In a SHM (Simple harmonic motion), the acceleration is directly proportional to

Question

Accepted Answer

The correct option is (3):

In Simple Harmonic Motion (SHM), the defining characteristic is that the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This relationship is often expressed by Hooke's Law: $F = -kx$, where $F$ is the restoring force, $k$ is the spring constant (a measure of stiffness), and $x$ is the displacement.

According to Newton's Second Law of Motion, force is also equal to mass times acceleration ($F = ma$). Combining these two equations, we get $ma = -kx$. Therefore, the acceleration ($a$) can be expressed as $a = -\frac{k}{m}x$. Since $k$ and $m$ are constants for a given system, we can see that the acceleration ($a$) is directly proportional to the displacement ($x$). The negative sign indicates that the acceleration is always directed towards the equilibrium position, opposite to the direction of displacement. This proportional relationship between acceleration and displacement is the fundamental condition for an object to execute SHM.

The incorrect option is (1) Time: Time is a variable over which the motion occurs, but acceleration itself is not directly proportional to time in SHM; it's a function of time, but not a direct proportionality like $a \propto t$.

The incorrect option is (2) Linear velocity: In SHM, acceleration is maximum when linear velocity is zero (at extreme positions), and acceleration is zero when linear velocity is maximum (at the equilibrium position). Thus, acceleration and linear velocity are not directly proportional.

The incorrect option is (4) Frequency: Frequency is a characteristic of the oscillation (how many cycles per second), and while it's related to the system's properties ($k$ and $m$), acceleration is not directly proportional to the frequency. The angular frequency $\omega = \sqrt{\frac{k}{m}}$, and $a = -\omega^2 x$, showing the relationship, but not a direct proportionality between $a$ and the linear frequency.