•  
agra,ahmedabad,ajmer,akola,aligarh,ambala,amravati,amritsar,aurangabad,ayodhya,bangalore,bareilly,bathinda,bhagalpur,bhilai,bhiwani,bhopal,bhubaneswar,bikaner,bilaspur,bokaro,chandigarh,chennai,coimbatore,cuttack,dehradun,delhi ncr,dhanbad,dibrugarh,durgapur,faridabad,ferozpur,gandhinagar,gaya,ghaziabad,goa,gorakhpur,greater noida,gurugram,guwahati,gwalior,haldwani,haridwar,hisar,hyderabad,indore,jabalpur,jaipur,jalandhar,jammu,jamshedpur,jhansi,jodhpur,jorhat,kaithal,kanpur,karimnagar,karnal,kashipur,khammam,kharagpur,kochi,kolhapur,kolkata,kota,kottayam,kozhikode,kurnool,kurukshetra,latur,lucknow,ludhiana,madurai,mangaluru,mathura,meerut,moradabad,mumbai,muzaffarpur,mysore,nagpur,nanded,narnaul,nashik,nellore,noida,palwal,panchkula,panipat,pathankot,patiala,patna,prayagraj,puducherry,pune,raipur,rajahmundry,ranchi,rewa,rewari,rohtak,rudrapur,saharanpur,salem,secunderabad,silchar,siliguri,sirsa,solapur,sri-ganganagar,srinagar,surat,thrissur,tinsukia,tiruchirapalli,tirupati,trivandrum,udaipur,udhampur,ujjain,vadodara,vapi,varanasi,vellore,vijayawada,visakhapatnam,warangal,yamuna-nagar

Kinematics of Rotational Motion, parameters, equations of motion, practice problems, FAQs

Kinematics of Rotational Motion, parameters, equations of motion, practice problems, FAQs

Have you ever wondered what kind of motion does see-saw, opening a door or turning a wrench involves. As the angle of rotation involved in such cases is often small, it might confuse you whether it is angular motion or not, but it actually is. To study such motion without considering the cause of motion, one must be aware of rotational kinematics that is analogous to translational kinematics in linear motion.

Table of contents

  • Circular Vs Rotational Motion
  • Rotational Parameters
  • Angular kinematic equations
  • Practice problems
  • FAQs

Circular Vs Rotational Motion

  • Circular motion is defined for a particle.
  • Rotational motion is for a collection of particles that form a rigid body.

Example:

Let the particles of Earth, namely, 1, 2, 3, and 4, be moving in a circular motion individually as shown in the figure. Here, the Earth is performing a rotational motion and the individual particles are performing a circular motion.

Please enter alt text

Axis of rotation

  • The loci of the centres of all the particles moving in a circular motion in a rigid body is known as the axis of rotation.
  • Axis of rotation is mandatory to represent the rotational motion.

Properties of axis of rotation

  1. It does not have to pass through the body.
  2. It does not have to be fixed.
  3. It does not have to be perpendicular to the surface plane of the object.

Rotational Parameters

In case of a rotational motion, the rotational equivalent parameters as in linear motion are:

  • Angular position
  • Angular displacement
  • Angular velocity
  • Angular acceleration

Angular position (θ)

The angle made by the position vector with respect to the origin with the reference line is known as the angular position.

Sign convention: The angle measured in the anticlockwise direction is to be taken as positive.

Angular displacement (Δθ):

It is the angle through which the position vector of the moving particle rotates in a given time interval with respect to a specified origin and axis. The SI unit of angular displacement is radian.

It is the difference between two angular positions/coordinates.

Δθ=θ2-θ1

Angular velocity (ω):

It is defined like its linear counterpart. It is the rate of change of angular displacement of a body with respect to time.

Average angular velocity (ωavg):

Let us say that a body has covered angle θ1 in time t1 and angle θ2 in time t2.

Then, the average angular velocity of the body is as follows:

ωavg=θ2-θ1t2-t1=ΔθΔt

Instantaneous angular velocity ():

It is defined as the rate of change of displacement over an infinitesimally small interval of time.

Let us say that a body has covered angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>&#x3B8;</mi></math> in time dt, as shown in the figure.

Then, the instantaneous angular velocity of the body is as follows:

ω=lim\belowΔt0ΔθΔt=dθdt

Angular acceleration: It is defined as the rate of change of angular velocity with respect to time.

Average angular acceleration (avg):

Let us say that a body has angular velocity ω1 at time t1 and its angular velocity changes to ω2 at time t2.

αavg=ω2-ω1t2-t1=ΔωΔt

Instantaneous angular acceleration ():

It is defined as the rate of change of angular velocity over an infinitesimally small interval of time.

α=lim\belowΔt0ΔωΔt=dωdt

The equations of motion for the rotational motion with a constant angular acceleration are similar to their linear counterparts.

Angular kinematic equations

The angular velocity of any point on the rigid body with respect to any other point on the rigid body always remains the same. Due to the same angular velocity of all the particles, we can define the angular velocity of the rigid bodies.

Following are the parameters and relations between them used in a circular motion. These terms are required in rotational motion as well.

The kinematic equations for constant magnitude of tangential and angular acceleration are as follows:

ω=ωo+αt

Δθ=ωot+12αt2

ω2=ωo2+2αΔθ

Practice problems

Q.  A rigid disc rotates with a uniform angular acceleration of 2.0 rad/s2 about its axis. How many revolutions will it make in the first 10 seconds, if the disc starts from rest?

A

Given,

Initial angular velocity, ⍵0 = 0

Angular acceleration, 𝛼 = 2.0 rad s-2

Time, t = 10 s

Let the disc make n revolutions in 10 s.

We know that,

θ=ωot+12αt2

θ=0+12×(2)×(10)2=100 rad

Number of revolutions are

n=θ2π=1002π=16

Thus, the disc makes 16 revolutions in the first 10 seconds.

Q. The wheel of a motor accelerates uniformly from rest and during the first second rotates through 5 radians. What will be the angle rotated during the next second?

A

Given,

Initial angular velocity, ⍵0 = 0

At time, t = 1 s, 𝜃1 = 5 radian

Let the angular acceleration be 𝛼.

We know that,

θ=ωot+12αt2

θ1=12×α×12=5 rad

α=10 rad/s2

At t=2 s

θ=ωot+12αt2

θ2=12×α×22=10×2=20 rad

Thus angle rotated from 1 sec to 2 sec is as follows

θ=θ2-θ1=20-5=15 rad

Q. A rigid block is moving downward into a well through a rope that is passing over a fixed pulley of radius 10 cm. Assume that there is no slipping between the rope and the pulley. Calculate the angular velocity and angular acceleration of the pulley at an instant when the block is going down at a speed of 20 cm/s and has an acceleration of 4.0 m s-2.

A

As there is no slipping between the rope and the pulley, velocity of the block, rope and the point of contact of pulley will be the same.

Given,

Radius of the pulley, r = 10 cm

Velocity of the block, v = 20 cm s-1

Acceleration, at = 4.0 m s-2

We know, angular velocity is as follows:

ω=vr

ω=0.20.1=2 rad/s

Similarly angular acceleration is:

α=atr

α=40.1=40 rad/s2

Q. The motor of an engine is rotating with an angular velocity of 120 rpm about its axis. It comes to rest in 10 s after the engine is switched off. Find the number of revolutions made by it before coming to rest assuming uniform angular deceleration.

A.

Given,

Initial angular velocity, ωo=120 rpm=120×2π60=4π rad/s

Final angular velocity, =0 rad/s

Time, t=10 s

Let be the constant angular acceleration.

Using first equation of angular kinematics,

ω=ωo+αt

0=4π+α×10

α=-4π10

Using third equation of angular kinematics,

ω2=ωo2+2αΔθ

02=(4π)2+2α×Δθ

-(4π)2=2-4π10×Δθ

Δθ=20π rad

Let n be the number of revolutions made by the engine before stopping

Therefore, number of revolutions, n=Δθ2π=20π2π=10

FAQs

Q. What is the instantaneous speed of the point of contact in pure rolling?
A.
In pure rolling, translational speed is balanced by rotational speed at the point of contact and therefore instantaneous speed of the point of contact is zero.

Q. If torque acting on a rotating lever arm is zero, what will be the angular acceleration of the arm?
A.
Newton's second law for rotational motion, τ=Iα

Since τ=0α=0

Therefore the lever arm will move at zero angular acceleration or constant angular velocity.

Q. What is the dimensional formula for angular velocity and angular acceleration?
A.
Angular velocity - [M0L0T-1]

Angular acceleration - [M0L0T-2]

Q. Do all the points on a rigid body performing pure rotational motion have the same angular velocity?
A.
Yes, all the particles that constitute the rigid body performing rotational motion will have the same angular velocity. Although their tangential velocity might vary depending upon the distance from the axis of rotation.

NEET Related Links

NEET Exam 2024

NEET 2024 Exam Dates

NEET 2024 Exam pattern

NEET 2024 Syllabus

NEET 2024 Eligibility Criteria

NEET 2024 Application

NEET UG Counselling

NEET FAQ

NEET UG Result

NEET 2024 Cut Off

Neet 2023 Toppers List Names & Rank

Neet Result 2023 Toppers list rank cut off

Neet Answer key Live Download PDF

Neet 2023 State Toppers List

JEE MAIN Related Links

JEE Main 2024

JEE Main Rank Predictor 2024

JEE Main College Predictor 2024

JEE Main 2024 Exam Dates

JEE Main 2024 Exam pattern

JEE Main 2024 Application

JEE Main 2024 Eligibility Criteria

JEE Main 2024 Syllabus

JEE Main 2024 Physics Syllabus

JEE Main 2024 Maths Syllabus

JEE Main 2024 Chemistry Syllabus

JEE Main 2024 Admit Card

JEE Main 2024 Counselling

JEE Main marks vs rank vs percentile

JEE Advanced Result 2023 live topper list

JEE Exam Preparation - How to calculate your rank jee

JEE Maths Syllabus - Important topics and weightage

JEE Advanced Related Links

JEE Advanced 2024 Exam Dates

JEE Advanced 2024 Application

JEE Advanced 2024 Eligibility Criteria

JEE Advanced 2024 Syllabus

JEE Advanced 2024 Maths Syllabus

JEE Advanced 2024 Physics Syllabus

JEE Advanced 2024 Chemistry Syllabus

JEE Advanced Exam Result

JEE Advanced Exam Dates

JEE Advanced Registration Dates

CUET Related Links

CUET 2024 Eligibility Criteria

CUET 2024 Admit Card

CUET 2024 Exam Pattern

CUET 2024 FAQs

CUET 2024 Counselling

CUET 2024 Syllabus

CUET 2024 Result

CUET 2024 Answer Key

CUET 2024 Preparation

CUET CUTOFF

CUET 2024 Application Form

Talk to Our Expert Request Call Back
Resend OTP Timer =
By submitting up, I agree to receive all the Whatsapp communication on my registered number and Aakash terms and conditions and privacy policy