Animated Solution for Physics - Oscillations: The maximum velocity of a particle, executing simple harmonic motion with an amplitude $7\text{ mm}$, is $4.4{\text{ ms}}^{-1}$. The period of oscillation is

Imagine a block attached to a spring, oscillating back and forth on a frictionless horizontal surface. As it moves, its velocity constantly changes. It momentarily stops at the extreme ends (where the displacement is equal to the amplitude, $A$) and zips past the central mean position ($x=0$) with its absolute maximum speed.

This maximum velocity is a crucial parameter in Simple Harmonic Motion (SHM) and is mathematically defined by a very elegant relationship:

Here, $A$ represents the amplitude of the oscillation, and $\omega$ is the angular frequency.

In our specific problem, we are given the maximum velocity $v_{\text{max}}=4.4{\text{ ms}}^{-1}$ and the amplitude $A=7\text{ mm}$. However, the question doesn't ask for the angular frequency; it asks for the time period ($T$).

We need a bridge to connect these concepts. The angular frequency $\omega$ tells us how many radians the phase of the oscillation changes per second, and it is directly related to the time period $T$ (the time taken for one complete cycle) by the formula:

Substituting this into our maximum velocity equation gives us our master working equation:

Before we rush into substituting the numbers, there is a classic trap waiting for us: inconsistent units. The velocity is given in SI units (meters per second), but the amplitude is given in millimeters.

We must convert the amplitude into meters to ensure our final answer is correct.

Now, we can safely substitute our raw values into the master equation:

Let's rearrange the equation to isolate the time period $T$:

Here is where a little mathematical foresight saves a lot of time. Instead of using the decimal approximation $3.14$ for $\pi$, let's use the fractional equivalent $\frac{22}{7}$. Notice how perfectly it pairs with the amplitude of $7$:

The $7$ in the numerator and the $7$ in the denominator cancel each other out beautifully. We are left with:

Since $44$ divided by $4.4$ is exactly $10$, the expression simplifies to:

Which gives us our final, clean answer:

By keeping a cool head, watching our units, and choosing the right mathematical approximations, a seemingly tedious calculation becomes a walk in the park!