When an object is moving in a simple harmonic motion, which of the following is at a minimum when the displacement from equilibrium is zero?
A. the magnitude of the velocity
B. the magnitude of the acceleration
C. the kinetic energy
D. the total mechanical energy
E. More than one of the above answers.
if x = a sin wt
v = aw cos wt
a = -w^2 x
when x = 0
|v| is MAX so Ke is MAX
total energy is CONSTANT
So only acceleration which is proportional to x
To determine which option is at a minimum when the displacement from equilibrium is zero in simple harmonic motion, we need to analyze the different quantities involved in the motion.
In simple harmonic motion, an object oscillates back and forth around an equilibrium position due to a restoring force. At the equilibrium position, the displacement from equilibrium is zero.
A. The magnitude of velocity: The velocity of an object in simple harmonic motion is maximum at the equilibrium position. As the object moves away from the equilibrium, the velocity decreases until it reaches zero at the furthest points from the equilibrium. Then, it changes direction and increases again towards the equilibrium. So, the magnitude of velocity is not at a minimum when the displacement from equilibrium is zero.
B. The magnitude of acceleration: The acceleration of an object in simple harmonic motion is maximum at the furthest points from the equilibrium, where the displacement is maximum. As the object moves towards the equilibrium, the acceleration decreases until it reaches zero at the equilibrium position. Then, it changes direction and increases again away from the equilibrium. So, the magnitude of acceleration is not at a minimum when the displacement from equilibrium is zero.
C. Kinetic energy: The kinetic energy of an object in simple harmonic motion is directly related to its velocity. Since the magnitude of velocity is not at a minimum when the displacement from equilibrium is zero, the kinetic energy (which is related to velocity) is also not at a minimum.
D. Total mechanical energy: The total mechanical energy in simple harmonic motion is the sum of potential energy and kinetic energy. At the equilibrium position, the potential energy is minimum, as the displacement from equilibrium is zero. Since the total mechanical energy is the sum of potential and kinetic energy, it reaches its minimum value when the displacement from equilibrium is zero.
Therefore, the correct answer is D. The total mechanical energy is at a minimum when the displacement from equilibrium is zero in simple harmonic motion.