A spring of negligible mass stretches 2.20 cm from its relaxed length when a force of 10.50 N is applied. A 1.100 kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x=5.0 cm and released from rest at t=0.

(c) What is the total energy of the system?

total energy is spring, E=1/2 kx^2

figure k, 10.5/.0022 N/m x=.05m

To find the total energy of the system consisting of the mass and the spring, we need to consider both the potential energy stored in the spring and the kinetic energy of the moving mass.

1. Potential Energy of the Spring (Us):
The potential energy of a spring is given by the equation: Us = (1/2)kx^2

Given:
- Spring displacement (stretch) = 2.20 cm = 0.022 m (from the relaxed length)
- Spring constant (k) is not given directly but can be calculated using Hooke's Law, which states that the force applied on a spring is directly proportional to its displacement:
F = -k * x
Rearranging this equation gives us:
k = -F / x

Using the given force (10.50 N) and the displacement (0.022 m), we can calculate the spring constant:
k = -10.50 N / 0.022 m ≈ -477.27 N/m (Note: The negative sign denotes the direction opposite to the applied force)

Now, we can calculate the potential energy of the spring:
Us = (1/2) * (-477.27 N/m) * (0.022 m)^2 ≈ 0.0115 J

2. Kinetic Energy of the Mass (K):
The kinetic energy of a moving object is given by the equation: K = (1/2)mv^2

Given:
- Mass of the particle (m) = 1.100 kg
- Displacement of the particle (x) = 5.0 cm = 0.050 m (from the equilibrium position)
- The particle is released from rest at t=0, meaning it has no initial velocity.

Since the particle is released from rest, we know that all the potential energy of the spring will be converted into the kinetic energy of the particle at the displacement x.

Using the conservation of energy, we can equate the potential energy to kinetic energy:
Us = K

Therefore:
K = 0.0115 J

3. Total Energy of the System:
The total energy of the system is the sum of the potential energy and the kinetic energy:
Total energy = Us + K
Total energy = 0.0115 J + 0.0115 J
Total energy = 0.023 J

Therefore, the total energy of the system is approximately 0.023 Joules.