Simple Harmonic Motion: A Special Periodic Motion
By the end of this section, you will be able to:
The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. They are also the simplest oscillatory systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator. If the net force can be described by Hooke’s law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in Figure 1. The maximum displacement from equilibrium is called the amplitude X. The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.
Find a bowl or basin that is shaped like a hemisphere on the inside. Place a marble inside the bowl and tilt the bowl periodically so the marble rolls from the bottom of the bowl to equally high points on the sides of the bowl. Get a feel for the force required to maintain this periodic motion. What is the restoring force and what role does the force you apply play in the simple harmonic motion (SHM) of the marble?
Figure 1 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude \(X\) and a period \(T\) . The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period \(T\) . The greater the mass of the object is, the greater the period \(T\) .
What is so significant about simple harmonic motion? One special thing is that the period \(T\) and frequency \(f\) of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant \(k\), which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.
In fact, the mass \(m\) and the force constant \(k\) are the only factors that affect the period and frequency of simple harmonic motion.
What must be the relationship between the restoring force and the displacement equilibrium for a body to undergo simple harmonic motion?
Which is not an example of simple harmonic motion?
The maximum magnitude of displacement from equilibrium of a periodic motion is called as:
Uniform circular motion, simple pendulum, and a block hanged on a spring moving repeatedly illustrate what type of motion?
Which of the following describes the simple harmonic motion in a horizontal-spring block oscillator system?
SIMPLE HARMONIC OSCILLATOR
The period of a simple harmonic oscillator is given by
and, because \(f = 1/T\), the frequency of a simple harmonic oscillator is
Note that neither \(T\) nor \(f\) has any dependence on amplitude.
What do sound waves, water waves, and seismic waves have in common? They are all governed by Newton’s laws and they can exist only when traveling in a medium, such as air, water, or rocks. Waves that require a medium to travel are collectively known as “mechanical waves.”
Find two identical wooden or plastic rulers. Tape one end of each ruler firmly to the edge of a table so that the length of each ruler that protrudes from the table is the same. On the free end of one ruler tape a heavy object such as a few large coins. Pluck the ends of the rulers at the same time and observe which one undergoes more cycles in a time period, and measure the period of oscillation of each of the rulers.
If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping (See Figure 1). Calculate the frequency and period of these oscillations for such a car if the car’s mass (including its load) is 900 kg and the force constant \((k)\) of the suspension system is \(6.53 \times 10^4 \thinspace \mathrm{N/m}\).
Strategy
The frequency of the car’s oscillations will be that of a simple harmonic oscillator as given in the equation \(f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}\). The mass and the force constant are both given.
Solution
Discussion
The values of \(T\) and f both seem about right for a bouncing car. You can observe these oscillations if you push down hard on the end of a car and let go.
If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in Figure 1. Similarly, Figure 2 shows an object bouncing on a spring as it leaves a wavelike "trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.
Figure 1 The bouncing car makes a wavelike motion. If the restoring force in the suspension system can be described only by Hooke’s law, then the wave is a sine function. (The wave is the trace produced by the headlight as the car moves to the right.)
Figure 2 The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.
The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by
where \(X\) is amplitude. At \(t = 0\), the initial position is \(x_{0} = X\), and the displacement oscillates back and forth with a period \(T\). (When \(t = T\), we get \(x = X\) again because \(\cos \thinspace 2 \pi = 1\).). Furthermore, from this expression for x, the velocity v as a function of time is given by:
where \(v _{\mathrm{max}} = 2 \pi X/T = X \sqrt{k/m}\). The object has zero velocity at maximum displacement—for example, \(v = 0\) when \(t = 0\), and at that time \(x = X\). The minus sign in the first equation for \(v (t)\) gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have \(x(t)\),\(v(t)\),\(t\), and \(a(t)\), the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is \(a = F/m = kx/m\). So, \(a(t)\) is also a cosine function:
Hence, \(a(t)\) is directly proportional to and in the opposite direction to \(x(t)\).
Figure 3 shows the simple harmonic motion of an object on a spring and presents graphs of \(x(t),v(t)\) and \(a(t)\) versus time.
Figure 3 Graphs of \(x(t),v(t)\) and \(a(t)\) versus t for the motion of an object on a spring. The net force on the object can be described by Hooke’s law, and so the object undergoes simple harmonic motion. Note that the initial position has the vertical displacement at its maximum value \(X;v\) is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point.
The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.
A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.
Click here to view the simulation page
Solve for the period of oscillation of a block with mass 7 kg hanging on a vertical spring. The spring constant is 154 N/m.
\(s\)
\(T=2\pi \sqrt {7\;kg \over 154\;N/m}\)
Answer: \(1.3396\;s\)
The period of oscillation of a 4-kg block hanging on a vertical spring is 6 seconds. Calculate the spring constant.
\(N \over m\)
\(k={4\pi^2m \over T^2}={4\pi^2(4\;kg) \over (6\;s)^2}\)
Answer: 4.3865 \({N\over m}\)
Which is true about the value of the period of the periodic motion?
Which spring-block oscillator set-up will have the longest period?
A block with a mass of 40 kg is attached at the end of two springs connected parallel to each other and hung on the wall. What is the period of oscillation in seconds of the block if the spring constants are 25 N/m and 40 N/m?
\(T=2\pi \sqrt{m \over k_1 + k_2}=2\pi\sqrt{40\;kg \over 25\;N/m + 40\;N/m}\)
Answer: 4.9289
A 2-kg mass at the end of a spring vibrates with a frequency of 4 Hz. What is the spring constant of the spring?
\(T=2\pi \sqrt{m \over k}\)
\( k=4\pi^2\;{m \over T^2}\;\;\;\text{where}\;\;\; T={1 \over f}={1 \over 4\;Hz}\)
\( k=4\pi^2\;{2\;kg \over 0.25^2}\)
Answer: 1263.3094 \({N \over m}\)
What is the mass of a block hanging on a vertical spring if the period of oscillation is 18 seconds and the spring constant is 16 N/m? Express your answer in kilograms.
\(m={kT^2 \over 4\pi^2}\)
\(m={(16\;N/m)(18\;s)^2 \over 4\pi^2}\)
Answer: \(131.3123\)
A particle of mass 100 g undergoes a simple harmonic motion. The restoring force is provided by a spring with a spring constant of \(40\; N \cdot m^{−1}\). What is the period of oscillation?
A 4-kg mass at the end of a spring vibrates with a frequency of 4 Hz. What is the spring constant of the spring?
\( k=4\pi^2\;{m \over T^2}\; \text{where}\; T={1 \over f}={1 \over 4\;Hz}\)
\( k=4\pi^2\;{4\;kg \over 0.25^2}\)
Answer: 2526.6187 \({N \over m}\)
Three blocks A, B, and C have masses 2 kg, 8 kg, and 4 kg, respectively. If the blocks are hanged on a vertical spring of the same spring constant, which of the following is the correct sequence of decreasing period?
Suppose the mass attached to a spring in a vertical spring-block oscillator is decreased, what will happen to the period of the simple harmonic motion?
In a vertical spring-block oscillator, what is the effect of a stiffer spring to the period of the simple harmonic motion?
Which two of the following relationships are correct for a vertical spring-block oscillator?