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# Simple Harmonic Motion

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<!-- @@image_id="IntPhysC-05-05-Foucault-pendulum" -->

A Foucault pendulum is a pendulum suspended from a long wire, that is sustained in motion over long periods. Due to the axial rotation of the earth, the plane of motion of the pendulum shifts at a rate and direction dependent on its latitude: clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere. At the poles the plane rotates once per day, while at the equator it does not rotate at all.

### Simple Harmonic Motion

Many objects vibrate or oscillate – an object on the end of a spring, a tuning fork, the balance wheel of a watch, a pendulum, the strings of a guitar or a piano.  When we speak of a vibration or oscillation, we mean the motion of an object that repeats itself, back and forth, over the same path.  This motion is also known as simple harmonic motion , often denoted as SHM.

A useful design for examining SHM is an object attached to the end of a spring and laid on a surface. The surface supports the object so its weight (the force of gravity) doesn’t get involved in the forces. The spring is considered to be weightless.

The position shown in the illustration is the equilibrium position . This position is the middle, where the spring is not exerting any force either to the left or to the right. If the object is pulled to the right, the spring will be stretched and exert a restoring force to return to the weight to the equilibrium position. Similarly, if the object is pushed to the left, the spring will be compressed and will exert a restoring force to return the object to its original position. The magnitude of the restoring force, $F$ , in either case must be directly proportional to the distance, $x$ , the spring has been stretched or compressed.  (A spring must be chosen that obeys this requirement.)

$F = -kx$

In the equation above, the constant of proportionality is called the spring constant .  The spring constant is represented by k and its units are N/m.  This equation is accurate as long as the spring is not compressed to the point that the coils touch nor stretched beyond elasticity.

Suppose the spring is compressed a distance $x = A$ , and then released.  The spring exerts a force on the mass pushing it toward the equilibrium position. When the mass is at the maximum displacement position, velocity is zero because the mass is changing direction. At the position of maximum displacement, the restoring force is at its greatest - the acceleration of the mass will be greatest. As the mass moves toward the equilibrium position, the displacement decreases, so the restoring force decreases and the acceleration decreases. When the mass reaches the equilibrium position, there is no restoring force. The acceleration, therefore, is zero, but the mass is moving at its highest velocity. Because of its inertia, the mass will continue past the equilibrium position, and stretch the string. As the spring is stretched further, the displacement increases, the restoring force increases, the acceleration toward the equilibrium position increases, and the velocity decreases. Eventually, when the mass reaches its maximum displacement on this side of the equilibrium position, the velocity has returned to zero and the restoring force and acceleration have returned to the maximum. In a frictionless system, the mass would oscillate forever, but in a real system, friction gradually reduces the motion until the mass returns to the equilibrium position and motion stops.

Imagine an object moving in uniform circular motion. Remember the yo-yo we spin over our heads? In your mind, turn the circle so that you are looking at it on edge; imagine you are eight feet tall, and the yo-yo's circle is exactly at eye level. The object will move back forth in the same way that a mass moves in SHM. It moves consistently from the far left to the far right until you stop spinning the yo-yo. Another example is to imagine a glowing light bulb riding a merry-go-round at night. You are sitting in a chair at some distance from the merry-go-round so that the only part of the system that is visible to you is the light bulb.  The movement of the light will appear to you to be back and forth in simple harmonic motion. Circular motion and simple harmonic motion have a lot in common.

The greatest displacement of the mass from the equilibrium position is called the amplitude of the motion. One cycle refers to the complete to-and-fro motion that starts at some position, goes all the way to one side, then all the way to the other side, and returns to the original position. The period , $T$ , is the time required for one cycle and the frequency , $f$ , is the number of cycles that occur in exactly 1.00 second. The frequency, in this case, is the reciprocal of the period.

$f=\frac{1}{T}$

Example Problem: When a 500. kg crate of cargo is placed in the bed of a pickup truck, the truck’s springs compress 4.00 cm.  Assume the springs act as a single spring.

(a) What is the spring constant for the truck springs?

(b) How far with the springs compress if 800. kg of cargo is placed in the truck bed?

Solution:

(a) $k=\frac{F}{x}=\frac{(500. \ \text{kg})(9.80 \ \text{m/s}^2)}{0.0400 \ \text{m}}=1.23 \times 10^5 \ \text{N/m}$

(b) $x=\frac{F}{k}=\frac{(800. \ \text{kg})(9.80 \ \text{m/s}^2)}{1.23 \times 10^5 \ \text{N/m}}=0.064 \ \text{m}=6.4 \ \text{cm}$

#### Summary

• Simple harmonic motion occurs in many situations, including an object of the end of a spring, a tuning fork, a pendulum, and strings on a guitar or piano.
• A mass oscillating on a horizontal spring is often used to analyze SHM.
• The restoring force for a mass oscillating on a horizontal spring is related to the displacement of the mass from its equilibrium position, $F = -kx$ .
• SHM is related to uniform circular motion when the uniform circular motion is viewed in one dimension.

#### Practice

Use this resource to answer the questions that follow.

1. What is the graph produced by a swinging pendulum's motion graphed over time?
2. How does the Exploratorium demonstrate the relationship between simple harmonic motion and circular motion? Is it convincing?
3. Why don't the pendulums all swing at the same rate?

The following website has a set of questions and answers about simple harmonic motion.

#### Review

1. In simple harmonic motion, when the speed of the object is maximum, the acceleration is zero.
1. True
2. False
2. In SHM, maximum displacement of the mass means maximum acceleration.
1. True
2. False
3. If a spring has a spring constant of 1.00 × 10 3 N/m, what is the restoring force when the mass has been displaced 20.0 cm?