Question

One example of simple harmonic motion is a block attached to a spring, pulled to one side, then released, so it slides back and forth over and over. In real experiments, friction and drag can't be eliminated, so both will do a small amount of negative work on the sliding mass, each cycle, slowly reducing its ME, until the block stops moving altogether.

a) As the system's ME decreases, the block's maximum speed gets smaller, but the oscillation period T doesn't change (at least, not until all motion ceases, at which point T becomes meaningless). Ignoring what happens when the block finally stops, explain why T remains constant as the block slows down. No equations or math. “T doesn't depend on the block's speed” restates the conclusion, but doesn't explain it. Try to explain, in physical terms, why T doesn't depend on the block's speed.

b) Now imagine conducting a real experiment of this nature: choose a mass for the block and an oscillation period – both physically realistic - then calculate the spring constant k required to make the block oscillate with that period.

c) Now choose a physically realistic initial amplitude for the block's motion and calculate the block's maximum speed before it loses any ME.

d) Calculate the amount of mechanical energy the system loses as the amplitude decreases to 2/3, 3/4, or 4/5 (your choice) of its initial value.

Answer 1

a) As the speed of the block reduces due to friction so does the amplitude of oscillation, so as speed decreases the maximum displacement (amplitude) also decreases proportionately, hence the time period remain constant.

b) let m = 1 Kg , T = 2 s then we have

c) let the amplitude be A = 1m so maximum speed is

d)The total mechanical energy is

A₁ = 2/3A

E₁ = 2.18 J

energy lost = 4.9 -2.18 = 2.72 J

A₂ = 3/4A

E₂ = 2.76 J

energy lost = 4.9 -2.76 = 2.14 J

A₃ = 4/5 A

E₁ = 3.14 J

energy lost = 4.9 -3.14 = 1.76 J