Question

A particle attached to a spring with k = 54 N/m is undergoing simple harmonic motion, and its position is described by the equation x = (5.5 m)cos(7.1t), with t measured in seconds (a) What is the mass of the particle? kg (b) What is the perlod of the motion? (c) What is the maximum speed of the particle? m/s (d) What Is the maximum potentlal energy? (e) What is the total energy?

Answer 1

a)
The given equation is, x = 5.5 cos(7.1t)
Comparing this with the standard equation, x = A cos($\omega$t),
Amplitude, A = 5.5 m,
Angular frequency, $\omega$ = 7.1 rad/s

From the equation, k = m$\omega$²,
Where k is the spring constant and m is the mass,
m = k/$\omega$²
= 54/(7.1)²
= 1.07 kg

b)
Period, T = 2$\pi$/$\omega$
= 2$\pi$/7.1
= 0.885 s

c)
Maximum speed, v_max = $\omega$A
= 7.1 x 5.5
= 39.05 m/s

d)
Maximum potential energy = 1/2 kA²
= 0.5 x 54 x (5.5)²
= 816.75 J

e)
Total energy = maximum potential energy = maximum kinetic energy
= 816.75 J