Please show all work, label all units, explain steps, and cite principles/laws u

Question

Please show all work, label all units, explain steps, and cite principles/laws used.

Sinusoidal Motion (a) An unknown mass is hung vertically from a spring with an unknown spring constant. When the system has reached equilibrium, the spring has been stretched by 9.8 cm. If the mass is set in motion, at what frequency will it oscillate around the new equilibrium position? (b) The mass is pulled 0.10 m below its new equilibrium position and given a "kick" in the downward direction so that its initial velocity is 1.0 m/s. What is the maximum displacement from equilibrium as the mass oscillates?

Explanation / Answer

a) The spring is stretched in the new equilibrium by length l = 9.8cm = 0.098m

Suppose the mass of the unknown mass is m

Suppose the spring constant of the spring is k.

In the new equilibrium position the spring exerts a force kl on the blocks upwards and the force of gravity, mg, is downwards on the block.

For the mass to be in equilibrium, mg = kl, or l = mg/k.

Now, let us take this position of the block in equilibrium as x=0 for the spring mass system.

If the block is moved by distance x downwards beyond this equilibrium, the force on the block is

F = [k(l + x) - mg] = kl + kx - mg = k(mg/k) + kx - mg = kx (upward)

So, the resultant force is proportional to the displacement x of the block in its new equilibrium too and so the motion is simple harmonic with spring constant k and mass m.

So the time period of oscillation will be T = 2[m/k] = 2[l/g] (since kl = mg)

or T = 2[(0.098m)/(9.8m/s2)] = /5s

or frequency of oscillation is f = 1/T = 5/Hz = 1.592Hz.

b) the angular frequency for the oscillation is:

= 2f = (2)(5/)rad/s = 10rad/s

For an SHM, the amplitude or the maximum displacemnt from equilibrium is related to maximum speed (which is at equilibrium position) by

v = A, A is the maximum displacement from equilibrium or amplitude.

The speed at equilibrium is v = 1m/s which is the maximum speed of the oscillation.

so we have A = v/ = (1m/s)/(10rad/s) = 0.1m

or A = 10cm

So, maximum displacement from equilibrium is A = 10cm

This concludes the answers. Check the answer and let me know if it's correct. If you need anymore clarification or correction I will be happy to oblige....

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