A Spring Analysis: Energy Conservation and Velocity Calculation
In this article, we will delve into a detailed explanation of the mechanics of a spring system using the principle of energy conservation. Specifically, we will examine a spring with a spring constant (k) of 300 N and a mass (m) of 5 kg, as it is stretched to a distance of 0.1 meters. This scenario will help us understand the energy transformations involved as the spring returns to its un-stretched position.
Understanding the Spring System
We are considering a spring that stretches downward, and our goal is to determine the velocity of the attached mass when the spring returns to its un-stretched position. Energy conservation principles will guide our analysis. We will begin by examining the potential and kinetic energy of the system during its various stages.
Initial Conditions
Initially, the spring is stretched a distance of 0.1 meters. At this position:
The stored energy of the spring is given by E 1/2 k Ax^2, where k is the spring constant (300 N/m) and Ax is the extension distance (0.1 m). The potential energy (PE) at this position, referenced to the stretched position, is set to zero. The kinetic energy (KE) is also zero because the mass is not moving at this point.Therefore, the total energy E at the stretched position is purely the stored energy in the spring, calculated as:
1/2 * 300 N/m * 0.1 m^2 15 J
Energy Transformation
When the spring returns to its un-stretched position:
The potential energy of the mass due to gravity is given by PE m * g * Ax, where m is the mass (5 kg) and g is the acceleration due to gravity (9.8 m/s^2). The potential energy of the mass is calculated as:5 kg * 9.8 m/s^2 * 0.1 m 4.9 J
Since the spring is now at its un-stretched position, the stored energy of the spring is zero. The total energy in the system remains constant and is now composed of the stored energy in the spring (0 J) plus the kinetic energy of the mass. The kinetic energy is the difference between the initial total energy and the current potential energy of the mass:
15 J - 4.9 J 10.1 J
Therefore, the kinetic energy of the mass is:
KE 10.1 J
Calculating the Velocity
The kinetic energy (KE) of the mass can be expressed as:
1/2 m V^2 10.1 J
Solving for the velocity (V) we get:
V sqrt(2 * 10.1 J / 5 kg) ≈ 2 m/s
This value represents the speed of the mass when the spring returns to its un-stretched position.
Conclusion
In summary, the application of energy conservation principles to a spring-mass system provides a powerful tool for understanding the dynamics of these systems. In the scenario described, the velocity of the attached mass when the spring returns to its un-stretched position can be accurately calculated using the stored and kinetic energies of the system.