Energy Transformation: Collision with a Spring in a Frictionless Environment

When a rigid body with a specific mass and velocity collides with a spring, an interesting phenomenon occurs. The kinetic energy of the moving object is transformed into the potential energy stored in the spring. This article will explore the detailed process and provide the calculation steps to determine how much the spring compresses.

Calculation of Initial Kinetic Energy

The kinetic energy of the block is calculated using the classical formula:

KE (1/2) * m * v2

Where m is the mass of the block and v is the velocity of the block.

Input Data

Mass of the block, m 0.5 kg Velocity of the block, v 6 m/s

Substituting these values into the formula:

KE (1/2) * 0.5 kg * (6 m/s)2

Calculating:

KE 1/2 * 0.5 * 36 J 9 J

Determination of Spring Compression

The potential energy stored in the spring is given by the formula:

PE (1/2) * k * x2

Where k is the spring constant and x is the compression of the spring.

Note: At maximum compression, the kinetic energy of the block equals the potential energy stored in the spring. Therefore, KE PE.

Given Data

Spring constant, k 400 N/m Kinetic energy, KE 9 J

Setting KE PE:

9 J (1/2) * 400 N/m * x2

Rearranging:

18 400 * x2

Solving for x:

x2 18 / 400

x2 0.045

x √0.045 ≈ 0.212 meters

Therefore, the spring compresses approximately 0.212 meters (or 21.2 centimeters).

Additional Considerations

When the block collides with the spring, several forms of energy can be generated:

Elastic Potential Energy: Most of the kinetic energy is converted into potential energy in the spring. Heat Energy: Some energy is converted into heat due to the friction or internal resistance of the spring material. If the mass of the spring is mo, the specific heat capacity of the material is c, and the rise in temperature of the spring is t, the heat generated is moct. Sounds Energy: Microscopic vibrations and movements can produce sound waves during collision. The amount of sound energy generated is denoted as s. Heat Loss to the Surroundings: Some heat is lost to the surroundings, and the amount of this heat is denoted as h.

In practice, if we neglect the heat energy and sound energy, the equation simplifies to:

1/2 mv2 1/2 kx2

Which leads to:

x v * √(m/k)

Substituting the given values:

x 6 * √(0.5/400) 0.212 meters

Conclusion

The spring compresses a distance of approximately 0.212 meters. This calculation assumes there are no dissipative forces such as friction, heat generation, or sound generation. Understanding the principles of energy transformation is crucial for solving problems in physics involving collisions with springs in idealized conditions.