Angular Displacement - Definition, Unit, Formula, Derivation, and FAQ
The concept of Angular Displacement teaches the students the concept of Displacement in the way it is discussed here later. It is specifically related to the Physics subject. Vedantu has provided the students with information on this topic to help them understand the same easily. The site also makes available many more study resources to let the students practice, learn, and enhance their level of preparation. The resources available on Vedantu’s website include revision notes, textbooks from various boards and institutes, worksheets, sample papers, previous year’s question papers, and others to facilitate improved exam preparation.
What Is Angular Displacement?
The motion of the body along a circular path is known as 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.
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.
\[\theta = wt + 1/2\alpha t^{2}\]
Where,
θ- Angular Displacement of the object
t- Time
α- Angular acceleration
Now, the formula for Angular Linear is
In Rotational, the kinetic equation is
\[\omega = \omega 0 + \alpha t\]
\[\triangle\omega = \omega_{0}t+1/2\alpha t^{2}\],
\[\omega^{2} = \omega_0^2 + 2\alpha\theta\],
In translational, the 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,
∫uvdv=a∫0tdt
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
∫uvvdv=a∫0sdx
\[V^{2}-u^{2} = 2as\]
From the equation -1 into equation – 2 by substituting the value of u, we get
\[V^{2}−(v−at)^{2}=2as\]
\[2vat–a^{2}t^{2}=2as\]
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.
\[s=ut+1/2at^{2}\]
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FAQs on Angular Displacement
1. What is meant by angular displacement in Physics?
Angular displacement is defined as the shortest angle through which an object rotates around a fixed axis or a fixed point. When a body moves along a circular path, its change in angular position from an initial point to a final point is its angular displacement. It is typically denoted by the Greek letter theta (θ).
2. What is the basic formula used to calculate angular displacement?
The formula for angular displacement (θ) relates the arc length (s) travelled by the object along the circular path to the radius (r) of the circle. The formula is given by:
θ = s / r
Here, 's' is the distance the object travels along the circumference, and 'r' is the radius of the circular path. This formula gives the displacement in radians.
3. What are the units and the dimensional formula for angular displacement?
Angular displacement can be measured in several units, but the standard units are:
- SI Unit: The standard international unit for angular displacement is the radian (rad).
- Other Common Units: It is also commonly measured in degrees (°) and revolutions (rev).
Since angular displacement is a ratio of two lengths (arc length 's' and radius 'r'), it is a dimensionless quantity. Its dimensional formula is represented as [M⁰L⁰T⁰].
4. How is angular displacement different from linear displacement?
Angular displacement and linear displacement both describe a change in position, but they do so in different contexts:
- Path of Motion: Linear displacement measures the shortest straight-line distance between the initial and final points, used for translational motion. Angular displacement measures the angle swept by an object moving in a rotational or circular motion.
- Units: Linear displacement is measured in units of length like metres (m), while angular displacement is measured in radians (rad) or degrees (°).
- Nature: Linear displacement is always a vector quantity. Angular displacement behaves like a vector only for infinitesimally small angles.
5. Is angular displacement a scalar or a vector quantity? Explain why.
Angular displacement is a complex case; it is treated as a vector quantity only for very small angles. For large or finite rotations, it is considered a scalar. This is because large angular displacements do not obey the commutative law of vector addition (i.e., A + B = B + A). If you perform two large rotations on an object in a different order, the final orientation will be different. However, for infinitesimally small angles, the order of rotation does not matter, and they can be treated as vectors.
6. Why is a complete circular rotation represented by an angular displacement of 2π radians?
A complete circular rotation is represented as 2π radians due to the relationship between the circle's circumference and its radius. The circumference of a circle is given by the formula C = 2πr. Using the angular displacement formula θ = s/r, if an object travels the full circumference (s = C), the angular displacement becomes:
θ = (2πr) / r = 2π radians.
Therefore, 2π radians corresponds to a full 360° rotation.
7. What is the significance of clockwise and anti-clockwise direction in angular displacement?
The direction of rotation is crucial for defining angular displacement, especially when treating it as a vector. By convention:
- Anti-clockwise rotation is considered positive (+).
- Clockwise rotation is considered negative (-).
This sign convention, determined by the right-hand thumb rule, helps in solving problems in rotational dynamics by providing a clear direction to the angular displacement, velocity, and acceleration.
8. Can you provide some real-world examples of angular displacement?
Certainly. Angular displacement is observed in many everyday scenarios:
- The movement of the hands of a clock. The minute hand has an angular displacement of π radians (180°) in 30 minutes.
- A spinning Ferris wheel where each passenger car undergoes angular displacement around the central axis.
- The rotation of a planet around the Sun. The Earth has an angular displacement of approximately 2π radians after one year.
- A CD or DVD spinning in a player.
9. How are angular displacement, angular velocity, and angular acceleration related in rotational kinematics?
These three quantities are the rotational analogues of linear displacement, velocity, and acceleration. They are related through the equations of rotational motion, similar to linear kinematics:
- Angular Velocity (ω) is the rate of change of angular displacement: ω = dθ/dt.
- Angular Acceleration (α) is the rate of change of angular velocity: α = dω/dt.
For constant angular acceleration, their relationship is described by the kinematic equations, such as θ = ω₀t + ½αt², where ω₀ is the initial angular velocity.
