Relation Between Linear Velocity and Angular Velocity

Angular velocity is the change in the angular position of an orbiting object. It denotes the rate of a particle while it is rotating, spinning and moving around a centre point. On the other hand, Linear velocity is the rate used to measure an object's movement from one place to another in a straight line. Angular velocity is measured in degrees and radians, and linear velocity is measured in m/s. Angular velocity is denoted by a symbol , and for linear velocity, it is v. Object moving in a circular motion has an angular velocity along with its axis and it always remains constant. In linear velocity, if an object is undergoing a circular motion, then it is measured along with the circumference of a circle, and it varies at every point. 

Key Terms: Angular velocity, Linear velocity, Vector quantity, average angular velocity, Instantaneous angular velocity, Constant linear velocity, Angular displacement

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What is Angular Velocity?

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Angular velocity is a vector quantity measurement of the rotation rate, which means how fast an object can travel from one point to another in a circular motion. The dimensional formula of angular velocity is:

[M⁰ L⁰ T^-1]

Where,

• M = Mass
• L = Length
• T = Time

If an object is rotating around an axis, then every point of the object has the same angular velocity. Every point farther from the axis of rotation is moving differently than the points closer to the axis.

The angular velocity of an object varies over time. It is expressed as follows:

ω = θ / t

Where ω represents angular velocity, θ represents angular displacement, and t represents the change in time.

According to the above-mentioned expression, positive angular velocity indicates anticlockwise rotation, while negative velocity indicates clockwise.

There are two types of angular velocity:

• The average angular velocity is defined by the ratio of the angular displacement to the time interval. 
• The instantaneous angular velocity is the limit of average angular velocity as the time interval approaches zero.

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What is Linear Velocity?

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Linear Velocity refers to the path change in an object’s position during a specific time range. It is a vector quantity. It also means that it is the rate of change in the movement of an object to time while the object is moving at a straight path. The dimension formula of linear velocity is [M]⁰[L]¹[T]^-1

One of the basic differences between Linear and Angular Velocity is, Linear velocity is applied to a moving object. Whereas angular velocity is applied to those that rotate, turn and spin. Example: Wheel and Earth’s revolution. 

Linear velocity depends on the distance travelled by an object to the time taken. The linear velocity formula is given by,

v = x / t

Where

• v = Linear velocity
• x = distance covered
• t = Time taken to cover the distance(x).

Any object that is turning or moving in a circular direction involves both linear and angular movement. The angular velocity refers to the ratio of the angle travelled to the amount of time it took to travel that angle. To calculate the linear velocity from angular velocity, you can apply the following formula:

V_r = r.ω

Where,

ω is expressed in radians/time and

r denotes the radius of the path taken.

Constant linear velocity a part of linear velocity, means an object is moving at a constant speed without changing its direction. Linear velocity shows that the object’s direction is not changed, and when it is said to be constant it means that the magnitude at which it is moving is constant.

Read More: Unit of Time

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Relationship between Angular Velocity And Linear Velocity

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There is a mathematical relationship that can be established between Angular and Linear Velocity. 

Let's consider an object of some weight in movement. The change in the movement is Δs with continuous change in time Δt.

According to the definition, the linear velocity of a particle v is,

ν = Δs / Δt

Similarly the angular velocity of a particular object with movement “s” in time “t” is given by,

ω = Δθ / Δt

An object is moving in a circular motion with radius “r”. The relation between linear and angular displacement can be given by the formula,

S = rθ - equation (1)

θ = S / r

And,

ν​ = Δr / Δt - for a particle performing linear displacement.

By Substituting the value of “S” from equation (1), we get the equation;

ν = Δs / Δt

Also,

According to the definition of angular velocity;

ν = Δθ  / Δt

Hence,

ν= r (Δθ / Δt)

As per the definition of angular velocity,

ω = Δθ / Δt

Hence,

ν = rω establishing the relationship between linear velocity and angular velocity.

Read More: Velocity Formula

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Solved Examples

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Ques. Find the angular velocity of the ball travelling at a speed of 10 m/s in a 20m radius circle. 

Ans. The relation between the angular and linear velocity of the ball is given by, 

|v| = rω 

Given: v = 10 m/s, r = 20m 

Find: ω =? 

v = rω

⇒ 10 = (20)ω ⇒ 0.5 m/s = ω

Ques. A planet is moving around its sun in a circular manner. The angular velocity of the planet is 0.5 rad/s. The distance of the planet from its sun is estimated at 1,00,000 Km. Find out the linear velocity of the planet. 

Ans. The relation between the angular and linear velocity of the ball is given by, 

|v| = rω 

Given: ω = 0.5 rad/s, r = 105 Km ⇒ r = 108 m 

Find: v =? 

v = rω

⇒ v = (108)(0.5) ⇒ 5 × 107 m/s = v

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Things to Remember

• Angular velocity refers to the rate at which a particular object is moving around a centre point.
• Linear velocity is the rate at which a particle gets displaced from one place to another in a straight line.
• Angular velocity is measured in degrees and radians whereas linear velocity is always measured in m/s or metres per second.
• Angular velocity is denoted by a symbol or and for linear velocity it is v.
• Angular velocity of an object is expressed as ω = θ/t
• There are two types of angular velocities: Average and Instantaneous.
• Formula for the linear velocity is: v = x/t
• Formula to calculate linear velocity from angular velocity is: Vr = r.ω