To find the potential energy stored in the mass-spring system at a specific time, we can use the principles of simple harmonic motion. The potential energy in a spring is given by the formula:
Potential Energy in a Spring
The potential energy (PE) stored in a spring is calculated using the equation:
PE = (1/2) k x²
where:
- k is the spring constant (in N/m),
- x is the displacement from the equilibrium position (in meters).
Finding Displacement at t = 1.42 s
First, we need to determine the displacement of the mass at t = 1.42 s. The displacement in simple harmonic motion can be expressed as:
x(t) = A cos(ωt)
where:
- A is the maximum displacement (amplitude),
- ω is the angular frequency (in rad/s),
- t is the time (in seconds).
Given:
- ω = 2.81 rad/s
- A = 0.232 m
- t = 1.42 s
Now, substituting the values into the displacement equation:
x(1.42) = 0.232 cos(2.81 * 1.42)
Calculating the argument of the cosine function:
2.81 * 1.42 ≈ 3.99 rad
Now, we find the cosine:
cos(3.99) ≈ -0.54 (using a calculator)
Now substituting back to find x:
x(1.42) = 0.232 * (-0.54) ≈ -0.125 m
Calculating Potential Energy
Now that we have the displacement at t = 1.42 s, we can calculate the potential energy:
PE = (1/2) k x²
Substituting the values:
PE = (1/2) * 45.2 * (-0.125)²
Calculating the square of the displacement:
(-0.125)² = 0.015625
Now substituting this back into the potential energy formula:
PE = (1/2) * 45.2 * 0.015625
PE ≈ 0.352 N·m or Joules
Final Result
The potential energy stored in the mass-spring system at t = 1.42 s is approximately 0.352 Joules.
