Introduction
When an object moving in a linear direction is different from a moving body in a circular dimension. Therefore, the displacement of the body in a linear direction will be different from that of the moving body in a circular path. Thus the displace of the body in circulation will be at a certain angle depending on various physical factors. The factors that directly affect the angular displacement calculator are the velocity and the acceleration of the rotating body in an axis at a given point in time. Alternatively, the angular displacement calculator changes its value with the change (increasing or decreasing) of the velocity of the rotating body and its acceleration at a particular time. Thus anybody that is undergoing angular displacement while rotating at a definite axis with a definite velocity and acceleration at a point of time is called angular displacement. This article deals with the angular displacement formula with its derivation and the angular displacement calculator relative to the physical factors.
What is Angular Displacement?
The definition of angular displacement is the displacement of a body at a definite angle that is measured in radian, that is in degree or revolution counts through which a body revolves a line or centre that is aligned with a specified axis. It is a vector quantity that measures the shortest angle between the initial and the final position of the revolving object at a definite axis. It is therefore a vector quantity. Thus it will have a magnitude as well as a direction that is represented by a circular arrow pointing towards the final position that can be either clockwise or anticlockwise. The unit of angular displacement is in degree or radian that is 360o is equal to 2π radian. Thus the SI unit of the angular displacement is in meters, which is a dimensionless quantity. Thus there is no dimensional formula of angular displacement. It can be proved by the following method.
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Considering the following diagram to explain the dimensional formula of the angular displacement, suppose the body or the particle is moving from point A to B along a fixed point O. therefore the displacement angle will be given by the ratio of the length of the arc to that of the radius of the circle. Hence θ = s / r = () / (L) = 1. Thus there is no unit for the dimensional formula of the angular displacement.
Angular Displacement Formula
The angular displacement of a particle from one point to another is given as:
angular displacement =θf - θi
And, θ = s / r
The angular displacement of an object in acceleration can be calculated on the basis of initial velocity and given time using formula as follows:
θ = ωt + 1/2ɑt2
Where, θ = Angular displacement of the object
s = Distance covered by the object in a circular path
r = The radius of curvature of the given path of the object
ω = Initial angular velocity
t = Time taken by the object to cover the circular distance
θ = Angular acceleration
Angular Displacement Formula Derivation
Consider a moving object accelerating in a linear motion from one point to another. Thus its acceleration is “a” and initial velocity is “u”. After a certain time “t” the final velocity, then the total displacement will be
\[ a = \frac{dv}{dx} \]
this is the change in the velocity which can also be written as
dv = a dt
Thus integrating both sides of the equation from initial to final velocity in RHS and time 0 to t in LHS we get,
\[ \int^{v}_{u} dv = a \int^{t}_{0} dt \]
V - u = at, also,
\[ a = v \frac{dv}{dx} \]
v dv = a dx
After integrating both sides of the equation from initial to final velocity in RHS and the distance covered from 0 to s, we get,
\[\int^{v}_{u} vdv = a \int^{s}_{0} dx \]
Therefore, \[ v^{2} - u^{2} = 2as \]
Substituting the value of equation one to equation 2, we get,
\[ v^{2} - (v - at)^{2} = 2as\]
\[ 2vat - a^{2}t^{2} = 2as \]
Divide both side of the equation by 2a, we get,
\[ s = vt - \frac{1}{2} at^{2} \]
Thus finally by substituting the value of V and u, the total displacement formula will be
\[ s = ut - \frac{1}{2} at^{2} \]
Solved Example of Angular Displacement Formula
1. Shyam went for a run on a circular track that has a diameter of 10 m.if he has covered a total distance of 60 m running on the circular path, then what will be the angular displacement of shyam?
Ans: shyam linear displacement (s) = 60 m
The path diameter = 10m
Therefore, radius = 10 / 2 = 5m
Thus the formula for angular displacement, θ = s / r = 60 / 5 = 12 radian
Therefore, the angular displacement angle will be 12 radian.
FAQs on Angular Displacement Formula
1. What is the angular displacement formula with time?
The angular displacement formula physics in respect to time is represented as
θ = ωt + 1/2ɑt2
Where, θ = angular displacement of the object
s = distance covered by the object in a circular path
r = the radius of curvature of the given path of the object
ω = initial angular velocity
t = time taken by the object to cover the circular distance
θ = angular acceleration
2. What does angular velocity depend on?
The angular velocity depends on the rotational angle of the revolving object. The more rotational the angle the more will be the angular velocity. Angular velocity is totally analogous to linear velocity and is measured in ω = radian / second.
