When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time (Figure 15.2). The string vibrates around an equilibrium position, and one oscillation is completed when the string starts from the initial position, travels to one of the extreme positions, then to the other extreme position, and returns to its initial position. We define periodic motion to be any motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by a child swinging on a swing. In this section, we study the basic characteristics of oscillations and their mathematical description.
Figure 15.2 When a guitar string is plucked, the string oscillates up and down in periodic motion. The vibrating string causes the surrounding air molecules to oscillate, producing sound waves. (credit: Yutaka Tsutano)
In the absence of friction, the time to complete one oscillation remains constant and is called the period (T) . Its units are usually seconds, but may be any convenient unit of time. The word ‘period’ refers to the time for some event whether repetitive or not, but in this chapter, we shall deal primarily in periodic motion, which is by definition repetitive.
A concept closely related to period is the frequency of an event. Frequency (f) is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is
f = 1 T . f = 1 T .The SI unit for frequency is the hertz (Hz) and is defined as one cycle per second:
1 Hz = 1 cycle s or 1 Hz = 1 s = 1 s −1 . 1 Hz = 1 cycle s or 1 Hz = 1 s = 1 s −1 .A cycle is one complete oscillation .
Ultrasound machines are used by medical professionals to make images for examining internal organs of the body. An ultrasound machine emits high-frequency sound waves, which reflect off the organs, and a computer receives the waves, using them to create a picture. We can use the formulas presented in this module to determine the frequency, based on what we know about oscillations. Consider a medical imaging device that produces ultrasound by oscillating with a period of 0.400 μ s 0.400 μ s . What is the frequency of this oscillation?
This frequency of sound is much higher than the highest frequency that humans can hear (the range of human hearing is 20 Hz to 20,000 Hz); therefore, it is called ultrasound. Appropriate oscillations at this frequency generate ultrasound used for noninvasive medical diagnoses, such as observations of a fetus in the womb.
A very common type of periodic motion is called simple harmonic motion (SHM) . A system that oscillates with SHM is called a simple harmonic oscillator .
In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.
A good example of SHM is an object with mass m attached to a spring on a frictionless surface, as shown in Figure 15.3. The object oscillates around the equilibrium position, and the net force on the object is equal to the force provided by the spring. This force obeys Hooke’s law F s = − k x , F s = − k x , as discussed in a previous chapter.
If the net force can be described by Hooke’s law and there is no damping (slowing down due to friction or other nonconservative forces), then a simple harmonic oscillator oscillates with equal displacement on either side of the equilibrium position, as shown for an object on a spring in Figure 15.3. The maximum displacement from equilibrium is called the amplitude (A) . 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.
in its motion. In figure (a) the mass is displaced to a position x = A to the right of x =0 and released from rest (v=0.) The spring is stretched. The force on the mass is to the left. The free body diagram has weight w down, the normal force N up and equal to the weight, and the force F to the left. (b) The mass is at x = 0 and moving in the negative x-direction with velocity – v sub max. The spring is relaxed. The Force on the mass is zero. The free body diagram has weight w down, the normal force N up and equal to the weight. (c) The mass is at minus A, to the left of x = 0 and is at rest (v =0.) The spring is compressed. The force F is to the right. The free body diagram has weight w down, the normal force N up and equal to the weight, and the force F to the right. (d) The mass is at x = 0 and moving in the positive x-direction with velocity plus v sub max. The spring is relaxed. The Force on the mass is zero. The free body diagram has weight w down, the normal force N up and equal to the weight. (e) the mass is again at x = A to the right of x =0 and at rest (v=0.) The spring is stretched. The force on the mass is to the left. The free body diagram has weight w down, the normal force N up and equal to the weight, and the force F to the left." width="727" height="546" />
Figure 15.3 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. In the above set of figures, a mass is attached to a spring and placed on a frictionless table. The other end of the spring is attached to the wall. The position of the mass, when the spring is neither stretched nor compressed, is marked as x = 0 x = 0 and is the equilibrium position. (a) The mass is displaced to a position x = A x = A and released from rest. (b) The mass accelerates as it moves in the negative x-direction, reaching a maximum negative velocity at x = 0 x = 0 . (c) The mass continues to move in the negative x-direction, slowing until it comes to a stop at x = − A x = − A . (d) The mass now begins to accelerate in the positive x-direction, reaching a positive maximum velocity at x = 0 x = 0 . (e) The mass then continues to move in the positive direction until it stops at x = A x = A . The mass continues in SHM that has an amplitude A 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 SHM? For one thing, the period T and frequency f of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, oscillates with the same frequency whether plucked gently or hard.
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 SHM. To derive an equation for the period and the frequency, we must first define and analyze the equations of motion. Note that the force constant is sometimes referred to as the spring constant.
Consider a block attached to a spring on a frictionless table (Figure 15.4). The equilibrium position (the position where the spring is neither stretched nor compressed) is marked as x = 0 x = 0 . At the equilibrium position, the net force is zero.
Figure 15.4 A block is attached to a spring and placed on a frictionless table. The equilibrium position, where the spring is neither extended nor compressed, is marked as x = 0 . x = 0 .
Work is done on the block to pull it out to a position of x = + A , x = + A , and it is then released from rest. The maximum x-position (A) is called the amplitude of the motion. The block begins to oscillate in SHM between x = + A x = + A and x = − A , x = − A , where A is the amplitude of the motion and T is the period of the oscillation. The period is the time for one oscillation. Figure 15.5 shows the motion of the block as it completes one and a half oscillations after release. Figure 15.6 shows a plot of the position of the block versus time. When the position is plotted versus time, it is clear that the data can be modeled by a cosine function with an amplitude A and a period T. The cosine function cos θ cos θ repeats every multiple of 2 π , 2 π , whereas the motion of the block repeats every period T. However, the function cos ( 2 π T t ) cos ( 2 π T t ) repeats every integer multiple of the period. The maximum of the cosine function is one, so it is necessary to multiply the cosine function by the amplitude A.
x ( t ) = A cos ( 2 π T t ) = A cos ( ω t ) . x ( t ) = A cos ( 2 π T t ) = A cos ( ω t ) .Recall from the chapter on rotation that the angular frequency equals ω = d θ d t ω = d θ d t . In this case, the period is constant, so the angular frequency is defined as 2 π 2 π divided by the period, ω = 2 π T ω = 2 π T .
graph to the next using a blue line, creating a graph of the position (horizontal) dependence on time (vertical). The x = 0 position is at the center of the horizontal surface. In the top graph, the mass is at x = +A, the net force is to the left and is equal to – k A. The spring is stretched the maximum amount. The time is t = 0. In the second graph, the mass is between x = +A/2 and x = A, the net force is to the left and smaller than in the previous graph. The spring is stretched less than at t=0. In the third graph, the mass is at x = 0, there is no net force. The spring is relaxed. The time is t = one quarter T. In the fourth graph, the mass is between x = -A/2 and x = -A, the net force is to the right. The magnitude of the force is the same as that in the second graph. The spring is somewhat compressed. In the fifth graph, the mass is at x = -A, the net force is to the right and is equal to + k A. The spring is compressed the maximum amount. The time is t = 1/2 T. In the sixth graph, the mass is between x = -A/2 and x = -A, the net force is to the right. The magnitude of the force is the same as that in the second graph. The spring is somewhat compressed. This graph is identical to the fourth graph. In the seventh graph, the mass is at x = 0, there is no net force. The spring is relaxed. The time is t = 3/4 T. This graph is identical to the third graph. In the eighth graph, the mass is between x = +A/2 and x = A, the net force is to the left. This graph is identical to the second graph. In the ninth graph, the mass is at x = +A, the net force is to the left and is equal to – k A. The spring is stretched the maximum amount. The time is t = 0. This graph is identical to the first (top) graph. The remaining four graphs repeat the second, third, fourth and fifth graphs, with the eleventh graph’s time at t = 1 and 1/4 T and the thirteenth at t = 1 and 1/2 T. The curve connecting the positions of the mass forms a vertical sinusoidal curve." width="961" height="672" />
Figure 15.5 A block is attached to one end of a spring and placed on a frictionless table. The other end of the spring is anchored to the wall. The equilibrium position, where the net force equals zero, is marked as x = 0 m . x = 0 m . Work is done on the block, pulling it out to x = + A x = + A , and the block is released from rest. The block oscillates between x = + A x = + A and x = − A x = − A . The force is also shown as a vector.
Figure 15.6 A graph of the position of the block shown in Figure 15.5 as a function of time. The position can be modeled as a periodic function, such as a cosine or sine function.
The equation for the position as a function of time x ( t ) = A cos ( ω t ) x ( t ) = A cos ( ω t ) is good for modeling data, where the position of the block at the initial time t = 0.00 s t = 0.00 s is at the amplitude A and the initial velocity is zero. Often when taking experimental data, the position of the mass at the initial time t = 0.00 s t = 0.00 s is not equal to the amplitude and the initial velocity is not zero. Consider 10 seconds of data collected by a student in lab, shown in Figure 15.7.
Figure 15.7 Data collected by a student in lab indicate the position of a block attached to a spring, measured with a sonic range finder. The data are collected starting at time t = 0.00 s, t = 0.00 s, but the initial position is near position x ≈ − 0.80 cm ≠ 3.00 cm x ≈ − 0.80 cm ≠ 3.00 cm , so the initial position does not equal the amplitude x 0 = + A x 0 = + A . The velocity is the time derivative of the position, which is the slope at a point on the graph of position versus time. The velocity is not v = 0.00 m/s v = 0.00 m/s at time t = 0.00 s t = 0.00 s , as evident by the slope of the graph of position versus time, which is not zero at the initial time.
The data in Figure 15.7 can still be modeled with a periodic function, like a cosine function, but the function is shifted to the right. This shift is known as a phase shift and is usually represented by the Greek letter phi ( ϕ ) ( ϕ ) . The equation of the position as a function of time for a block on a spring becomes
x ( t ) = A cos ( ω t + ϕ ) . x ( t ) = A cos ( ω t + ϕ ) .This is the generalized equation for SHM where t is the time measured in seconds, ω ω is the angular frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters, and ϕ ϕ is the phase shift measured in radians (Figure 15.8). It should be noted that because sine and cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or sine function.
Figure 15.8 (a) A cosine function. (b) A cosine function shifted to the left by an angle ϕ ϕ . The angle ϕ ϕ is known as the phase shift of the function.
The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the position equation:
v ( t ) = d x d t = d d t ( A cos ( ω t + ϕ ) ) = − A ω sin ( ω t + ϕ ) = − v max sin ( ω t + ϕ ) . v ( t ) = d x d t = d d t ( A cos ( ω t + ϕ ) ) = − A ω sin ( ω t + ϕ ) = − v max sin ( ω t + ϕ ) .
Because the sine function oscillates between –1 and +1, the maximum velocity is the amplitude times the angular frequency, v max = A ω v max = A ω . The maximum velocity occurs at the equilibrium position ( x = 0 ) ( x = 0 ) when the mass is moving toward x = + A x = + A . The maximum velocity in the negative direction is attained at the equilibrium position ( x = 0 ) ( x = 0 ) when the mass is moving toward x = − A x = − A and is equal to − v max − v max .
The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:
a ( t ) = d v d t = d d t ( − A ω sin ( ω t + ϕ ) ) = − A ω 2 cos ( ω t + φ ) = − a max cos ( ω t + ϕ ) . a ( t ) = d v d t = d d t ( − A ω sin ( ω t + ϕ ) ) = − A ω 2 cos ( ω t + φ ) = − a max cos ( ω t + ϕ ) .
The maximum acceleration is a max = A ω 2 a max = A ω 2 . The maximum acceleration occurs at the position ( x = − A ) ( x = − A ) , and the acceleration at the position ( x = − A ) ( x = − A ) and is equal to a max a max .
In summary, the oscillatory motion of a block on a spring can be modeled with the following equations of motion:
