## Simple Harmonic Motion

13.1 Simple Harmonic Motion

### Learning Objectives

By the end of this section, you will be able to:

- Define the terms period and frequency
- List the characteristics of simple harmonic motion
- Explain the concept of phase shift
- Write the equations of motion for the system of a mass and spring undergoing simple harmonic motion
- Describe the motion of a mass oscillating on a vertical spring

### Periodic Motion

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.

**Practice!**

Practice 13.1.1 |
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The three sinusoidal functions in the figure show how the position changes in time for a mass moving with periodic motion. Which curve represents motion with the longest period? |

(a) They all have the same period. |

(b) Curve 3 |

(c) Curve 2 |

(d) Curve 1 |

The curve that represents motion with the longest period is **curve 3**. The period is defined as the time to complete one cycle, or one round-trip. To measure the period, you need to measure the time it takes the mass to move from some initial position back to that initial position. In curve 3, the time to go from the initial position and back is the longest time interval.

The motion represented by curve 1 has the shortest period and the motion represented by curve 2 has a period greater than that of curve 1 and less than that of curve 3.

The SI unit for period is the second (s), but sometimes the period is also expressed in units of seconds per cycle.

Practice 13.1.2 |
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The three sinusoidal functions in the figure show how the position of a mass moving with periodic motion changes in time. Which curve represents motion with the greatest frequency? |

(a) Curve 1 |

(b) Curve 2 |

(c) Curve 3 |

(d) They all have the same frequency. |

The curve that represents motion with the greatest frequency is **curve 1**. The frequency is defined as the number of cycles per unit time. Curve 1 completes a cycle in the shortest amount of time so it has the greatest frequency.

Curve 3 completes a cycle in the longest amount of time so it has the lowest frequency of the three curves shown. Curve 2 completes a cycle in a time between that of curve 1 and curve 3, so its frequency is lower than that of curve 1 and greater than that of curve 3.

The SI unit for frequency is the hertz (Hz) but sometimes the frequency is also expressed with units of cycles per second (cycles/s) or inverse seconds (1/s).

Practice 13.1.3 |
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The three sinusoidal functions in the figure show how the position of a mass moving with periodic motion changes in time. Which curve represents motion with the largest amplitude? |

(a) They all have the same amplitude. |

(b) Curve 1 |

(c) Curve 2 |

(d) Curve 3 |

**All three curves represent motion with the same amplitude**. The amplitude is defined as the magnitude of the maximum displacement from the equilibrium position, where the equilibrium position is at *x* = 0. Each curve in the figure has the same maximum value and the same minimum value, so they all must have the same amplitude.

Practice 13.1.4 |
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The three sinusoidal functions in the figure show how the position of a mass moving with periodic motion changes in time. Which curve represents motion with the longest period? |

(a) Curve 3 |

(b) Curve 2 |

(c) Curve 1 |

(d) They all have the same period. |

**All three curves represent motion with the same period**. The period is defined as the time to complete one cycle, or one round-trip. To measure the period, you need to measure the time it takes the mass to move from some initial position back to that initial position. In all three curves shown, the time to go from the initial position and back is the same time interval.

Practice 13.1.5 |
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The three sinusoidal functions in the figure show how the position of a mass moving with periodic motion changes in time. Which curve represents motion with the greatest frequency? |

(a) They all have the same frequency. |

(b) Curve 1 |

(c) Curve 2 |

(d) Curve 3 |

**The three curves represent motion with the same frequency**. The frequency is defined as the number of cycles per unit time. All three curves complete a cycle in the same amount of time so they have equal frequencies.

Practice 13.1.6 |
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The three sinusoidal functions in the figure show how the position of a mass moving with periodic motion changes in time. Which curve represents motion with the greatest amplitude? |

(a) Curve 1 |

(b) Curve 2 |

(c) Curve 3 |

(d) They all have the same amplitude. |

The curve that represents motion with the greatest amplitude is** curve 3**. The amplitude is defined as the magnitude of the maximum displacement from the equilibrium position, where the equilibrium position is at *x* = 0. Curve 3 has the greatest maximum value and the greatest minimum value, so it must have the greatest amplitude.

Curve 1 has the lowest maximum and minimum values so it represents motion with the smallest amplitude. Curve 2 represents motion with amplitude between that of curve 1 and curve 3.

Practice 13.1.7 |
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For an object moving with periodic motion, how is the frequency, f, of the object’s motion related to the period, T, of its motion? |

(a) The frequency is equal to the amplitude, A, divided by the period: f = A/T. |

(b) The frequency is equal to the product of the amplitude, A, and the period: f = AT. |

(c) The frequency is equal to the period: f = T. |

(d) The frequency is equal to the inverse of the period: f = 1/T. |

For an object moving with periodic motion, **the frequency is equal to the inverse of the period: f = 1/T**. The period is the measure of the amount of time to complete one cycle, or seconds per cycle. The inverse of the period is cycles per second, or frequency.

The frequency and period cannot be equal to each other since they are measurements of different properties of the periodic motion. The amplitude, *A*, of the periodic motion does not affect the frequency or the period of the motion so it is not included in the calculation of either *f* or *T*.

Practice 13.1.8 |
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For an object moving with periodic motion, how is the frequency, f, of the object’s motion related to the angular frequency, , of its motion? |

(a) The frequency is equal to the angular frequency: f = . |

(b) The frequency is equal to the angular frequency divided by 2π: [insert: f = /2π . |

(c) The frequency is equal to 2π times the angular frequency: f = 2π. |

(d) The frequency is equal to the inverse of the angular frequency: f = 1/. |

### Simple Harmonic Motion

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**.

**Discuss!**

In a physics lab, you attach a 0.200-kg air-track glider to the end of an ideal spring of negligible mass and start it oscillating. The elapsed time from when the glider first moves through the equilibrium point to the second time it moves through that point is 2.60 s.

Find the spring’s force constant (k).

**Practice!**

Practice 13.1.9 |
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Identical springs with identical masses hanging from them. System 1 is pulled 10 cm from the equilibrium position and released. System 2 is pulled 20 cm from equilibrium and released. Which will have the longer period? |

(a) System 1 will have the longer period. |

(b) System 2 will have the longer period. |

(c) They will have the same period. |

(d) Cannot be determined |

You can try out this experiment yourself, using this simulation: https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html

Practice 13.1.10 |
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If an object of mass m attached to a light spring is replaced by one of mass 9m, the frequency of the vibrating system changes by what factor? |

(a) 1/9 |

(b) 1/3 |

(c) 3.0 |

(d) 9.0 |

(e) 6.0 |

### SHM Equations of Motion

**Practice!**

Practice 13.1.11 |
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A mass oscillates up and down on a spring. Its position as a function of time is shown below. At which of the points shown does the mass have positive velocity and negative acceleration? |

(a) |

(b) |

(c) |

Pause & Predict 13.1.1 |
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What is the mass’s velocity 1.7 s after it is released? |

(a) -0.92 m/s |

(b) -1.6 m/s |

(c) 1.8 m/s |

(d) 0.50 m/s |

Pause & Predict 13.1.2 |
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At what time after it is released will the mass have an acceleration of 6.5 m/s^{2}? |

(a) 0.16 s |

(b) 0.44 s |

(c) 2.3 s |

(d) 0.35 s |

Practice 13.1.12 |
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A particle vibrates in simple harmonic motion with a frequency of 2.60×10^{9} Hz and an amplitude of 1.70×10^{−8} m. What is its maximum speed? |

(a) 88.4 m/s |

(b) 442 m/s |

(c) 278 m/s |

(d) 139 m/s |

What I like about this previous question is how it is a real-life application of simple harmonic motion. If you think of atoms and molecules, you know that they are not static and they actually vibrate. In a material, the bonds between atoms and molecules are not rigid but rather elastic. Because the bonds are elastic and the particles vibrate, we model materials with a mass-spring system in which the masses are the atoms/molecules and the the springs are the bonds. Using this model, we can determine how quickly atoms vibrate in a material, just like you did in this problem.

Practice 13.1.13 |
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A particle on a spring moves in simple harmonic motion along the x axis between turning points at x_{1} = 100 cm and x_{2} = 140 cm. At which of the following positions does the particle have maximum speed? |

(a) 100 cm |

(b) 110 cm |

(c) 120 cm |

(d) none of these positions |

Practice 13.1.14 |
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A particle on a spring moves in simple harmonic motion along the x axis between turning points at x_{1} = 100 cm and x_{2} = 140 cm. At which position does it have maximum acceleration? |

(a) 100 cm |

(b) 110 cm |

(c) 120 cm |

(d) none of these positions |

**Discuss!**

A mass *m* = 2 kg on a spring oscillates with an amplitude of 10 cm. At *t* = 0 its speed is maximum, and is *v* = +2 m/s.

(a) What is the angular frequency of oscillation ω?

(b) What is the spring constant *k*?

You see a graph of the displacement of an object moving in SHM.

Determine the:

(a) Amplitude

(b) Period

(c) Phase constant

(d) Use that information to write an equation for the displacement of the object as a function of time.

A ball moves in SHM, and its acceleration is defined by the equation: *a*(*t*)=−28*x*(*t*) where *x*(*t*) is the time-dependent function of position. What is the period of the ball’s motion?