# Periodic Motion

Periodic motion is a type of motion that repeats itself after a certain amount of time. For instance, a rocking chair moving back and forth or a swinging pendulum are both examples of periodic motion. The time it takes for the pendulum to swing back and forth once is called the period 'T', which is measured in seconds. The frequency 'f', on the other hand, tells us how many times the event happens in one second (measured in Hertz).

All events that repeat after a certain amount of time are said to be periodic, and they all have a frequency. This relationship is expressed in an equation. While periodic motion can be seen in various events, it's helpful to distinguish between specific types of motion such as circular motion and simple harmonic motion.

Circular motion happens around a central point with a constant radius and speed. Examples of circular motion include the blades of a ceiling fan rotating. The distance of rotation, also known as radius, stays constant at all times. The object's velocity changes continuously because velocity depends on both speed and direction.

If you want to calculate the force of circular motion, you can use the following equation:

F = m*v^2/r

Where F is the centripetal force in Newtons, m is the mass of the object in kg, v is the velocity of the object in m/s, and r is the radius of the object's orbit. Understanding periodic motion is crucial to understanding how things move in our daily lives. By using keywords like "periodic motion" and "circular motion", this article can rank well in search engines.

When an object moves in a circular motion with a radius of r and a period of T, its angular velocity (ω) can be determined using the formula below:

ω = 2πf = 2π/T

Here, f is the frequency in Hertz. The speed (v) of the object and its angular velocity are related, and this relation can be expressed as:

v = ωr

For example, if a tennis ball is attached to a stick with a 1.2m long rope and is moving at a constant velocity of 28.2 m/s in a circular motion around the stick, we can use the centripetal force equation to calculate the force acting on the ball. The equation is:

F = mv^2/r

The mass of the ball is 60g, and the length of the rope (r) is 1.2m. To calculate F, we can convert the mass to kilograms (0.06 kg) and substitute the values into the equation:

F = (0.06 kg) x (28.2 m/s)^2 / 1.2 m

After solving this equation, we get the centripetal force acting on the tennis ball, which is approximately 4.73 N. these equations concepts calculate the acting on objects in circular motion, which is useful in many real-world applications.

**Simple harmonic motion**

If there is no friction nor any other external force acting on the object, it will oscillate with equal displacement on either side of the position the oscillator takes when there are no forces acting on it.

In simple harmonic motion, the oscillation period remains constant regardless of the force applied. For example, when you pluck the strings of a guitar with different levels of force, they will oscillate with the same frequency. The relationship between the period (T), mass (m), and spring constant (k) is given by the formula below:

T = 2π√(m/k)

For instance, let's say we have a cube with a mass of 4kg attached to a string with a force constant of 2 N/m. If a force of 10N is applied to compress the string, we can calculate the period and frequency of the cube's simple harmonic

We can use the equation for finding the period of simple harmonic motion:

T = 2π√(m/k)

Adding the known variables, we get:

T = 2π√(4kg/2N/m) = 4π seconds

To find the frequency of the motion, we can either use the equation for finding the frequency:

f = 1/T = 1/(4π) Hz

Or, we can use the shortcut that the frequency is equal to 1/T. Therefore, the frequency of the cube's simple harmonic motion is approximately 0.079 Hz.

Understanding simple harmonic motion and its relationship between mass, spring constant, and period is crucial in many fields, including physics, engineering, and music.

**The simple pendulum**

A simple pendulum is an object with a certain mass suspended from a string or wire.

In a simple pendulum, the net force acting on the object can be expressed as F = -mg sin(θ) or F = -kx, where m is the mass of the object, g is the acceleration due to gravity, k is the spring constant, x is the displacement from equilibrium, and θ is the angle of displacement. For small angles of oscillation, sin(θ) can be approximated as θ, and k can be expressed as mg/L, where L is the length of the pendulum.

The period of a simple pendulum can be determined using the formula:

T = 2π√(L/g)

For example, let's say we have a simple pendulum with a length of 50cm and a period of 1.4576s. We can use the formula to find the acceleration due to gravity (g).

First, we can square the formula for finding the period:

T² = 4π²(L/g)

Then, we can solve for g:

g = 4π²(L/T²)

Substituting the given values, we get:

g = 4π²(0.5m)/(1.4576s)² = 9.770 m/s²

Therefore, the acceleration due to gravity for this simple pendulum is approximately 9.770 m/s².

Understanding periodic motion and its various forms, including simple harmonic motion and simple pendulums, is important in many fields, including physics, engineering, and even music.

## Periodic Motion

**Are all periodic motions simple harmonic motions?**

No, periodic motions are not all simple harmonic motions. They can also be circular motions.

**How is time measured using periodic motion?**

We can measure time by using periodic motion if we know the period of the motion, which is the time it takes for the motion to complete one cycle. After that, we can count the periods and multiply the counted value with the value of the period to measure time in seconds.

**Are all periodic motions oscillatory?**

No, all periodic motions are not oscillatory, but all oscillatory motions are periodic motions because they repeat themselves at certain time intervals called periods.