Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit τ → 0) for overdamping. Sketch the response functions.

Most harmonic oscillators are damped and, if undriven, eventually come to a stop. Why?

Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E=12mv2+12kx2, and use Equation 12.28.

A mass m0is attached to a spring and hung vertically. The mass is raised a short distance in the vertical direction and released. The mass oscillates with a frequency f0. If the mass is replaced with a mass nine times as large, and the experiment was repeated, what would be the frequency of the oscillations in terms of f0?

(b) An oscillating pendulum has length 0.3 m and 240 g bob. If the total energy is 0.06 J, calculate the amplitude of the oscillation. Satu handul

amass 1 =0.3Kg related with horizontalsp oscilltes with srig constant 30 g /s AL £=05ec the mass it X (1) = it veloity () =2 /566 in @ gl frequency o) period. © freauency. {d) amplitude A and initial phasc. (6 maimum veloity and maxismum acclertion, () total energy., (@ poieiiaenesgy and Kineic energy at he posiion = 1) =413

(III) Consider two objects, A and B, both undergoingSHM, but with different frequencies, as described by the equations andwhere is in seconds. After find the next threetimes at which both objects simultaneously pass throughthe origin.

(I) How long must a simple pendulum be if it is to make exactly one swing per second? (That is, one complete oscillation takes exactly 2.0 s.)

(4) Due to the shape of the earth - in fact, it is not a perfect sphere - the acceleration due to gravity increases when latitude increases. If the pendulum is taken from the equator (latitude = 0°) to the North or South Pole (latitude = 90°), describe the changes that would take place with respect to the period of the pendulum and the accuracy of the clock (i.e., would the clock run too fast or too slow) and why.