In this article, you will learn more about periodic motion, especially springs and pendulums in order to provide the best exam preparation.

Table of Contents

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Springs 

Definition of Springs

A spring is an elastic object used to store mechanical energy in its coils through its ability to stretch and/or be compressed. They are usually referred to as coil springs. As a spring is compressed from its resting (non-mobile) position, it has the capability of exerting an opposing force in relation to the change in length it has undergone.

Restorative Force Capability

Springs

“Springs have restorative forces” Image created by Lecturio

Mechanical energy can be kinetical or potential energy with the former being energy when the spring is not stretched and the latter is energy when the spring is stretched. It is true, therefore, that while kinetic energy increases elastic potential energy will decrease.

The opposing force existing in the mechanical aspects of the spring is referred to restorative force or Hooke’s Law. This is the force required to restore the spring to its resting position or equilibrium position.

The restorative force is dependent on the spring constant and the distance the spring has been displaced from the spring’s equilibrium position. The spring constant is dependent on the strength of the spring. It takes into consideration various physical properties of the skin, such as the material the spring is made from and the diameter of the spring’s thickness. The thicker the spring, the higher the spring constant. In reverse, the thinner the spring, the lower the spring constant.

The equation for the restorative force is:

Fspring = – k x

k = spring constant

x = displacement of the spring from equilibrium

Fspring = restorative force

The calculation requires a negative sign since the spring is trying to pull the spring back to its equilibrium after it is stretched a certain distance or after it is compressed a certain distance. The force is in the opposite direction of the initial stretch or compression. That is the direction in which the spring stretches is opposite to the direction of the force it exerts.

The k (spring constant) has no negative sign as only the magnitude is considered. It is measured in Newton per centimeter or Newton per millimeter.

Springs have an elastic limit referring to the extent to which a spring can be stretched without altering its ability to return to its original length. As long as this limit is not exceeded the spring has elastic behavior but once it is exceeded it portrays plastic behavior. Hooke’s law holds true if the spring remains elastic.

Oscillating Ability of a Spring

Springs have the ability to oscillate, in other words, to produce a repetitive variation in motion between two extremes: stretched and compressed from a midpoint (the resting point or the equilibrium point).

When a spring is stretched and held in place, the object at the end of the spring has no velocity since it is not moving. The force will be in the direction that is opposite the stretch creating the negative sign in Hooke’s Law since the force is trying to restore the position of the spring towards equilibrium. The displacement is referred to as x but the maximum displacement the spring can undergo is denoted as maximum amplitude (A). When the object at the end of the spring is released, the object will be pulled towards equilibrium with a velocity v.

Spring-passes-equilibrium

“Spring passes equilibrium” Image created by Lecturio

At the equilibrium point, x would equal 0 since the spring is passing through this point as it is being compressed to oppose the stretch. The spring continues to compress due to the velocity it has and passes equilibrium until it reaches minimum amplitude where again the velocity is 0. The spring will then have restorative force in the opposite direction to return to equilibrium.

This will continue until the spring finally stops at the resting point. This motion occurring back and forth between stretch and compress is termed an oscillation. Once the spring completes many oscillations and stops moving, it returns to the resting point where x = 0 and the velocity will end up being 0.

Spring Period & Frequency Calculations

One can determine various aspects based on the spring oscillations and its properties. The frequency f indicates the number of cycles per second the spring undergoes. The period P is the time between the oscillating motions. The two traits are inversely related as follows:

P = 1/f

P = period

f = frequency

Let us look at the frequency to determine what properties affect this variable. The frequency is affected by the spring constant k and the spring mass m.

f = (1 / 2) ( )

Thus, P = (2n) ( )

Energy Conservation of a Spring

The spring oscillations create a similar scenario with kinetic and potential energy. When a spring is stretched and subsequently is released, it moves through the equilibrium point. The energy in the beginning before release equals the energy as the spring moves through the equilibrium point. This is the law of energy conservation of a spring.

Thus,

Einitial = Efinal

Energy is a combination of kinetic energy K and potential energy U.

So,

Kinitial + Uinitial = Kfinal + Ufinal

Since there is no kinetic energy before the spring is released after being stretched, Kinitial = 0. Also, since there is no potential energy at equilibrium, Ufinal = 0. This helps simplify the equation:

Uinitial = Kfinal

(1/2) k A2 = (1/2) m v2

This is true since potential energy is dependent on the stretched spring, whereas kinetic energy is dependent on the velocity of the spring motion.

Solving for velocity:

v = A

So, in other words, the stretched spring contains potential energy only. After it is released, the restorative forces convert the potential energy to kinetic energy and, when the spring reaches equilibrium; all the energy is kinetic without any potential energy left. This will occur as it compresses to the minimal amplitude where all is potential energy with no kinetic energy. This is similar to the potential and kinetic energy changes that occur due to gravity’s effect on an object.

Energy-Conservation-of-Springs

“Energy Conservation of Springs” Image created by Lecturio

Applications

This knowledge is used in the making of kitchen scales as well devices that use a spring to measure, car’s suspensions and retractable pens.

Pendulums 

Definition

A pendulum is a weight suspended from a frictionless pivot so that is can swing freely. Initially, before swinging, the pendulum is straight downward which is the resting position or equilibrium position. Once it is displaced sideways, the swinging starts with oscillating motions back and forth. The pendulum swings in this manner since there is a restorative force to return it towards the equilibrium position.

Restorative Force Capability

Pendulums

“Pendulums” Image created by Lecturio

Restorative force of pendulum is the force that brings it back to its equilibrium point after it has been displaced. Similar to a spring, the pendulum has a similar restorative force and an oscillating pattern. The pendulum, once it starts swinging, will pass through the resting position and swing until it reaches a distance past the point, stop, and swing back in the opposite direction.

This oscillating motion will continue until the pendulum eventually stops swinging.

The action of the pendulum acts in this manner since there is no loss to friction or heat. Unlike a spring, the restorative force is dependent on gravity and the angle (ɵ) of motion from the midpoint rather than the mass that is suspended by the pendulum. The pendulum is like the spring since the restorative force for each is dependent on displacement.

F = m g sin ɵ

F = restorative force to return to equilibrium

M = mass of swinging object

g = gravity

ɵ = swing angle from the midpoint

However, for mathematical calculations, sin ɵ = ɵ for very small angles causing:

F = m g ɵ

Pendulum Period & Frequency Calculations

One can determine various aspects based on the pendulum, oscillations and its properties. The frequency f indicates the number of cycles per second the spring undergoes. The period P is the time between the oscillating motions. The two traits are inversely related as follows:

P = 1/f

P = period

f = frequency

Let us look at frequency to determine what properties affect this variable. The frequency is affected by the gravitational energy g and the length of the pendulum L.

f = (1 / 2) ( )

Thus,

P = (2n) ( )

Note that the pendulum, unlike the spring, is not dependent on the mass.

Energy Conservation of a Pendulum

Energy-Conservation-of-Pendulum

“Energy Conservation of Pendulums” Image created by Lecturio

The pendulum oscillations create a similar scenario with kinetic and potential energy. When a pendulum is displaced and subsequently is released, it moves through the equilibrium point.

The energy in the beginning before release equals the energy as the pendulum moves through the equilibrium point. This is the law of energy conservation of a pendulum.

Thus, Einitial = Efinal. However, since there is a difference in the vertical location of the pendulum initially and finally, the equation depicts this: Etop = Ebottom.  Etop represents the energy prior to releasing the pendulum from its displaced position. Ebottom represents the energy at the equilibrium point.

Energy is a combination of kinetic energy K and potential energy U.

So,

Ktop + Utop = Kbottom + Ubottom

Since there is no kinetic energy before the pendulum is released after being stretched, Ktop = 0. Also, since there is no potential energy at equilibrium, Ubottom = 0. This helps simplify the equation:

Utop = Kbottom

mgh = (1/2) m v2

This is true since potential energy is dependent on the displaced pendulum, whereas kinetic energy is dependent on the velocity of the pendulum motion.

Solving for velocity:

v =  √(2gh)

So, in other words, the displaced spring contains potential energy only. After it is released, the restorative forces convert the potential energy to kinetic energy and, when the pendulum reaches equilibrium; all the energy is kinetic without any potential energy left. This will occur as it swings back and forth. This is similar to the potential and kinetic energy changes that occur due to gravity’s effect on a free-falling object or due to the compression/stretching oscillation of a spring.

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