SAT Physics Simple Harmonic Motion - Terms Related To SHM

SAT Physics Simple Harmonic Motion - Terms Related To SHM

Period, T, is the time in seconds for an object in SHM to complete one full oscillation. An example of a full oscillation would be the movement of a pendulum from its initial starting position back to that position again. When more than one oscillation is given, the period can be determined by dividing the total time by the number of oscillations:
Note that the number of oscillations has no units. It is simply the count of an event.

Frequency, f, is the number of full oscillations that occur in 1 second. Divide the number of oscillations by the time interval to determine the frequency.
Frequency is the reciprocal of period and therefore has units of 1 /seconds, which is also known as Hertz (Hz). The relationship between period and frequency can be expressed as follows:
Period and frequency are inversely proportional. As one increases, the other decreases.

Determining the Period from the Frequency
A plucked guitar string vibrates at a frequency of 100 Hertz. What is the period of vibration of the string?


Period is the reciprocal of frequency.

Amplitude, A, is the magnitude of the maximum displacement, A - xmax, of an oscillating particle or wave relative to its rest position. Amplitude is a measure of intensity of the oscillation and is directly proportional to the energy imparted into the oscillating system. The amplitude does not affect the period or the frequency.
The amplitude can be assigned as either a positive or negative value, depending on which side of the equilibrium position is being described. For example, the amplitude of a plucked guitar string can either be above or below the equilibrium position of the string at rest. These two maximum amplitudes are individually assigned values of +A and -A, respectively.

Determining the Spring Constant
Imagine a mass suspended by a spring, as represented in Figure 14.1. The force of gravity pulls the mass toward Earth while the restorative force of the spring pulls the mass upward in an effort to restore the spring to its original, unstretched position. At equilibrium, the mass will be at rest and the spring will be stretched by an amount proportional to the force of gravity upon the mass.
Figure 14.1. Hooke’s law

The restorative force of a spring, Fs, is represented by the following equation, known as Hooke’s law:
Fs = kx
In Hooke’s law, x represents the displacement of the spring from its unstretched position, in meters, and k represents the spring constant, in newtons per meter. The spring constant, k, is specific to each spring, regardless of the mass suspended from the spring.
The equilibrium position is reached when the restorative force of the spring is equal in magnitude but opposite in direction to the gravity force acting on the mass. This can be described by setting the force of gravity equal to but opposite in sign to the restorative force of the spring.
Fs = Fg
Kx = mg

Determining the Spring Constant

A 4.0-kilogram mass is suspended by a spring. In its equilibrium position, the mass has extended the spring 0.10 meters beyond its unstretched length. What is the spring constant of this spring?


Equilibrium is reached when the force of gravity is equal in magnitude but opposite in direction to the restorative force of the spring.
Remember that this value for the spring constant will be the same for this particular spring regardless of any other masses suspended by the spring. This concept will become important in an example later in the chapter.

Determining the Period of a Spring in SHM
The period of a spring in SHM is the amount of time, in seconds, in which the mass on a spring moves from an initial position back to that same initial position. The actual motion of a spring-mass system follows the same linear path up and down or back and forth. However, if the motion of an oscillating spring is graphed versus time, the resulting function is a sine wave. Figure 14.2 is a graph of position versus time. The positions and stretch of a vertically oriented spring-mass system have been superimposed on the graph at five key locations.

Figure 14.2. Position versus time

The sine wave represents the predictable,periodic nature of an oscillationg spring-mass system during a time interval. If the mass hanging from a spring were released at position 1 in Figure 14.2, the mass would descend vertically, passing through equilibrium at position 2. The mass would reach maximum displacement at position 3, where it would reverse direction. On the return trip, the mass would pass through equilibrium at position 2 a second time and would return to its starting location at position 1. One complete cycle occurs when the oscillator returns to its initial position (position 1) for the first time. The period is the time of one complete cycle. For a spring-mass system, the period, Ts,can be determined as follows:
You should note that the period depends on only the mass attached to the spring, m, and the spring constant, k. The period DOES NOT depend on the displacement of the spring or the gravity in which the spring is oscillating.

Determining Period and Frequency of an Oscillating Spring

A 4.0-kilogram mass is suspended from an unstretched spring. When released from rest, the mass moves a maximum distance of 0.20 meters before reversing direction. What are the period and frequency of this spring-mass oscillator?


For problems involving a spring, you must determine the spring constant first.
The mass moves a distance of 0.20 meters before it reverses direction. This is the entire up-down motion. The equilibrium position is in the middle of this motion.So it occurs at 0.10 m.
Apply the spring constant to the formula for the period of a spring-mass oscillator.
Frequency is simply the reciprocal of period.


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