A 0.5kg mass connected to a light spring of constant 20N/M occilates on a frictionless track. Calculate the total energy of the system

To calculate the total energy of the system, we need to consider both the potential energy and the kinetic energy.

1. Potential Energy (PE):
The potential energy of a spring is given by the formula: PE = 0.5 * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.
In this case, the spring constant is 20 N/m. However, to find the displacement x, we first need to calculate the spring's elongation.

The elongation of a spring is given by the formula: elongation = mass * acceleration / spring constant.
In this case, the mass is 0.5 kg and the acceleration can be determined using Hooke's Law.

Hooke's Law states that the force exerted by a spring is proportional to its elongation from the equilibrium position. The formula for Hooke's Law is: F = -k * x, where F is the force exerted by the spring and x is the displacement from the equilibrium position.
Since the mass is connected to the spring, the force exerted by the spring equals the force experienced by the mass, which is given by F = m * a, where m is the mass and a is the acceleration.
Therefore, -k * x = m * a.
Rearranging this equation, we get: a = -k * x / m.

Substituting the given values, we have:
a = (-20 N/m) * x / (0.5 kg).
Simplifying further:
a = -40 * x m/s^2.

Now, we can use the formula for elongation to find x:
0.5 kg * (-40 * x m/s^2) / 20 N/m = x.
Simplifying the equation:
-20 x = x.
Rearranging and solving for x, we find x = 0.

Since the displacement x is zero, it means the mass is at the equilibrium position, and thus the potential energy of the system is zero.

2. Kinetic Energy (KE):
The kinetic energy of the mass is given by the formula: KE = 0.5 * m * v^2, where m is the mass and v is the velocity.
The mass is given as 0.5 kg, but we need to find the velocity.

Since the system is frictionless, there is no external force acting on the mass. As a result, the total mechanical energy is conserved, meaning that the total energy at any point is equal to the total energy at any other point.

At the equilibrium position, the potential energy (PE) is zero. Therefore, the total energy of the system is equal to the kinetic energy (KE).

To find the velocity, we can use the formula for kinetic energy:
KE = 0.5 * m * v^2.
Rearranging and solving for v, we find:
v = sqrt(2 * KE / m).

Since we know that KE =