Content Curator
Angular displacement can be defined as “the angle in degrees or radians through which any line can be rotated in a specified sense about a specified axis.” Angular displacement is the circular equivalent of this. It can be understood as the angle of the change in position of a body moving in a circular path.
Read More: Rotational Inertia
| Table of Content |
Key Terms: Angular Displacement, Rotational Motion, Circular Movement, Rest, Motion, Initial Angular Velocity, Final Angular Velocity
What is Angular Displacement?
[Click Here for Sample Questions]
Angular Displacement is the angle in radians via which any line or point rotates in a particular sense about a specified axis. Curvilinear motion is an integral concept in physics. Studying displacement in terms of movement along a curved path is different from that of a linear path. In linear motion, the distance between the starting point and ending point is displacement.
To fully understand angular displacement, one must have a grasp of the concept of rotational motion. Rotational motion is the circular movement of an object around a fixed axis. The earth orbiting around the sun is an example of rotational movement.
Using a compass to draw a circle is also an example of rotational movement. When a rigid body orbits around a fixed axis, all the particles rotate through the same angle in the same intervals of time. The angular displacement would then be, the shortest angle from the point of rest of the body to any point of motion, or the final point of motion.
Unit of Angular Displacement
[Click Here for Previous Year's Questions]
Since angular displacement involves curved movement, it is measured in degrees or radians. 360° is equal to two times pi multiplied by the radian, that is, 2r = 360°. The vector quantity has a direction that is specified by the axis of rotation, and it has magnitude, which is the rotation in radians around the axis. This is true for 3 dimensions.
However, angular displacement does not obey the commutative law for addition and is hence not defined as a vector quantity. Angular displacement is represented by the Greek letter θ and the formula is as follows:
| θ = sr |
Here,
- s is equal to the distance travelled by the object and,
- r is equal to the radius of the circle along which the object is moving.
It is hence the distance travelled by it around the circumference of a circle divided by its radius.
How to Calculate Angular Displacement?
[Click Here for Sample Questions]
Angular displacement can also be calculated as the difference between the initial point and the final point of movement, that is, θf - θi.
If we are aware of the acceleration of the object (α), the initial angular velocity (ω) and the time (t) at which the displacement is to be calculated, we can use the formula:
| θ = wt + 12αt2 |
Derivation of the Formula
Consider an object with the initial velocity “u”, final velocity “v”, the initial acceleration “a” at a point in time “t”, with the total displacement “s”. Acceleration is the rate of change of velocity,
⇒ a = \(\frac{dv}{dt}\)
⇒ dv = adt
When we integrate both sides, we get,
⇒ \(\int\limits_u^v dv = a \int\limits_0^t dt\)
⇒ v – u=at
Also,
⇒ a = \(\frac{dv}{dt}\)
⇒ a =\(\frac{dv}{dx}\) = \(\frac{dx}{dt}\)
Since v = dx/dt, the equation can become,
⇒ a =\(\frac{vdv}{dx}\)
Then, vdv = adx
Once again, integrate both sides,
⇒ \(\int\limits_u^v vdv = a \int\limits_0^s dx\)
⇒ v2 – u2 = 2as
Substituting the value of u from the first equation into the second equation, we get,
⇒ v2 − (v − at)2 = 2as
⇒ 2 vat – a2t2 = 2as
If we divide both sides by 2a, we get
⇒ S = vt – 12at2
Finally, when v is substituted with u, we get
⇒ S = ut + 12at2
That is,
| θ = wt + 12αt2 |
Previous Year Questions
- The moment of inertia of a uniform cylinder of length … [JEE Mains 2017]
- A circular disc D1 of mass M and radius R has two identical discs … [JEE Mains 2019]
- A long cylindrical vessel is half filled with a liquid … [JEE Mains 2019]
- A metal coin of mass 5g and radius 1cm is fixed … [JEE Mains 2019]
- A particle of mass m is fixed to one end of a light spring … [JEE Mains 2020]
- A rectangular solid box of length … [JEE Mains 2019]
- A rigid massless rod of length … [JEE Mains 2019]
- A roller is made by joining together … [JEE Mains 2016]
- A slender uniform rod of mass … [JEE Mains 2017]
- A solid sphere of mass M and radius R … [JEE Mains 2019]
- A string is wound around a hollow cylinder of mass … [JEE Mains 2019]
- A tennis ball (treated as hollow spherical shell) … [JEE Mains 2013]
- A thin bar of length L has a mass per unit length … [JEE Mains 2014]
- A thin circular plate of mass M and radius R … [JEE Mains 2019]
- A thin disc of mass MM and radius RR has mass per … [JEE Mains 2019]
- A uniform rectangular thin sheet … [JEE Mains 2019]
- A uniform solid cylindrical roller of mass … [JEE Mains 2015]
Things to Remember
- An angle is formed at the intersecting point of two lines or two planes. Angles define the orientation of these lines or planes relative to each other.
- Angular displacement represents the angle between the final position and the initial position.
- Similar to linear displacement, angular displacement has a direction linked with it. A clockwise rotation is negative and a counterclockwise rotation is positive.
- In angular kinematics, the movement of a segment or rigid body is determined as it rotates about a point.
- Angular position/displacement is expressed in rads (not degrees or revolutions) for this relationship to be precise.
Also Check:
Sample Questions
Ques. What are the Examples of Angular Displacement? (2 Marks)
Ans. Angular displacement example—A dancer’s angular rotation will be 360° 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 which means it contains both magnitude and direction. For example, a displacement of 360° done clockwise is very different compared to anticlockwise.
Ques. What is the rate of angular displacement called? (2 Marks)
Ans. Angular velocity is the rate of velocity at which an object or a particle is rotating around a centre or a specific point in a given time period. It is also known as rotational velocity. Angular velocity is measured in angle per unit time or radians per second (rad/s). The rate of change of angular displacement is angular velocity.
Ques. Why doesn't Angular Displacement in a Simple Pendulum go above 4 Radians? (3 Marks)
Ans. 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.
Ques. How Angular Displacement is a Vector? (4 Marks)
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.
Ques. How is Angular Displacement Measured? (4 Marks)
Ans. In radians and degrees, the angular displacement will be measured. Considering radians since it provides a very simple relationship between the distances travelled around the circle and the distance r from the centre.
⇒ θ = s/r
For example, if the body rotates around a circle of radius r at 360°, then the angular displacement is found by the distance travelled around the circumference. This is found by 2πr, divided by radius θ = 2πr/r.
In simplistic terms, it can be denoted as θ =2πr, where 1 revolution is 2n radians. When a particle travels from point P to point Q, as it does in the picture to the left, the radius of the circle goes through a change in angle θ = θ2 - θ1 which equals the angular displacement.
Ques. A body moving along a circular path of radius 14 m describes an angle 120° at the centre of the circle while moving from A to B. Calculate the displacement and distance of the body. (4 Marks)
Ans. Calculating the value of d we-
sin 60 = d/14
d = 14 x sin 60
d = 7√3 m
- Displacement = 2d = 14√3m
- Distance = Length of the arc
Arc = θ.R = (2π/3) . 14 m
Where θ is angular displacement and R = Radius
- Distance = 28π/3 m
Ques. How is Angular Displacement represented? (1 mark)
Ans. Angular displacement is usually represented by the Greek letter θ and the formula is \(\theta\) = sr.
Ques. What is rotational motion? (1 mark)
Ans. Rotational motion can be defined as the rotation of bodies which remains constant throughout the entire duration of rotation over a fixed axis.
Check Out:
Comments