Lesson 16 Harmonic Motion

Newton’s second law F = m a combined with a restoring force leads to a motion that repeats itself over a period of time. This is known as harmonic motion. Galileo studied a lamp in a church and timed the lamp’s swing. Galileo observed that each swing took the same amount of time even as the motion died down. This observation led to the development of the first accurate clock. In other words, pendulums can be used to keep time.

Time must be “kept” in music which is also an example of harmonic motion. Musical instruments share a special property with bobbing weights. Both the weights and instruments create vibrations at a certain frequency which produces a certain pitch that doesn’t change as the motion decreases. There are many factors that determine the tone of a musical instrument including: length of string, size, shape, and the technique with which the instrument is played. In the physics behind music, one factor never varies. Once the tone has been produced the pitch is the same even though the vibrations diminish. The pitch remains constant because of force, the F in the equation F = m a.




In a certain position all forces are balanced, however, when a spring is stretched it attempts to pull the mass back to its original position. The farther the mass moves the greater the force to pull the mass back to its original position. Conversely, when a spring is compressed it tries to push the mass back, which ever direction the mass moves, a force emerges to counter the displacement. The combination of this force and the mass’s inertia will be the key to keeping time.

At each point in its motion, the net force is proportional to the distance x of the mass from its equilibrium position and in the opposite direction.

The equation is F = -k x where the value of k depends on how stiff the spring is.

In earlier periods of history, there was a constant need for objects that keep accurate time. Cycles in nature, lunar eclipses, solar eclipses, and solar cycles were all used to tell time. However, with the discovery of harmonic motion, clocks were developed to keep accurate time because all clocks oscillate with precision following the most inexorable law in physics F = m a 

F = m a
F = m d^2x/dt^2
F = -k x
d^2x/dt^2 = -k/m (x)

This equation describes the simple harmonic motion of not only a mass on a spring but any system that when disturbed, returns to equilibrium with a force proportional to the distance.

x = A sin(omega(t))

dx/dt = A omega cos(omega(t))

d^2x/dt^2 = -A (omega)^2 sin(omega(t))

d^2x/dt^2 = -w^2 x

d^2x/dt^2 = -k/m (x)

(omega)^2 == k/m

(omega) = sqrt (k/m)

Frequency = omega = 2pif where omega (radians per second) is equal to 2pi (radians cycle) multiplied by the frequency (cycles per second).

Principles of uniform circular motion and harmonic motion are closely related as viewed in the video below:

Potential energy of harmonic motion can be modeled by a curved shaped bowl.

Potential energy = du/dx = -F

integral(du/dx) = kx

U = k(½)x^2 while K = (½) m v^2

As a ball rolls in the curved shaped bowl there is  a continual interchange between potential and kinetic energy.




U = k(½)x^2                                K = (½) m v^2

x = A sin(omega(t))                     v = omega A cos(omega(t))

U =  k(½) A^2 sin^2(omega(t))

K = (½) m omega^2 A^2 cos^2(omega(t))

omega^2 = k/m

m(omega)^2 = k

E = U + K

U+K = ½ k A^2

E =  ½ k A^2 where the total energy is constant although potential and kinetic energy change.
It takes the same time for a pendulum to complete a swing even if motion dies down. All pendulums of the same length oscillate at the same rate regardless of their mass. In the case of a falling body, if all bodies fall at the same rate, regardless of their mass, then all pendulums should oscillate at the same rate regardless of mass. All bodies fall to earth with the same constant acceleration. The force of gravity, which depends on the mass of the pendulum, will cancel the mass in the equation F = m a. In general, if an object is disturbed it exhibits simple harmonic motion. Below is a clip from the Mechanical Universe that explains harmonic motion: