Angular Displacement

What Is Angular Displacement?

The motion of the body along a circular path is known as the rotational motion. The displacement done through such a type of motion is different from the displacement done on linear motion; it is usually in the form of an angle, and hence it is known as angular displacement. Below we discuss angular displacement along with the formula, let us define it with the help of examples.

While moving in a circular path, the angle made by the body is known as angular displacement. Before discussing further on the topic, we have to understand what rotational motion is. The motion ceases to become a particle when a rigid body is rotating about its axis. Due to the motion in the circular path, change in the acceleration and velocity can happen at any time. Rotational motion is defined as the motion of the rigid bodies that will remain constant throughout the rotation over a fixed axis.

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Angular Displacement Definition – 

To define angular displacement, let's suppose a body is moving in a circular motion, the angle made by a body from its point of rest at any point in rotational motion is known as angular displacement.

The shortest angle between the initial and the final position for an object in a circular motion around a fixed point is known as the angular displacement; it is considered a vector quantity. 

Unit of Angular Displacement

The unit of angular displacement is Radian or Degrees. 360o is equal to two pi radians. Meter is the SI unit for displacement. Since angular displacement involves the curvilinear motion, the SI unit for angular displacement is Degrees or Radian. 

Angular Displacement Formula

The Formula of Angular Displacement

For a point the angular displacement is as follows:

Angular displacement = θf​−θi​

The displacement will have both magnitudes as well as the direction. The circular arrow pointing from the initial position to the final position will indicate the direction. It can either be clockwise or anticlockwise in direction.

It can be measured by using a simple formula. The formula is:

θ=s/r, where,

θ is angular displacement,

s is the distance traveled by the body, and

r is the radius of the circle along which it is moving.

Simplistically, the distance traveled by an object around the circumference of a circle divided by its radius will be its displacement.

Derivation

The angular displacement can be calculated by the below formula when the value of initial velocity, acceleration of the object, and time are shared.

 θ= wt + 1/2αt^2

Where,

θ- Angular displacement of the object

t- Time

α- angular acceleration

Now, formula for Angular Linear is

In Rotational, kinetic equation is

ω=ω0+αt,  

Δθ=ω0t+1/2αt^2, 

ω^2=ω02+2αθ, 

In translational, kinetic equation is

v=u+at

or s=ut+1/2at^2

v2 = vo2 + 2ax

Where,

ω- Initial angular velocity

Considering an object having a linear motion with initial acceleration a and velocity u, when time t and the final velocity of the object is with the total displacement s then,  

a = dv​/dt

The change in velocity

 The rate which can be written as 

dv = a dt

Integrating both the sides, we get,

∫uv​dv=a∫0t​dt

v – u = at

Also,

a=dv/dt​

a=dxdv​/dtdx​

As we know v=dx/dt​, we can write,

a=vdv/dx​

 v dv=a dx

The equation we get after integrating both sides

∫uv​vdv=a∫0s​dx

V^2–u^2=2as

From the equation -1 into the equation – 2 by substituting the value of u, we get

V^2−(v−at)^2=2as

2vat–a^2t^2=2as

By dividing the equation of both sides by 2a, we have

s=vt–1/2at^2

And at last, the value of v being substituted by u, we will get.

s=ut+1/2​at^2

FAQ (Frequently Asked Questions)

1. How is Angular Displacement Measured?

Ans In radians and degrees, the angular displacement will be measured. Considering radians as it provides a very simple relationship between the distances traveled around the circle and the distance r from the center.

θ = s/r 

For example, if the body rotates around a circle of radius r at 360o, then the angular displacement is found by the distance traveled around the circumference. This is found by 2πr, divided by radius  θ = 2πr/r. In simplistic terms, it can be denoted as θ=2π, where 1 revolution is 2π radians. 

When a particle travels from point P to point Q over δt, as it does in the picture to the left, the radius of the circle goes through a change in angle Δθ = θ₂ - θ₁ which equals the angular displacement.

2. How Angular Displacement is a Vector?

 Ans Angular displacement is not a vector.

A quantity that has direction and magnitude also follows the rules of vector algebra; it can be known as a vector.

Although angular displacement may appear to be a quantity directly represented in one direction, you can specify directions to specify conventions such as the rule of thumb of the right hand. Magnitude means the amount of rotation.

However, all the rules of vector algebra are not obeyed by it, particularly the commutative law: u+v = v+u for the vector and u and v. Pick a 3D object like a cell phone and the screen facing towards you upright. Rotate it clockwise, so the screen still faces you but is still horizontal (landscape adjustment). This time rotate again so that the screen faces the ceiling. This is caused due to the addition of 2 angular displacements. When the order of rotation is switched, the final orientation will be different. Different results violating commutability will be achieved by angular displacement in a different order.

3. What are the Examples of Angular Displacement?

Angular displacement example–A dancer angular rotation will be 360o if they are dancing around a pole in a full rotation. If the rotation is half, the displacement will be 1800. 

This will be a vector quantity that means it contains both magnitude and direction.

For example, a displacement of 360o done clockwise is very different compared to anticlockwise. 

4. Why doesn’t Angular Displacement in a Simple Pendulum go above 4 Radians?

You perhaps mean 4°. 4 radians is more than 1800 and makes sense for a pendulum angular displacement. The reason angle should be small (40 being a sensible through random limit), meaning you need a small angle to be able to approximate the force with an elastic one, which ends up by harmonic oscillations. If the displacement becomes too large, the harmonic approximation will no longer exist, and you will get a complex undesirable system that will make it difficult and strange to perform. For example, the oscillation period is not independent of amplitude.