When a mass M is attached to the spring of force constant K, then the spring stretches by L. If the mass oscillates with amplitude L, what will be maximum potential energy stored in the spring?

To find the maximum potential energy stored in a spring when a mass M is attached, we need to apply Hooke's Law and the formula for potential energy in a spring. The spring stretches by a distance L when the mass is attached, and if the mass oscillates with an amplitude equal to L, we can derive the maximum potential energy stored in the spring during the oscillation.

Understanding the Spring System

When a mass is attached to a spring, the spring exerts a restoring force proportional to the displacement from its equilibrium position. This relationship is described by Hooke's Law, which states:

F = -Kx

Here, F is the force exerted by the spring, K is the spring constant, and x is the displacement from the equilibrium position. When the mass M is at rest, the spring stretches to a new equilibrium position L due to the weight of the mass:

Mg = KL

Where g is the acceleration due to gravity. This equation shows that the force due to the mass equals the force exerted by the spring when the mass is at rest.

Calculating Maximum Potential Energy

Now, when the mass begins to oscillate with an amplitude L, the maximum potential energy (PE) stored in the spring occurs when the spring is at its maximum displacement, which is equal to the amplitude. The potential energy stored in a spring is given by the formula:

PE = (1/2) K x²

In this case, since the maximum displacement x is equal to L, we can substitute L into the equation:

PE_max = (1/2) K L²

Example Calculation

Let's say you have a spring with a spring constant K of 200 N/m and the mass stretches the spring by L = 0.5 m. The maximum potential energy would be calculated as follows:

PE_max = (1/2) * 200 N/m * (0.5 m)²

Calculating this gives:

PE_max = (1/2) * 200 * 0.25 = 25 J

Thus, the maximum potential energy stored in the spring would be 25 Joules.

Final Thoughts

The maximum potential energy in a spring system, when the mass oscillates with an amplitude equal to its stretch, is crucial for understanding energy transformations in mechanical systems. It illustrates how energy is stored and released as the mass moves, providing insight into the principles of oscillatory motion.