Chapter 4, Problem 4.25P

Calculate the energy, corrected to first order, of a harmonic oscillator with potential:

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?

Consider a simple harmonic oscillator consisting of a one kilogram mass m' on a spring with spring/forceconstant k and length L'. If the mass of the spring ms is 9% of the attached mass, and k = 66 N/m,and if we determined the attached body is displaced 3 cm and given a downward velocity of 0.4 m/s -calculate,→ the frequency ω of the motion, ,→ and the amplitude A of the motion

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.

Let f(x)= 4xex - sin(5x). Find the third derivative of this function. Note ex is denoted as e^x below. Select one: (12+4x^3)e^x + 125sin(5x) 12e^x + 125cos(5x) not in the list (12+4x)e^x + 125cos(5x) (8+4x)e^x + 25sin(5x)

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 simple harmonic oscillator consists of a spring with a force constant of k = 59.5 N/m connected to a block with a mass of m = 1.05 kg. Assume that the surface supporting the block is flat and frictionless, and there is no air resistance. At time t = 0, the spring is stretched 16.5 cm from its natural length and the block is moving at a speed of 1.95 m/s in the −x direction, as shown below. (a) Find x(t), the displacement of the block from equilibrium as a function of time. Hint: you’ll need to find the constants ω (in rad/s), A (in cm), ? (in radians) for the function: x(t) = Acos(ωt + φ). (b) What is the velocity of the block (in m/s) at t = 2.00 s? (c) What is the acceleration of the block (in m/s2) at t = 2.00 s?

In the above question let us consider the position of mass when the spring is relaxed as x = 0, and the left to right direction as the positive direction of the x-axis.Provide x as a function of time t for the oscillating mass, if at the moment we start the stopwatch (t = 0), the mass is:( i ) at the mean position,( ii ) at the maximum stretched position, and( iii ) at the maximum compressed position.

In this limit, what is the effective spring constant k for the pendulum in the limit of small angles? In this limit, what is the effective angular frequency ω of a pendulum in the limit of small angles?