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]]> When a guitar string is plucked or a spring moves up and down, the time interval between each oscillation is defined as **periodic motion**. The time that one oscillation is completed is called the **period**. In addition to indicating repeated oscillations, a period can also represent one event. The unit of measurement for a period is typically indicated in seconds.

When a period is repeated, the number of oscillations per unit of time is indicated as **frequency**. Mathematically, the frequency is expressed by the following formula:

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

Where,

f = frequency

T = Time period

1 = Constant ** **

The SI unit for frequency is in cycles per second, also known as **hertz (Hz)**. One cycle is equal to one oscillation. Oscillations are repetitive for a number of cycles.

**Review Questions**

**1.** A ______ causes a disturbance in a system that activates an oscillation.

**2.** Waves created by oscillations carry energy.** T/F **

**3.** The restoring force of an object ______ when the deformation is increased.

- is increased
- is decreased
- stays the same

**4.** One cycle is equal to how many oscillations?

**5.** The term _______ refers to the number of oscillations per unit of time.

Oscillations are very common in nature and by human made objects because they occur in so many different ways. One type of oscillation is **simple harmonic motion**. It refers to oscillatory motion that is directly proportional to displacement, and the system in which oscillations occur is called a **simple harmonic oscillator**.

If there is no friction or other nonconservative forces that dampen oscillations, a simple harmonic oscillator will continue to oscillate indefinitely with equal displacement on either side of the equilibrium position. The maximum displacement from equilibrium is called **amplitude**. Amplitude and displacement of objects such as a metal coiled spring are in **meters**, whereas sound oscillations are indicated in units of **pressure**.

A significant fact about simple harmonic motion is that the period *t* and frequency *f* are independent of amplitude. For example, guitar strings will oscillate at the same frequency whether it is plucked gently or hard. For this reason, simple harmonic oscillations are used to operate clocks because the period remains constant.

The only factors that affect the period and frequency of simple harmonic motion are **mass and the force constant k**. Whenever a harmonic oscillator is stiff, a large force k is required to activate it. Also, it will have a smaller time period than an object that is less stiff. The period of a harmonic oscillator is impacted by its mass. The more massive the system is, the longer its period.

All simple harmonic motions are related to sine and cosine waves. The displacement is a function of time in any harmonic motion as oscillations occur with a period T. The velocity of the motion is also a function of time. At maximum displacement from equilibrium, velocity and time are zero.

One type of simple harmonic oscillator is a **simple pendulum**. A simple pendulum is an object that has a small mass, which is suspended by a light wire or string.

When a simple pendulum is displaced from equilibrium, it swings in an arc. The length of the displacement is called the *arc length* and is identified as s. When displacement occurs, a restoring force is created that is in the direction towards the equilibrium position. This restoring force is directly proportional to the displacement.

**Two factors** affect the period of a simple pendulum, which is the time duration at which one oscillation takes place. **One factor** is the length of the string or wire, and the **second factor** is the acceleration due to gravity. The period T is nearly independent of amplitude and mass.

Fig.1: Simple Pendulum – Harmonic Oscillator

**Review Questions**

**6.** A simple harmonic motion is never capable of oscillating indefinitely. **T/F**

**7.** A significant fact about simple harmonic motion is that _____ is independent of amplitude.

- the period
- frequency
- Both a and b

**8.** Which factor is true about affecting the period and frequency of simple harmonic motion?

- The less stiff an object is, the smaller its time period.
- Whenever a harmonic oscillator is stiff, a large force is required to activate.
- The more massive a system is, the longer the period.
- All of the above

**9.** When an object oscillates and reaches its maximum displacement, velocity and time are ______.

**10.** List two factors that affect the time period of a simple pendulum.

**Energy and the Simple Harmonic Oscillator**

A simple harmonic oscillator has both potential energy and kinetic energy. When an object is deformed and at the moment it is not moving, it has stored potential energy.

Because a simple harmonic oscillator has no dissipative forces, it has kinetic energy. Therefore, as an undamped simple harmonic motion takes place, the energy oscillates back and forth between kinetic and potential energy. An example is the oscillations of a spring. When it is completely compressed and is not moving, all energy is stored as potential energy. When the spring decompresses, the elastic potential energy is converted to kinetic energy. At the equilibrium position, the entire energy of the spring is kinetic. As it moves passed equilibrium, the energy in the spring is converting back to potential energy.

When a simple harmonic oscillation has reached its maximum displacement position, the velocity is zero. In this position, all of the energy is in the potential form and there is no kinetic energy. As the restoring force causes the oscillation to move towards equilibrium, the potential energy decreases, and the kinetic energy increases. Energy is shared by both, but the total energy does not change. When the oscillation reaches the equilibrium position, its velocity is at a maximum level.

**Maximum velocity depends on three factors:**

- Maximum velocity is directly proportional to amplitude.
- Maximum velocity is greater for stiffer objects.
- Maximum velocity is smaller for objects that have larger masses.

When an object moves in a circular path with a constant angular velocity and uniform circular motion, a simple harmonic motion takes place. The motion is back and forth on the x-axis. The period T of an oscillator is the time it takes for the object to make one complete revolution.

When viewing a merry-go-round from a distance, any object exhibits simple harmonic motion when it goes back and forth between left and right positions as it turns to create uniform circular motion.

**Review Questions**

**16.** Give an example of when the damping of an oscillator is desirable.

**17.** As the oscillations of harmonic motion slow down due to damping, the net force _____.

- increases
- decreases
- stays the same

**18.** _______ damping refers to a system that is slow and sluggish.

- Over
- Under
- Critical

**19.** When driving an object with a frequency equal to its natural frequency, a condition called ______ occurs.

**20.** Whenever the damping of a harmonic oscillator becomes smaller, the amplitude of the oscillator also becomes smaller. **T/F**

**Review Answers**

**force****T****a****One****Frequency****F****c****d****zero****Length and acceleration due to gravity****c****zero****T****It remains constant****A****The shocks on an automobile****b****c****resonance****F**

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]]>The post What is Hooke’s Law | Definition Formula appeared first on Electrical Academia.

]]>There are many examples of things that oscillate. Oscillations refer to an object or system that moves back and forth between two points and creates waves. A guitar string vibrates back and forth when plucked; the oscillations of the cone inside an audio speaker makes it produce sound waves that travel through the air; a child that plays on a swing goes back and forth is an oscillatory motion; even some atoms vibrate, especially when exposed to elevated temperatures.

All types of oscillations involve force and energy. A force causes a disturbance that activates the oscillation, and as the waves move from one location to another, energy is carried by them.

**Hooke’s Law Example: Stress and Strain**

If the end of a plastic ruler used for measuring is held in one hand and the other end is bent, it will oscillate when the bent end is allowed to return to its original position. The oscillations take place because of the **restoring force** that opposes displacement. This force is from the ruler being deformed. As the ruler reaches its original shape, called its equilibrium, its momentum carries it passed this position during the first oscillation, causing the opposite deformation and movement. This process is repeated until dissipative forces dampen the motion and the oscillations.

The most basic oscillations occur when the restoring force is directly proportional to displacement. The relationship between force and displacement is referred to as Hooke’s law.

Mathematically, it is expressed by the following formula:

\[\text{F=-kx}\]

**Where, **

F = Restoring Force

x = Deformation (Displacement from equilibrium)

k = Constant (related to the difficulty in deforming the object)

The minus sign indicates the restoring force is in the opposite direction of the displacement. The constant value indicated by k is dependent on the stiffness, or rigidity of the object. The greater the constant value, the larger the restoring force and the stiffer the object. The unit of measurement for k is Newton’s per meter (N/m).

** **The deformation of an object requires work. Work equals force times distance. Therefore when plucking a guitar string or compressing a spring, a force must cause an object to move through a distance. If the deformed object does not return to its equilibrium position, such as a compressed spring, potential energy is stored. Mathematically the potential energy stored in the spring is:

\[\text{P}{{\text{E}}_{\text{EL}}}\text{=}\frac{\text{1}}{\text{2}}\text{k}{{\text{x}}^{\text{2}}}\]

**Where, **

PE_{EL} = Elastic potential energy

x = displacement from equilibrium

k = Constant (related to the difficulty in deforming the object)

Graphically, we can prove the above formula for potential energy using a simple triangle:

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