# Relationship between Linear and Angular Motion

In this tutorial, we will learn about the types of motion, the physical quantities which define them, and their mathematical equations. Also, you will learn about the basic principles of physics which are used in the analysis of Kinematics and relationship between linear and angular motion.

Contents:

## Types of Motion

The process of moving or relocating a physical object or the movement of a body with respect to time is called motion. The motion of a body is not feasible without considering time. If an object performs any motion without considering time, that would mean that the initial and final position of the body has not changed. Motion is basically categorized into the following parts:

• Plane Motion: Plane Motion is defined as the movement of a body in one plane only. It is also known as motion in two dimensions.
• Representing any physical object such as a tree, car, or a machine tool, requires a minimum of three co-ordinates. But its motion can be confined within one plane or by using two-coordinate geometrical system.
• For example, a ring spanner (which is used to tighten down nuts) is a 3-dimensional object but it can be used in only one plane and its motion is confined to a fixed hinge/point.
• Rectilinear Motion: It is the simplest type of plane motion. An object is said to be in Rectilinear Motion if it translates in one plane and in one line.
• The tool of a shaper performs to and fro rectilinear motion. Other examples of rectilinear motion are longitudinal feed in a lathe or action of a hacksaw blade or a carpenter’s jack planer (used to produce smooth wooden surfaces).
• Curvilinear Motion: The motion of a body in a curved path in a plane is known as Curvilinear Motion.
• The motion in curvilinear motion can be completed in complete circular paths or in incomplete arcs.
• There are various examples of inversions in four-bar mechanism which perform a combination of rectilinear and curvilinear motion.

In practical experiments or applications of certain kinematic elements, the motion may not be completely rectilinear or completely rotational. Rather, it may be a combination of both motions.

## Linear Quantities

The quantities which are associated with linear motion are known as Linear Quantities. These quantities are used in the analysis of objects moving in the same plane in a same line. Linear Quantities are generally vector in nature (having both magnitude and direction).

• Linear Displacement: It is defined as the shortest distance travelled by a body from an initial point to a final position with at least one point fixed.
• If a body is either travelling in a straight line or in a curve, and it returns to its initial position, then the displacement of the body will be zero.
• Displacement is a quantity which involves both magnitude and direction, hence it is a vector quantity.
• In Mechanical Engineering, any part or kinematic element of a machine moves under the following paths:
• Straight line
• Curved path with fixed radius
• Curved part with varying radius

In any of the path followed, the displacement chosen is between the closest points for minimum distance. A Shaper Machine’s tool performs linear to and fro reciprocating motion as shown below:

The above diagram shows a Shaper machine which is used in machining processes. The key component of the shaper is the cutting tool which moves to and fro in a line machining various components.

• Linear Velocity: It is defined as the rate of change of displacement with respect to time.
• Since velocity is a time-derivative of displacement, it is also a vector quantity.
• For an object moving in a circular or curved path, the velocity is given by the tangential component at a point at that particular instant of time.
• Mathematically,
Velocity = $$\frac{Rate \,of \,change \,of\, Displacement }{dT }$$=$$\frac{dS}{dT }$$
• Linear Acceleration: It is defined as the rate of change of velocity with respect to time.
• Acceleration is also a vector quantity since it is a time-derivative of velocity.
• Mathematically,
Acceleration = $$\frac{d^2s}{dt^2}$$
• From the above mathematical observation, we can also conclude that acceleration is the double derivative of displacement with respect to time.

## Equations of Linear Motion

For a body moving with a uniform acceleration, three equations provide relationship between initial velocity (u), final velocity (v), acceleration (a), the time taken (t), and distance travelled (S).
Consider the following graph below:

The above graph is known as Velocity-Time Graph. A velocity-time graph is used to show the velocity and direction of a body moving under uniform acceleration.

• First Equation
Consider linear motion of a rigid body moving with an initial velocity ‘u’ . The body performs uniform acceleration and in time ‘t’ , acquires a final velocity ‘v’.
As evident from Figure 2,
Initial velocity (at t=0) = OA = u
Final velocity (at time t) = OC = v
Acceleration a = Inclination of the line AB
The first equation obtained is:
v = u + at
• Second Equation
Let a body in velocity-time graph travels a distance ‘S’ in time ‘t’.
The second equation used to calculate linear distance is given below:
S = ut + ½ at2
• Third Equation
Let the body travelling with initial velocity ‘u’ and then acquiring a final velocity ‘v’ travels a distance ‘S’.
The third equation provides a relation between initial velocity, final velocity, acceleration of the body, and the distance travelled by it as given below:
v2 – u2 = 2aS
• Special Cases
There are some special cases in kinematics where these equations take a special form and can be applied directly to calculate the required parameters.
• For a body starting from rest
• If a body starts from rest, then its initial velocity ‘u’ = 0. Therefore,

• 1st Equation
v = at
In this case, the initial velocity (u) becomes zero. Hence, the final velocity of the body is the product of acceleration and time.
• 2nd Equation
S = ½ × at2
In this case, the first term of the equation is eliminated. Here, acceleration and time taken are sufficient to find out the distance travelled by the object.
• 3rd Equation
v2 = 2aS
In this case, the final velocity is a square root of the product of the acceleration and the distance travelled.
• For a body free falling under gravity (g)
• 1st Equation
v = gt
• 2nd Equation
S = ½ × gt2
• 3rd Equation
v2 = 2gH [ where H = height of the free-fall]
• For a body moving under negative acceleration or in case of braking of an object
In case of retardation, the acceleration is negative. Hence, the equations take the following forms:
• 1st Equation
v = u − at
• 2nd Equation
S = ut – (½ × at2)
• 3rd Equation
v2 = u2 – 2aS

## Angular Quantities

The quantities which deal with the study of circular motion are known as Angular Quantities. They are also known as Angular Magnitudes. These quantities have magnitude as well as direction about a fixed axis. Hence, all Angular Quantities are vector in type.

• Angular Displacement: Consider the following diagram below:

The above diagram shows an object travelling in a circular path from position A to B. The object covers a distance ‘δθ’ in time δt.

• The angle ‘δθ’ is called ‘Angular Displacement’.
• Angular Displacement can also be referenced as the angle traced by an object moving in a curve with respect to a fixed axis.
• Since, it has both magnitude (δθ) and direction, angular displacement is also a vector quantity.
• It is measured in ‘radians.’
• Angular Velocity: It is defined as the rate of change of angular displacement with respect to time. It is denoted by ‘ω’.
• Mathematically,
ω = $$\frac{δθ}{δt}$$
• Angular Velocity is also vector in type similar to Linear Velocity.
• Angular Acceleration: It is defined as the rate of change of angular velocity with respect to time. It is denoted by the Greek letter ‘α’.
• Mathematically,
α = $$\frac{δω}{δt}$$
• Since, α is a time-derivative of ω, it is also a vector quantity.
• It is to be noted that the direction of angular acceleration need not be same as the angular velocity or displacement.

## Equations of Angular Motion

There are four equations which are primarily required in Kinematics. These equations form a typical analogy with equations of linear motion.

• Equation involving ω, ω0 , α, and t
This equation involving ω, ω0 , α, and t is in analogy with the equation ‘ v = u + at’ as shown below:
ω = ω0 + αt
Here,
ω = Initial angular velocity of the body in rad/s.
ω0 = Final angular velocity of the body in rad/s.
α = Angular acceleration of the body in rad2/s.
t = time taken by the body in seconds.
• Equation involving ω, ω0 , α, and θ
This equation involving ω, ω0 , α, and θ is in analogy with the equation ‘ v2 = u2 + 2aS ‘ as shown below:
ω2 = ω02 + 2αθ
Here,
θ = Angular Displacement of the body in radians.
• Equation involving ω0, α, θ, and t
This equation involving ω0, α, θ, and t is in analogy with the equation ‘S = ut + ½ at2 ‘ as shown below:
θ = ω0t + ½ αt2
• Equation involving θ, t, ω, and ω0
This equation involving θ, t, ω, and ω0 is used to find the angular displacement when the initial velocity, final velocity, and the time taken are given as shown below:
θ = [(ω0 + ω)t]/2
• Equation to convert RPM into Angular Velocity
Consider the diagram below:

The above diagram shows two frictional discs rotating at N rpm. To convert RPM into Angular Velocity the following relation is used:

## Relation between Angular and Linear Quantities

Consider the diagram below:

The above diagram depicts a body moving in a circular path from A to B. Let the angular velocity of the body be ω rad/s.

• Let
ω = Angular Velocity of the body
v = Linear Velocity
s = Linear Displacement
α = Angular Acceleration of the body
r = Radius of the circular path
θ = Angular Displacement of the body in radians
From these assumptions, three relationships between angular and linear quantities will be obtained.
• Relationship between linear and angular displacement
The relation between Linear and Angular Displacement is given below:
s = r . θ
This relation is used to calculate the linear displacement of a body when the radius of the circular path and its angular displacement are given.
• Relationship between linear and angular velocity
The relation between linear and angular velocity is given below:
v = r × ω
This relation is used in the calculation of linear velocity of a body when radius and angular velocity are given.
• Relationship between Linear and Angular acceleration
The relation between Linear and Angular acceleration is given below:
a = r × α
This relation is used to calculate linear acceleration of a body when radius and angular acceleration are given.

## Relation between Angular and Linear Motion

The linear quantities form a typical analogy with angular quantities. Apart from the variable change, there is no change in the mathematical approach in both cases.

The following table describes the relation between Angular and Linear Motion.

Parameters Linear Angular
Initial Velocity u ω0
Final Velocity v ω
Displacement S θ
Acceleration a α
Relation for Final Velocity v = u + at ω = ω0 + αt
Relation for Distance Travelled S = ut + ½ at2 θ = ω0t + ½ αt2
Relation for Final Velocity v2 = u2 + 2a ω2 = ω02 + 2αθ

The above table describes about the variations while one changes from angular to linear quantities or vice-versa. However, we must take care of the units and dimensions while performing numerical calculations.

## Uses of Linear and Angular Quantities

Linear and Angular quantities are used as a prime tool in each and every engineering subject. Some uses are listed below:

• Linear quantities are used to calculate speed, time, and distance problems in physics.
• Angular Quantities are used in finding out Centrifugal and Centripetal Forces.
• Derivations of energy and mass conservations extensively use linear quantities.
• These equations are also applied in the field of Mathematics.
• Certain angular quantities are used in Momentum and Torque calculation in engineering subjects.
• Subjects like Strength of Materials and Machine Design cannot be imagined without the use of these quantities.
• Quantities like angular velocity are used to derive secondary properties like Angular Momentum which form the basis of Rotatory Kinetics.
• These equations are also used in the study of quantum and particle physics.

## Key Points to Remember

Here is the list of key points we need to remember about “Relationship between Linear and Angular Motion”.

• Rotation is defined as the movement of a body in circular path of fixed radius.
• Rectilinear motion is also known as Translatory Motion.
• Curvilinear rotational motion is defined as the motion o the body in circular path whose radius is not constant.
• Speed is a scalar quantity whereas velocity is a vector quantity.
• Negative Acceleration is also known as Retardation.
• If a body is moving in a circular path of fixed radius, then its velocity will be zero irrespective of its speed.
• The equations of linear motion are only applicable for uniform acceleration. If acceleration is variable, the functions S, v, t must be used for expressing and then integrated.
• If the rate of change of Angular Displacement is constant with respect to time, then it is known as Angular Speed of that object.

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