Chapter 7, Problem 7.2P

A pendulum has length 94 centimeters and bob mass 700 grams and is dropped from an angle of 31 degrees from the vertical. a. Using the law of conservation of mechanical energy and considering only kinetic energy and gravitational potential energy, what is the maximum speed of the bob? Include units in your answer.  b. We can treat the pendulum as a mass on a spring on a horizontal surface with an effective spring constant of (w/L), oscillating back and forth with an amplitude equal to the maximum horizontal distance from the equilibrium position. Using the law of conservation of mechanical energy and considering only kinetic energy and elastic potential energy, what is the maximum speed of the bob? Include units in your answer.   PLEASE ANSWER IN HANDWRITING

Consider a bob attached to a vertical string. At t = 0, the bob is displaced to a new position where thestring makes −8° with the vertical and is then released from rest. If the angular displacement θ(t) iswritten as a cosine function, then the phase constant φ is:

In the case of a damped pendulum, how would the dyanmics change as a fixed point varied from being a stable spiral, to a stable degenerate node, to a stable node? I know that all the trajectories continue to lose altitude, and that the pendulum goes from whirling clockwise over the top, loses energy, settles to a small oscillation, and eventually comes to rest at the bottom, but wasn't sure if this general description changes based on the variation of fixed points.

A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass is slowly lifted along a circular arc to a height h. a. Assuming the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin u. b. Noting that an element of length along the path of the pendulum is ds = L du, evaluate an integral in u to show that the work done in lifting the mass to a height h is mgh.

A bee of mass 1 g lands on the end of a horizontal twig, which starts to osci llate up and down with a period of 1 s. Treat the twig as a massless spring, and estimate its force constant.

Function y 2(sin(4x – 7) – 4) , determine its amplitude, phase shift and period.

A physical pendulum composed of a solid sphere with radius R = 0.500m, is hanged from a ceiling by string of length equal to radius. What are the (a) angular frequency, (b) period, (c) frequency of the system for small angles of oscillation? For solid sphere Icm = 2/5 mr2. Also, why is the distance of the center of mass of the system from the point of oscillation 3R/2?

A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙  proportional to its velocity with the Lagrangian  L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion. (a) Consider the following generating function: F = eγtqP - QP.Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t). (b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency  ω and note that the transformed Hamiltonian yields a constant of motion.  Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).

(a) The Hamiltonian for a system has the formH = 1/2 (1/q2+ p2q4)..Find the equation of motion for q. (b) Find a canonical transformation that reduces H to the form of harmonic oscillator. Show that the solution for the transformed variables is suchthat the equation of motion found in part (a) is satisfied.