For an oscillating spring, the restoring force, and consequently the acceleration, are greatest and positive at
. These quantities decrease as
x approaches the equilibrium position and are zero at
x =
0. The restoring force and acceleration—which are now negative—increase in magnitude as
x approaches
and are maximally negative at
.
Important Properties of a Mass on a Spring
There are a number of important properties related to the motion of a mass on a spring, all of which are fair game for SAT II Physics. Remember, though: the test makers have no interest in testing your ability to recall complex formulas and perform difficult mathematical operations. You may be called upon to know the simpler of these formulas, but not the complex ones. As we mentioned at the end of the section on pulleys, it’s less important that you memorize the formulas and more important that you understand what they mean. If you understand the principle, there probably won’t be any questions that will stump you.
Period of Oscillation
The period of oscillation, T, of a spring is the amount of time it takes for a spring to complete a round-trip or cycle. Mathematically, the period of oscillation of a simple harmonic oscillator described by Hooke’s Law is:
This equation tells us that as the mass of the block, m, increases and the spring constant, k, decreases, the period increases. In other words, a heavy mass attached to an easily stretched spring will oscillate back and forth very slowly, while a light mass attached to a resistant spring will oscillate back and forth very quickly.
Frequency
The frequency of the spring’s motion tells us how quickly the object is oscillating, or how many cycles it completes in a given timeframe. Frequency is inversely proportional to period:
Frequency is given in units of cycles per second, or hertz (Hz).
Potential Energy
The potential energy of a spring (
) is sometimes called elastic energy, because it results from the spring being stretched or compressed. Mathematically,
is defined by:
The potential energy of a spring is greatest when the coil is maximally compressed or stretched, and is zero at the equilibrium position.
Kinetic Energy
SAT II Physics will not test you on the motion of springs involving friction, so for the purposes of the test, the mechanical energy of a spring is a conserved quantity. As we recall, mechanical energy is the sum of the kinetic energy and potential energy.
At the points of maximum compression and extension, the velocity, and hence the kinetic energy, is zero and the mechanical energy is equal to the potential energy,
Us= 1/2
.
