Suppose that a 200g mass (0.20kg) is oscillating at the end of a spring upon a horizontal surface that is essentially friction-free. The spring can be both stretched and compressed and have a spring constant of 240N/m. It was originally stretched a distance of 12cm (0.12m) from its equilibrium (un-stretched) position prior to release. Ignoring friction, what are the values of the potential energy, kinetic energy, and velocity of the mass, when the mass is 6cm from the equilibrium position?

To calculate the potential energy, kinetic energy, and velocity of the mass when it is 6cm from the equilibrium position, we need to consider the conservation of mechanical energy.

1. Potential Energy:
The potential energy of the mass attached to the spring is given by the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.

In this case, the spring constant (k) is 240N/m and the displacement (x) is 6cm or 0.06m. Plugging these values into the formula, we get:
Potential Energy = (1/2) * 240 * (0.06)^2 = 0.432 Joules

The potential energy of the mass when it is 6cm from the equilibrium position is 0.432 Joules.

2. Kinetic Energy:
Since no external forces are acting on the mass and there is no friction, the total mechanical energy (potential energy + kinetic energy) of the system remains constant.

Therefore, at any position, the kinetic energy can be calculated using the formula:
Kinetic Energy = Total Mechanical Energy - Potential Energy

The total mechanical energy can be found by considering the initial potential energy when the spring is stretched by 12cm from equilibrium.

Initial Potential Energy = (1/2) * 240 * (0.12)^2 = 1.728 Joules

Since the total mechanical energy remains constant, at the position when the mass is 6cm from equilibrium, the kinetic energy is:
Kinetic Energy = 1.728 Joules - 0.432 Joules = 1.296 Joules

The kinetic energy of the mass when it is 6cm from the equilibrium position is 1.296 Joules.

3. Velocity:
The velocity of the mass can be determined using the equation for kinetic energy:
Kinetic Energy = (1/2) * m * v^2
where m is the mass and v is the velocity.

In this case, the mass (m) is 0.20kg and the kinetic energy is 1.296 Joules. Plugging these values into the equation, we can solve for v:
1.296 = (1/2) * 0.20 * v^2

Rearranging the equation, we get:
v^2 = (2 * 1.296) / 0.20
v^2 = 12.96
v ≈ √12.96
v ≈ 3.60 m/s

The velocity of the mass when it is 6cm from the equilibrium position is approximately 3.60 m/s.