**Correct Option: B. Total mechanical energy** ### Detailed Explanation: In simple harmonic motion (SHM), an object oscillates back and forth around an equilibrium position. The key quantities involved in SHM are kinetic energy (KE), potential energy (PE), and total mechanical energy (TME). 1. **Total Mechanical Energy (TME)**: - The total mechanical energy in a system undergoing SHM is the sum of its kinetic energy and potential energy. - Mathematically, this can be expressed as: \[ TME = KE + PE \] - In SHM, the total mechanical energy remains constant because there are no non-conservative forces (like friction) doing work on the system. The energy simply transforms between kinetic and potential forms as the object oscillates. 2. **Kinetic Energy (KE)**: - Kinetic energy is given by the formula: \[ KE = \frac{1}{2}mv^2 \] - As the object moves through its oscillation, its speed changes, which means its kinetic energy also changes. At the maximum displacement (amplitude), the speed is zero, and thus the kinetic energy is zero. Conversely, at the equilibrium position, the speed is maximum, and so is the kinetic energy. Therefore, kinetic energy is not constant. 3. **Potential Energy (PE)**: - The potential energy in SHM is given by: \[ PE = \frac{1}{2}kx^2 \] - Here, \(k\) is the spring constant (for a spring-mass system), and \(x\) is the displacement from the equilibrium position. As the object moves, the displacement changes, which means the potential energy also changes. At maximum displacement, potential energy is at its maximum, while at the equilibrium position, it is zero. Thus, potential energy is not constant. 4. **Displacement**: - Displacement refers to the position of the object relative to the equilibrium position. In SHM, displacement varies continuously as the object oscillates back and forth. It reaches maximum values at the amplitude and is zero at the equilibrium position. Therefore, displacement is not constant. ### Why Other Options Are Wrong: - **A. Kinetic energy**: As explained, kinetic energy varies with the speed of the object, which changes throughout the motion. It is maximum at the equilibrium position and zero at the maximum displacement. - **C. Potential energy**: Potential energy also varies with displacement. It is maximum at the maximum displacement and zero at the equilibrium position. - **D. Displacement**: Displacement changes continuously as the object oscillates. It is not constant, as it varies from maximum positive to maximum negative values. ### Summary of Key Points: - The total mechanical energy in simple harmonic motion remains constant throughout the motion. - Kinetic energy and potential energy vary as the object oscillates, depending on its position. - Displacement changes continuously, reflecting the oscillatory nature of the motion. - Understanding the conservation of energy in SHM is crucial for analyzing the motion of oscillating systems. This thorough understanding of the energy transformations in simple harmonic motion is essential for solving related problems in physics.