IMAT 2013 Q53 [Harmonic Systems]

A mass is connected to a spring and it vibrates up and down, forming a simple harmonic system.

Which of the following are correct?

  1. The kinetic energy of the mass is at a maximum half way up.
  2. The potential energy of the system is at a maximum at the top of the mass’s motion.
  3. The potential energy of the system is at a maximum at the bottom of the mass’s motion.

A. 1, 2 and 3
B. 1 and 2 only
C. 2 only
D. 3 only
E. 1 only

Let’s think about this in terms of the relevant energy at a maximum- we ask ourselves:

In a simple harmonic system:

When is kinetic energy at a maximum?

In simple harmonic motion, kinetic energy is at it’s maximum in equilibrium positions. Since the motion is up and down, the equilibrium is halfway up, or at the halfway point.

When is potential energy at a maximum?

In simple harmonic motion, potential energy is at it’s maximum at the extremes of the system. In this case, that would be the top most and bottom most part of the spring- since it “vibrates up and down”.

Now, we are able to review the statements made, and ask ourselves if they are consistent with our statements:

  1. “The kinetic energy of the mass is at a maximum half way up.” This statement is correct, consistent with our first answer.
  1. The potential energy of the system is at a maximum at the top of the mass’s motion. This statement is correct, consistent with our second answer.
  1. The potential energy of the system is at a maximum at the bottom of the mass’s motion. This statement is correct, consistent with our second answer.
\fcolorbox{red}{grey!30}{Therefore all 3 statements are correct, and our answer is A.}
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