Question:

A 2.0 kg mass is initially at rest on a frictionless horizontal surface. It is connected to a spring with a spring constant of 80 N/m. The mass is pulled 0.4 m to the right and released from rest.

a) Calculate the potential energy stored in the spring as the mass is pulled 0.4 m to the right.

b) As the mass oscillates back and forth, determine the maximum kinetic energy it possesses.

c) Using the principle of conservation of energy, calculate the frequency of oscillation of the mass-spring system.

Assume no energy losses due to damping or air resistance throughout the entire experiment.

Answer:

a) To calculate the potential energy stored in the spring, we can use the formula for elastic potential energy:

Where:

• PE_{\text{spring}} is the potential energy stored in the spring,
• k is the spring constant, and
• x is the displacement of the mass from its equilibrium position.

Given:

• Mass (m) = 2.0 kg
• Spring constant (k) = 80 N/m
• Displacement (x) = 0.4 m

Plugging these values into the formula, we find:

Therefore, the potential energy stored in the spring as the mass is pulled 0.4 m to the right is 6.4 J.

b) The maximum kinetic energy of the mass can be determined when it passes through the equilibrium position. At this point, all the potential energy is converted into kinetic energy. Therefore, the maximum kinetic energy is equal to the potential energy stored in the spring.

Thus, the maximum kinetic energy possessed by the mass is also 6.4 J.

c) The frequency of oscillation can be calculated using the formula for the frequency of a mass-spring system:

Where:

• f is the frequency,
• k is the spring constant, and
• m is the mass.

Given:

• Mass (m) = 2.0 kg
• Spring constant (k) = 80 N/m

Plugging these values into the formula, we find:

Therefore, the frequency of oscillation of the mass-spring system is approximately 1.003 Hz.