An object is attached to a horizontal spring. It is initially displaced by distance Δx from the equilibrium position and then released from rest. At what point of its motion along the x-axis reaches the object its highest total energy
a) At the equilibrium position
b) At the position where it is released
c) At midway between the amplitude and equilibrium positions
d) The total energy is the same at all positions
e) The object reaches its highest total energy only during a later cycle, not during the first oscillation
My ans: KE is highest at the equilibrium position and PE is highest at maximum displacement.. TE is a linear relation and should be constant (no damping - simple harmonic motion) so I believe answer is d) ?
Your understanding is partially correct. The total energy of an object undergoing simple harmonic motion is indeed constant throughout its motion, assuming there is no damping. However, the correct answer to the question is b) At the position where it is released.
When the object is released from rest, it has a maximum potential energy (PE) due to its displacement from the equilibrium position. As it starts to move, the potential energy is gradually converted into kinetic energy (KE) when the object passes through the equilibrium position. At this point, the object has zero potential energy and maximum kinetic energy.
As the object continues its motion towards the opposite extreme position, the kinetic energy gradually decreases and is converted back into potential energy. At the extreme position, the object has maximum potential energy and zero kinetic energy.
Therefore, the point where the object reaches its highest total energy is at the position where it is released, when it has maximum potential energy and zero kinetic energy.
Your understanding is mostly correct. In a system undergoing simple harmonic motion without any damping, the total mechanical energy is conserved and remains constant throughout the motion. So, the correct answer is indeed d) The total energy is the same at all positions.
To explain why this is the case, let's break down the motion of the object attached to the spring.
When the object is at the equilibrium position (option a), it has no potential energy (PE) but a maximum kinetic energy (KE) since it is moving at its maximum speed.
When the object is at the position where it is released (option b), it has a maximum potential energy due to its maximum displacement from the equilibrium position but zero kinetic energy since it is momentarily at rest.
At the midway point between the amplitude and equilibrium positions (option c), the object has an equal amount of kinetic and potential energy since it is equally displaced from the equilibrium position in both directions. However, the total energy at this point is not higher than at any other position along its motion.
While the object will reach its highest total energy during each oscillation, it is important to note that the same maximum value of total energy will be reached in subsequent cycles as well. So, the statement in option e) is not correct, as the highest total energy is not exclusive to a later cycle.
Overall, the correct answer is d) The total energy is the same at all positions.