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Understanding Circular Motion at ease


Understanding Circular Motion at ease

It is common to observe circular motion; it is a particular case of elliptical motion, such as when planets orbit with gravity. Once you understand linear motion in one or two dimensions, it is a natural extension to consider circular motion.

What is Circular Motion?

Whenever a body moves along the circumference of a circle with a constant speed, it moves in a circular motion.

For example, consider a car traveling in a circular motion at a speed of 8 m/s around the circumference of 24 meters.

A single cycle will take 3 seconds at a uniform speed. This means every circle around the 24-meter circumference of the circle would take 4 seconds. Hence, this relationship between the circumference of a circle, the time required for one revolution, and the body's speed can be described as average.

So, Average speed = distance/time

= circumference/time

= 2 * π * r/T

As, circumference = 2 * π * r

Circular Motion Dynamics

Imagine a body moving in a paper plane following a circular path of radius r in a clockwise direction. Consider a circular motion whose axis passes through the center of a paper, perpendicular to its plane.

From point P to point Q, the angle is called the angular displacement, which is given by,

θ = PQ/r = S/r

This is a vector quantity.

You can determine its direction by using the right-hand rule.

The right-hand rule

When the fingers are curled as if gripping an axis of rotation, they will slip. An angular displacement vector can be founded by holding the thumb perpendicular to the curvature of the fingers.

The motion of the circle is circular, and the velocity of the circle is angular. It is the change in angular displacement that determines the angular velocity. It is represented by ω.

Where ω = v/r

This is a vector quantity. For example, a thumb represents angular velocity according to the right-hand rule.

According to the right-hand rule, for an anticlockwise rotating body, * is along the axis of a circular path, and upward, while it is downward for a clockwise rotating body, it is downward.

Uniform Circular Motion Dynamics

It is the natural tendency of the body to move uniformly in a straight line. When we apply a centripetal force, it is forced to move along the circle. When a pseudo-force is applied to a body, it will move uniformly around the circle's circumference.

Centripetal force is a pseudo-force that acts along the radius and is directed towards the center. According to Newton's first law of motion, an external force cannot shift a body's direction to maintain circular motion.

However, the direction of motion of this body is constantly changing by itself, as well as its velocity, which is why it is accelerating, known as the radial centripetal acceleration.

a = v2/r

We know that F = ma

F = mv2/r

The Dynamics of Uniform Circular Motion

The body returns to its natural linear path when moving along a circle. Consequently, a centrifugal force is created.

The centripetal force can be viewed as a reaction to it. Therefore, the centrifugal force is always equal and opposite to the centripetal force.

Thus, centrifugal force = mv 2 /r acts along the radius but away from the circle's center.

Acting forces are centripetal, and reacting forces are centrifugal.

The dynamics of non-uniform circular motion

Imagine a body moving with an angular velocity, ω. While the axis of rotation remains fixed, either its direction (clockwise or anticlockwise) or its magnitude can change.

As a result, the position vector 'r' remains constant.

Since v = rω

When we differentiate it according to time, we get,

dv/dt = ωdr/dt + rdω/dt

As a = dvdt, dr/dt = v, α (angular acceleration) = dω/dt

= vω + rα

a = ac + at

Here, ac = radial or centripetal acceleration, calculated as the rate at which the particle's velocity changes in the radial direction.

a t = tangential acceleration, which measures the change in velocity magnitude in the tangential direction.

If circular motion is taken, the resultant acceleration is,

a = /a/ = √ac^2 + at ^2

Observing circular motion

Consider the following examples of objects moving in a circular path:

  • A hammer thrower swings a hammer

  • Drying clothes in a spin dryer

  • Separating chemicals in a centrifuge

  • Cornering on a bike or in a car

  • A string is used to whirl the stone

  • Looping a plane

  • Turntable spinning a DVD, CD, or record

  • Earth-orbiting satellites

  • Almost circular orbits of planets around the sun

  • Attractions at the fairgrounds

  • An electron in orbit around a nucleus

The motion in a circle can be viewed as a particular case of the motion in a curve. So what makes us study it? First, it's common, and the circular motion is easy to understand!


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