Question

rana writes a summary about a mass on a spring in simple harmonic motion as it moves upward from the equilibrium position toward the maximum positive displacement.
as the mass moves upward, the velocity is positive and decreasing while the acceleration is negative and decreasing.
what error did rana make?
the acceleration should be
increasing, not decreasing.
the acceleration should be
positive, not negative.
the velocity should be
increasing, not decreasing.
the velocity should be
negative, not positive.

Explanation:

In simple harmonic motion (SHM) for a mass - spring system, the acceleration \(a = -kx/m\) (where \(k\) is the spring constant, \(x\) is the displacement from equilibrium, and \(m\) is the mass). When the mass moves upward from equilibrium towards maximum positive displacement, the displacement \(x\) is positive (assuming upward is positive). So the acceleration \(a=-kx/m\) is negative (since \(x\) is positive, \(k\) and \(m\) are positive). Now, as the mass moves towards maximum displacement, the magnitude of \(x\) is increasing. Since \(a = -kx/m\), the magnitude of acceleration is \(|a|=\frac{kx}{m}\), so as \(x\) increases, the magnitude of acceleration increases, meaning the acceleration (which is negative) is becoming more negative, i.e., the acceleration is increasing in magnitude (or we can say the acceleration is increasing in the negative direction). Rana said the acceleration is negative and decreasing, but it should be increasing (in magnitude, or becoming more negative). The velocity: as the mass moves from equilibrium to maximum displacement, it is slowing down, so velocity is positive (since it's moving upward) and decreasing (which is correct). So the error is that the acceleration should be increasing, not decreasing.

Answer:

The acceleration should be increasing, not decreasing.