Question

An oscillating block-spring system has a mechanical energy of 2.46 J, an amplitude of 12.5 cm, and a maximum speed of 1.29 m/s. Find (a) the spring constant, (b) the mass of the block and (c) the frequency of oscillation.

Answer 1

Mechanical Energy is defined as:
E = K + U
(K is the kinetic energy and U the potential energy).
(a)
In the point where the block stops the kinetic energy is zero, thus E = U, (x is the amplitude):
U = k*x²/2
k = 2*U/x² = 2*2.46 / 0.125^2 = 314.88 N/m
(b)
In the equilibrium point the block has zero potential energy, thus E = K, (v is the maximum speed):
K = m*v²/2
m = 2*K/v² = 2* 2.46 /1.29^2 = 2.9565 kg
(c)
The frequency in a harmonic oscillator is
f = (1/(2*π))*(k/m)^(1/2) = [1/6.28] * [ sqrt (314.88/2.9565)]

f = 1.64 Hz