Question

Consider a mass spring system. The spring has a constant k=30N/m and mass=3kg. The mass oscillates w/amplitude of 10cm. a.)what is the frequency of oscillation b.) what is the displacement at time t=0 c.) when is the first time the mass is at maximum displacement? (t=?) d.) what is the maximum acceleration felt by the mass? Where in the motion does this occur? e.)what is the minimum acceleration felt by the mass? Where in the motion does this occur? f.)What is the max potential energy stored in the spring (PE=1/2kx^2)? g.) since energy is conserved, what is the max velocity of the mass (KE=1/2mv^2)?

Answer 1

a) The frequency of oscillation is given by , which on substituting given values, gives 0.503 Hz.

b) The displacement, in general, can be written as , where is the initial phase difference. For the problem at hand, . So at t=0, the displacement is . If we assume , then the bob is at mid position with 0 displacement.

c) The maximum displacement occurs when x(t) is maximum, which is true when sin(wt)=1. This happens for the first time when . So t is around 0.497 s.

d) The equation of motion of the bob is . Therefore, the maximum acceleration occurs when x is maximum, that is A (10 cm). so, maximum acceleration is 1 m/s².

e) The minimum acceleration experienced is at the mean position, i.e. when x=0. Here the acceleration too is 0 m/s².

f) The maximum potential energy stored in the spring is , since the maximum x is the amplitude. This, in numbers, is equal to 0.15 J.

g) Since energy is conserved, the maximum kinetic energy(at the mean position) should be equal to the maximum potential energy (at the far ends). So, , which gives us that the maximum velocity is 0.316 m/s.