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10.1: Simple Harmonic Motion

Created by: CK-12

The development of devices to measure time, like the pendulum, led to the analysis of periodic motion. Such motion repeats itself in equal intervals of time (called periods) and is also referred to as harmonic motion. When an object moves back and forth over the same path in harmonic motion it is said to be oscillating. If the distance such an object travels in one oscillation remains constant, it is called simple harmonic motion (SHM). A grandfather clock’s pendulum and the quartz crystal in a modern watch are examples of SHM.

Key Concepts

  • In harmonic motion there is always a restorative force, which attempts to restore the oscillating object to its equilibrium position. The restorative force changes during an oscillation and depends on the position of the object. In a spring the force is given by Hooke’s Law: F = -kx; in a pendulum it is the component of gravity along the path.
  • Objects in simple harmonic motion do not obey the “Big Three” equations of motion because the acceleration is not constant. As a spring compresses, the force (and hence the acceleration) increases. Similarly, as a pendulum swings, the tangential component of the force of gravity changes. The equations of motion for SHM are given in the Key Equations section.
  • The kinetic energy and the speed are at a maximum at the equilibrium point, but the potential energy and restorative force is zero there.
  • At the end points the potential energy is at a maximum, while the kinetic energy and speed are zero. However at the end points the restorative force and acceleration are at a maximum.
  • In SHM since energy is conserved, often, the most fruitful method of calculating position and velocity is to set the total energy equal to the sum of kinetic and potential energies. Similarly force and acceleration are best calculated by using \sum F = ma.

Key Equations and Definitions

\text{Period~Equations}\begin{cases}T = \frac{1}{f} & \text{Period is the inverse of frequency}\\T_{\text{spring}} = 2\pi\sqrt{\frac{m}{k}} & \text{Period of mass m} \text{on a spring with constant k}\\T_{\text{pendulum}} = 2\pi\sqrt{\frac{L}{g}} & \text{Period of a pendulum of length L}\end{cases}

\text{Kinematics of SHM}\begin{cases}x(t) = x_0 +  A\cos{2\pi f(t-t_0)} & \text{Position of an object in SHM of Amplitude A}\\v(t) =  -2\pi f A\sin{2\pi f(t-t_0)} & \text{Velocity of an object in SHM of Amplitude A}\end{cases}



Simple Harmonic Motion Problem Set

    1. What is the force that is leading to simple harmonic motion?
    1. Find the value of the spring constant.
    2. Find the mass of the block of wood.
  5. Give some everyday examples of simple harmonic motion.
  8. The effective k of the diving board shown here is 800\;\mathrm{N/m}. (We say effective because it bends in the direction of motion instead of stretching like a spring, but otherwise behaves the same.) A pudgy diver is bouncing up and down at the end of the diving board, as shown. The y vs t graph is shown below.
    1. What is the distance between the lowest and highest points of oscillation?
    2. What is the y-position of the diver at times t = 0\;\mathrm{s}, t = 2\;\mathrm{s}, and t = 4.6\;\mathrm{s}?
    3. Estimate the man’s period of oscillation.
    4. What is the diver’s mass?
    5. Write the sinusoidal equation of motion for the diver.

    1. Estimate the period T in years.
    3. How long is each cycle?
  13. A mass on a spring on a frictionless horizontal surface undergoes SHM. The spring constant is 550\;\mathrm{N/m} and the mass is 0.400\;\mathrm{kg}. The initial amplitude is 0.300\;\mathrm{m}.
    1. At the point of release find:
      1. the potential energy
      2. the horizontal force on the mass
      3. the acceleration as it is released
    2. As the mass reaches the equilibrium point find:
      1. the speed of the mass
      2. the horizontal force on the mass
      3. the acceleration of the mass
    3. At a point .150\;\mathrm{m} from the equilibrium point find:
      1. the potential and kinetic energy
      2. the speed of the mass
      3. the force on the mass
      4. the acceleration of the mass
    4. Find the period and frequency of the harmonic motion.
  14. A pendulum with a string of 0.750\;\mathrm{m} and a mass of 0.250\;\mathrm{kg} is given an initial amplitude by pulling it upward until it is at a height of 0.100\;\mathrm{m} more than when it hung vertically. This is point P. When it is allowed to swing it passes through point Q at a height of .050\;\mathrm{m} above the equilibrium position, the latter of which is called point R.
    1. Draw a diagram of this pendulum motion and at points P, Q, and R draw velocity and acceleration vectors. If they are zero, state that also.
    2. At point P calculate the potential energy.
    3. At point R calculate the speed of the mass.
    4. At point Q calculate the speed of the mass.
    5. If the string were to break at points P, Q, and R draw the path the mass would take until it hits ground for each point.
    6. Find the tension in the string at point P.
    7. Find the tension in the string at point R.
    8. Find the period of harmonic motion.

Answers to Selected Problems

    1. Buoyant force and gravity
    1. 9.8 \times 10^5 \;\mathrm{N/m}
    2. 0.5 \;\mathrm{mm}
    3. 22 \;\mathrm{Hz}, no,
  1. 3.2 \times 10^3 \;\mathrm{N/m}
  3. .
  4. .
    1. 0.0038 \;\mathrm{s}
    2. 0.0038 \;\mathrm{s}
  5. .
  6. .
  7. 4 times
  8. 0.04 \;\mathrm{m}
    1. 16 \;\mathrm{Hz}
    3. 0.063 \;\mathrm{s}
    1. 24.8 \;\mathrm{J},165 \;\mathrm{N}, 413 \;\mathrm{m/s}^2
    2. 11.1\mathrm{m/s}, 0, 0
    3. 6.2 \;\mathrm{J}, 18.6 \;\mathrm{J}, 9.49 \;\mathrm{m/s}, 82.5 \;\mathrm{N}, 206 \;\mathrm{m/s}^2
    4. .169 \;\mathrm{sec}, 5.9 \;\mathrm{Hz}
  9. b. .245 \;\mathrm{J} c. 1.40\mathrm{m/s} d. 1.00 \;\mathrm{m/s} f. 2.82 \;\mathrm{N} g. 3.10 \;\mathrm{N}

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