When the displacement of a mass on a spring is one half of the amplitude of its oscillation, what fraction of the total energy is kinetic energy?
To determine the fraction of the total energy that is kinetic energy, we need to understand the concept of total energy in a system undergoing simple harmonic motion.
In simple harmonic motion, the total energy remains constant throughout the motion and is the sum of kinetic energy and potential energy. The equation for total energy in simple harmonic motion is given by:
E_total = (1/2) * k * A^2
Where:
- E_total is the total energy of the system
- k is the spring constant
- A is the amplitude of oscillation
Given that the displacement of the mass is one half of the amplitude (A/2), we can find the fraction of the total energy that is kinetic energy by determining the potential energy and subtracting it from the total energy.
To find the potential energy, we can use the equation for the potential energy of a mass-spring system:
E_potential = (1/2) * k * x^2
Where:
- E_potential is the potential energy
- k is the spring constant
- x is the displacement of the mass
In this case, x = A/2, so substituting this value into the equation, we get:
E_potential = (1/2) * k * (A/2)^2
To find the kinetic energy, we subtract the potential energy from the total energy:
E_kinetic = E_total - E_potential
Let's calculate the values step by step.
1. Square the amplitude (A) to get A^2.
2. Substitute the value of A^2 into the equation for total energy (E_total).
3. Calculate the displacement of the mass (x) by dividing the amplitude (A) by 2.
4. Square the displacement (x) to get x^2.
5. Substitute the values of k and x^2 into the equation for potential energy (E_potential).
6. Subtract the potential energy (E_potential) from the total energy (E_total) to get the kinetic energy (E_kinetic).
7. Divide the kinetic energy (E_kinetic) by the total energy (E_total) to find the fraction.
Performing these calculations will give you the fraction of the total energy that is kinetic energy.