Find the smallest positive phase of the mass on a spring that undergoes simple harmonic motion if at time t = 0.1 s the position x = 8 cm, the spring constant k = 16 N/m, the mass m = 4 kg and the amplitude A = 10 cm.

A. 1.57 rad
B. 0.79 rad
C. 0.20 rad
D. 0.64 rad
E. 0.44 rad

Explanation:

Given a mass spring system

At t = 0.1s, the positions is at x=8cm

x(0.1) = 8cm = 0.08m

Spring constant k=16N/m

Mass attached m = 4kg

Amplitude of oscialltion A=10cm = 0.1m

The angular frequency can be calculated using

w = √k/m

Where k is spring constant in N/m

m is mass attached object to the spring in kg

w = √16/4 = √4

w = 2rad/s.

Generally, the equation of a spring is given as

x = ACos(wt+Φ)

Where,

A is amplitude in metre

w is angular frequency in rad/sec

Φ is phase in radian

x(t) = 0.1Cos(2t + Φ)

At t=0.1 x = 0.08

0.08 = 0.1Cos(0.2+Φ)

Cos(0.2+Φ) = 0.08/0.1

Cos(0.2+Φ) = 0.8

0.2+Φ = ArcCos(0.8), note angle in radiant

0.2+Φ = 0.644

Φ = 0.644 — 0.2

Φ = 0.444 rad.

The phase of the SHM is 0.444rad

The answer is E = 0.44 rad

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