A body is suspended at the end of a spring kept vertical . The periodic time on increasing its mass 1]will decrease 2]may decrease or increase 3]will remain constant 4]will increase

When a body is suspended from a spring and its mass is increased, the effect on the periodic time of the spring system is quite interesting. The correct answer to your question is that the periodic time will increase. Let's break down why this happens.

The Basics of Spring Motion

A spring-mass system exhibits simple harmonic motion (SHM), where the periodic time (T) is influenced by both the mass (m) attached to the spring and the spring constant (k), which measures the stiffness of the spring. The formula for the period of a mass-spring system is given by:

T = 2π√(m/k)

Understanding the Variables

• T is the period of the motion, which is the time taken for one complete cycle.
• m is the mass attached to the spring.
• k is the spring constant, a measure of how stiff the spring is.

Impact of Increasing Mass

When you increase the mass (m) of the body suspended from the spring, you can see from the formula that the period (T) is directly proportional to the square root of the mass. This means that as the mass increases, the period will also increase. Specifically, if you double the mass, the period will increase by a factor of √2, which is approximately 1.41 times longer than it was before.

Why Does This Happen?

To visualize this, think of a heavier object being attached to a spring. A heavier mass will take longer to complete a cycle because it has more inertia. Inertia is the property of matter that resists changes in motion; thus, a heavier mass will not accelerate as quickly as a lighter one when the spring exerts a force on it. This results in a longer time to complete each oscillation.

Practical Example

Imagine you have a spring with a spring constant of 100 N/m and you initially attach a 1 kg mass. The period can be calculated as follows:

T = 2π√(1/100) ≈ 0.628 seconds

Now, if you increase the mass to 4 kg, the new period would be:

T = 2π√(4/100) ≈ 1.257 seconds

This example clearly shows that increasing the mass results in a longer period of oscillation.

Conclusion

In summary, when you increase the mass of a body suspended from a spring, the periodic time will indeed increase. This relationship is a fundamental aspect of the physics of oscillatory motion and is crucial for understanding how different factors affect the behavior of spring systems.