Simple Harmonic Motion

The average energy in a simple harmonic oscillator is given by: This is the correct expression for the average energy in the system over one full oscillation.

The period of a simple pendulum is given by: where: is the length of the pendulum, and is the acceleration due to gravity. If the length is doubled, the period increases by a factor of , so the new period will be .

The displacement equation of simple harmonic motion obtained by combining the motions is the result of vector addition of the displacements.
We first need to sum the two displacements: The displacements and have the same phase and can be added directly: Next, has a phase shift of , and we use the trigonometric identity to combine it with the sum of the other two: Now, we combine and using the trigonometric identity: The combined displacement is approximately:
Thus, the correct answer is (c).

The average kinetic energy of a particle in SHM is given by: where is the angular frequency. Substituting the expression for , the average kinetic energy becomes: Hence, the correct answer is (d).