The m ’s and the k ’s cancel. The result is , or about 86% of the amplitude.(b) The total energy is ½kA 2 . At some position x , the potential energy will be ½ of its maximum
value. At that point, ½kx 2 = ½(½kA 2 ). Canceling and solving for x , it is found thatThis works out to about 70% of the maximum amplitude.
(c) Since we solved in terms of A , we can just look at our answers to (a) and (b). The velocity is half
maximum at , or 86%, of A ; the potential energy is half maximum at or 71%, of A .Therefore, the mass is farther from equilibrium when velocity is half maximum.Rapid Review
• An oscillation is motion that regularly repeats itself over the same path. Oscillating objects are acted
on by a restoring force.
• One repetition of periodic motion is called a cycle. The maximum displacement of an oscillating object
during a cycle is the object’s amplitude. The time it takes for an object to go through a cycle is the
period of oscillation.
• Period is related to frequency: T = 1/f , and f = 1/T .
• When solving problems that involve springs or simple pendulums, be on the lookout for ways to apply
conservation of energy. Not every simple harmonic motion problem will require you to use
conservation of energy, but many will.
• The position–time graph of an object in simple harmonic motion is a cosine graph. Specifically, the
position of the object is found by the equation x = A cos(ωt ), where