A body of mass 0.4 kg when suspended by an ideal spring increases the length of the spring by 2 cm. What will the time period when a body of 2 kg is suspended by this spring?

To determine the time period of a mass-spring system, we first need to understand how the spring behaves under different loads. The time period of oscillation for a mass-spring system is influenced by the mass attached to the spring and the spring constant, which describes how stiff the spring is. Let's break this down step by step.

Understanding Spring Constant

When a mass is suspended from a spring, it stretches the spring according to Hooke's Law, which states that the force exerted by a spring is proportional to its extension. Mathematically, this is expressed as:

F = kx

Where:

• F is the force applied (in Newtons)
• k is the spring constant (in N/m)
• x is the extension of the spring (in meters)

In your case, a mass of 0.4 kg causes the spring to extend by 2 cm (or 0.02 m). The force due to the weight of the mass can be calculated using:

F = mg

Where:

• m is the mass (0.4 kg)
• g is the acceleration due to gravity (approximately 9.81 m/s²)

Calculating the force:

F = 0.4 kg × 9.81 m/s² = 3.924 N

Calculating the Spring Constant

Now, we can find the spring constant k using the extension:

3.924 N = k × 0.02 m

Rearranging gives us:

k = 3.924 N / 0.02 m = 196.2 N/m

Finding the Time Period for a Different Mass

The time period T of a mass-spring system is given by the formula:

T = 2π√(m/k)

Now, we want to find the time period when a mass of 2 kg is suspended from the same spring. Plugging in the values:

T = 2π√(2 kg / 196.2 N/m)

Calculating the Time Period

First, calculate the fraction:

2 kg / 196.2 N/m ≈ 0.01019

Now, take the square root:

√(0.01019) ≈ 0.10095

Finally, multiply by 2π:

T ≈ 2π × 0.10095 ≈ 0.634 s

Final Result

The time period for the 2 kg mass suspended by the spring is approximately 0.634 seconds. This means that the mass will complete one full oscillation in about 0.634 seconds. Understanding these principles helps in analyzing various mechanical systems and their behaviors under different conditions.