A mass is attached to the end of a spring and set into oscillation on a horizont

Question

A mass is attached to the end of a spring and set into oscillation on a horizontal frictionless surface by releasing it from a stretched position. The position of the mass at any time is described by x - (8.0 cm)cos[2Tt/(4.08 s)]. Determine the following. (a) period of the motion How may the period of oscillation be determined from an expression for the position of the oscillating mass at any time? s (b) frequency of the oscillations How is the frequency of oscillation related to the period of the oscillation? Hz (c) first time the mass is at the position x0 What fraction of a period does it take for the oscillating mass to go from the position of maximum positive displacement to the equilibrium position? s (d) first time the mass is at the site of maximum compression of the spring What fraction of a period does it take for the oscillating mass to go from the position of maximum positive displacement to the position of maximum negative displacement? s

Explanation / Answer

Comparing the given equation with general equation

x = A*cos(w*t)

a) period is T = 2*pi/w

w = 2*pi/T

w = 2*pi/4.08

so T = 4.08 sec

b) f = 1/T = 1/4.08 = 0.245 Hz

c) cos(pi/2) = cos(2*pi*t/4.08)

pi/2 = 2*pi*t/4.08

1/2 = 2*t/4.08

t = 1.02 sec

d) for maximum compression x = 8 cm

8 = 8*cos(2*pi*t/4.08)

1 = cos(2*pi*t/4.08)

pi = 2*pi*t/4.08

t = 2.04 sec

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