Animated Solution for Physics - Oscillations: A mass $M$ attached to a spring oscillates with a period of $2\text{ s}$. If the mass is increased by $2\text{ kg}$ the period increases by one sec. Find the initial mass $M$ assuming that Hooke's law is obeyed.

Imagine a block of mass $M$ suspended from a ceiling by a spring.

When you pull it down and release it, it doesn't just return to its original position; it dances.

It moves up and down in a beautiful, rhythmic pattern known as Simple Harmonic Motion (SHM).

This motion is governed by Hooke's law, which states that the restoring force is directly proportional to the displacement.

In this problem, we are given a fascinating puzzle: by observing how the rhythm of this dance changes when we add more mass, we can determine the original mass of the block without ever placing it on a scale!

Let's dive into the physics and mathematics that make this possible.

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The time period $T$ of a spring-mass system is the time taken for one complete oscillation.

It is determined by two competing factors: the inertia of the mass and the stiffness of the spring.

Mathematically, this relationship is expressed by the classic formula:

Where: - $m$ is the total suspended mass. - $k$ is the spring constant (stiffness of the spring).

Notice that the time period is directly proportional to the square root of the mass ($T\propto \sqrt{m}$).

This means that if you increase the mass, the inertia of the system increases, making it sluggish and slow, which in turn increases the time period of oscillation.

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Let's break down the problem into two distinct cases.

The Initial Setup

Initially, a mass $M$ is attached to the spring.

The system oscillates with a time period $T_1=2\text{ s}$.

Using our formula, we can write:

The Increased Mass

Next, the mass is increased by $2\text{ kg}$, making the new mass $M+2$.

This extra mass slows down the oscillations, increasing the time period by $1\text{ s}$.

Therefore, the new time period $T_2$ is:

We can write the equation for this second scenario as:

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Instead of dealing with the unknown spring constant $k$ and the constant $2\pi$, we can elegantly eliminate them by taking the ratio of the two time periods:

Simplifying this ratio, we get:

Now, let's substitute the known values of $T_1=2\text{ s}$ and $T_2=3\text{ s}$ into this equation:

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To find the value of $M$, we need to solve this algebraic equation step-by-step.

First, let's eliminate the square root by squaring both sides of the equation:

Next, we cross-multiply to clear the fractions:

Expanding the right side:

Now, let's isolate the terms containing $M$ by subtracting $4M$ from both sides:

Finally, dividing both sides by $5$ gives us the initial mass:

Thus, the initial mass of the block is exactly $1.6\text{ kg}$.

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This elegant problem demonstrates how we can use the principles of Simple Harmonic Motion to probe the properties of a system.

By simply measuring the change in the rhythm of oscillation when a known mass is added, we were able to calculate the unknown initial mass without needing to know the stiffness of the spring.

This concept of using frequency or time period changes to measure mass is widely used in modern technology, such as in quartz crystal microbalances, which can measure extremely tiny masses (down to nanograms) by observing changes in vibration frequencies!