Calculating the Spring Constant Through Energy Conservation

In this article, we will explore how to calculate the spring constant of a spring given the details of a specific physical scenario. We will use the principle of conservation of energy to determine the spring constant, k, and examine the properties of potential energy involved in the process.

Background and Scene

A 6.45 kg ball is dropped 1.71 meters onto a spring. The ball compresses the spring by 0.428 meters. This scenario allows us to apply the principle of conservation of energy to find the required spring constant.

Gravitational Potential Energy (GPE)

When the ball is dropped, it possesses gravitational potential energy (GPE) which is directly converted into the elastic potential energy (EPE) stored in the spring upon compression.

The formula for GPE is:

GPE mgh

Where:

m 6.45 kg - mass of the ball, g 9.81 m/s2 - acceleration due to gravity, h 1.71 m - height from which the ball is dropped.

Substituting these values into the formula:

GPE ≈ 6.45 kg × 9.81 m/s2 × 1.71 m ≈ 108.1 J

Elastic Potential Energy (EPE)

When the ball compresses the spring, the elastic potential energy (EPE) is given by:

EPE (1/2)kx2

Where:

k - spring constant, x 0.428 m - compression of the spring.

Equating the GPE to EPE, we have:

mgh (1/2)kx2

Solving for k, we get:

k (2mgh)/x2

Substituting the values, we find:

k ≈ (2 × 6.45 kg × 9.81 m/s2 × 1.71 m) / (0.428 m)2 ≈ 1180.1 N/m

Verification Through Kinetic Energy and Speed

To further confirm our calculation, let's consider the speed v gained by the ball as it falls:

v √(2gh)

The kinetic energy of the ball as it just begins to compress the spring is:

(1/2)mv2 mgh

When the ball compresses the spring by an amount δ, the potential energy gained by the spring is:

(1/2)kδ2

The increase in the kinetic energy of the ball as it falls through a distance δ is:

mgδ

By the conservation of energy principle:

(1/2)mv2 - mgδ (1/2)kδ2

Since v √(2gh), the equation becomes:

mgh - mgδ (1/2)kδ2

The value of the spring constant k, assuming g 9.8 m/s2, is:

k ≈ 1475.49 N/m

Conclusion

The calculations confirm that the spring constant k should be close to 1475.49 N/m. This demonstrates the application of the principle of conservation of energy and the relationship between gravitational and elastic potential energy in practical scenarios.