of two first order equations for the position and velocity of themass. Find the equilibrium points an

Consider an ideal pendulum consisting of a "bob" of mass m hanging from a light (massless) string of length L. The pendulum swings back and forth in simple harmonic motion (SHM). You may assume that the oscillations are small, so that the motion is "ideal" SHM. Which of the following statements are true?
1. If the string len

A 3.0 kg rod of length 5.0 m has at opposite ends pointmasses of 4.0 kg and 6.0 kg(a) Will the center of mass of this system be (1) nearer to the 4.0 kg mass, (2) nearer to the 6.0 kg mass, or (3) at the center of therod? Why?
(b) Where is the center of mass of thesystem?

Please see the attached files for full description.
15. The displacement of a mass oscillating on a spring is given by x(t) = x_m * cos(wt + ph). If the initial displacement is zero and the initial velocity is in negative x direction, then the phase constant phi is:
16. Mass m, oscillating on theend of a spring with sprin

A slender rod is pivoted at 0 which is a frictionless bearing. The circular frequency in rad/s at therod for a small oscillation is?
(See attached file for full problem description)

Please see attached.
1. The two spheres of equal mass m are able to slide along the rotating horizontal black rod. If they are initially latched in position a distance r from the axis of rotation with the assembly rotating freely with an angular velocity of Wo.
Determine the new angular velocity Wf after the spheres are rele