To calculate the kinetic and potential energy of the oscillating mass-spring system, we can use the formulas for these energies.
Kinetic Energy (KE): The kinetic energy of an object is given by the formula: KE = (1/2) * m * v^2 where m is the mass and v is the velocity of the object.
Potential Energy (PE): The potential energy stored in a spring is given by Hooke's Law: PE = (1/2) * k * x^2 where k is the force constant (spring constant) and x is the displacement from the equilibrium position.
Given: Mass (m) = 3 kg Force constant (k) = 60 N/m Amplitude (A) = 10 cm = 0.1 m
To find the kinetic energy, we need to determine the maximum velocity of the oscillating mass. At the extreme points of the motion (maximum displacement), all the potential energy is converted to kinetic energy, and the velocity is at its maximum. At maximum displacement, the entire potential energy is converted to kinetic energy and vice versa.
- Kinetic Energy: The maximum velocity (v) is related to the amplitude (A) by the formula: v = ω * A where ω is the angular frequency of the oscillation. The angular frequency is given by: ω = sqrt(k / m)
Substituting the values: ω = sqrt(60 N/m / 3 kg) ≈ 5.48 rad/s
v = ω * A = 5.48 rad/s * 0.1 m = 0.548 m/s
Now, we can calculate the kinetic energy: KE = (1/2) * m * v^2 = (1/2) * 3 kg * (0.548 m/s)^2 ≈ 0.448 J
- Potential Energy: The potential energy is given by: PE = (1/2) * k * x^2
At maximum displacement (amplitude), x = A = 0.1 m.
PE = (1/2) * 60 N/m * (0.1 m)^2 = 0.3 J
Therefore, the kinetic energy of the system is approximately 0.448 Joules, and the potential energy is approximately 0.3 Joules.