a.) We know the spring constant and the mass, so we can find the oscillation frequency and then the period of oscillation
This means that the time t = 1.25 s is two complete cycles. Let's describe a cycle; to answer the question, we just get two full cycles!
b.) The initial PE = 1/2 k y2, where k = 100 N/m and the initial position is at the maximum displacement, or y = ymax = 5.0 cm = 0.05 m. This means the initial PE = 1/2 (100 N/m) (0.05 m)2 = 0.12 J
- first quarter cycle the mass starts from rest at its initial position and speeds up, moving towards the equilibrium position. Its direction corresponds to a negative velocity. At equilibrium it has its maximum negative velocity.
- second quarter cycle from equilibrium, the mass continues to move with a negative velocity, but now it is slowing down. It continues to do so until it reaches its maximum negative displacement, the same distance from the origin as its initial position.
- third quarter cycle from its maximum negative displacement where it is at rest for an instant, the mass speeds up, moving with a positive velocity back towards equilibrium. At equilibrium, it has its maximum positive velocity.
- fourth quarter cycle from equilibrium, the mass continues with a positive velocity, but it slows down as it approaches its maximum positive displacement (its initial position).
c.) At equilibrium, the PE is zero, the energy has been changed into KE. If there's not too much damping, the KE at equilibrium = 0.12 J; i.e., the same as the initial PE.
d.) We can relate the KE at equilibrium to the speed at equilibrium, using the formula for KE
e.) If the spring had twice the stiffness (i.e., k = 200 N/m) but the same initial displacement, then the initial PE would be 0.25 J. This means as the mass passes through equilibrium, it has 0.25 J of KE. This is twice the value used in d.). So the speed would be 1.4 times greater.
We'll give an explanation both in equations and in words...
- the period of oscillation depends only on the spring constant, k, and the mass, M. It does not depend on how much energy the oscillator has.
In words, this means the period gets longer as the mass increases or as the spring is replaced by one that is not as stiff (smaller k).
- the amplitude of oscillation is related to the spring constant, k, and the PE at maximum displacement. It does not depend on the mass.
- the speed of the mass as it goes through equilibrium depends on the energy stored and the mass. The KE at equilibrium = PE at the maximum displacement. So
