Discussion
Consider a spring-mass system with mass \(m\) and spring stiffness \(k\) as shown in the illustration. At time \(t = 0\), the mass is displaced by \(P\) units and the velocity of the mass is zero. The displacement of the mass, \(x(t)\), is measured from the equilibrium position.
[The illustration shows a horizontal spring \(k\) fixed to a wall at its left end and attached to the mass \(m\) at its right end, with \(x(t)\) measured to the right.]
Which one of the following functions represent \(x(t)\)?
Answer: D
The motion is \(x(t) = P\cos\left(\sqrt{k/m}\,t\right)\).
An undamped spring-mass system obeys \(m\ddot{x} + kx = 0\), whose general solution is a sum of a cosine and a sine at the natural frequency \(\omega_{n} = \sqrt{k/m}\):
\[ \begin{aligned} x(t) &= A\cos\omega_{n}t + B\sin\omega_{n}t \\ x(0) &= A = P \\ \dot{x}(0) &= B\omega_{n} = 0 \Rightarrow B = 0 \end{aligned} \]
So only the cosine term survives, with amplitude P.
The sine-only answer starts at zero displacement, not at P. The sum \(P\cos + P\sin\) starts at P but with velocity \(P\omega_{n}\), not zero. The answer with P/2 on each term starts at only half the given displacement and also has a non-zero initial velocity.
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