Problem 56 - University Physics with Modern Physics

Question

When the displacement of a mass on a spring is half of the amplitude of its oscillation, what fraction of the mass's energy is kinetic energy?

Step-by-Step Solution

Answer

Answer: When the displacement is half the amplitude, the fraction of the mass's energy that is kinetic energy is 3/4 or 75%.

Step 1: Understand the relation between displacement and potential energy

The potential energy of a mass on a spring is given by the formula: \[PE = \frac{1}{2}kx^2,\] where \(PE\) is the potential energy, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position.

Step 2: Find the potential energy at half amplitude

The problem states that the displacement is half the amplitude. Let \(A\) represent the amplitude, so the displacement is \(\frac{1}{2}A\). We can plug this into the potential energy formula to get: \[PE = \frac{1}{2}k\left(\frac{1}{2}A\right)^2 = \frac{1}{8}kA^2.\]

Step 3: Understand the relation between total energy and amplitude

Total mechanical energy for a mass on a spring in simple harmonic motion is the sum of potential and kinetic energies. At the maximum displacement (amplitude), all energy is potential energy. Therefore, the total energy can be expressed as: \[E_{total} = \frac{1}{2}kA^2.\]

Step 4: Find the kinetic energy at half-amplitude

At the given displacement, we know the total energy and potential energy. The kinetic energy, \(KE\), can be found by subtracting the potential energy from the total energy: \[KE = E_{total} - PE = \frac{1}{2}kA^2 - \frac{1}{8}kA^2 = \frac{3}{8}kA^2.\]

Step 5: Calculate the fraction of kinetic energy in total energy

To find the fraction of kinetic energy in total energy, divide the kinetic energy by total energy: \[\frac{KE}{E_{total}} = \frac{\frac{3}{8}kA^2}{\frac{1}{2}kA^2} = \frac{3}{4}.\] Hence, when the displacement of the mass on a spring is half of the amplitude of its oscillation, the fraction of mass's energy that is kinetic energy is \(\frac{3}{4}\) or 75%.