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Tangential Velocity Formula

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  • Last Updated : 01 Feb, 2022

The linear component of any object’s speed when moving on a circular path is called tangential velocity. When an object moves in a circular path at a distance of r from the centre, the velocity of the body is always tangential. Tangential velocity is the word for this. It can also be stated that the linear velocity is equal to the tangential velocity at any given time. The tangential velocity formula will be explained with examples in this article. 

What is the Tangential Velocity?

Tangential velocity explains the motion of an object along the circle’s edge whose direction is always at the tangent to any given point on the circle.

Hence, Tangential velocity is the component of motion along the edge of a circle measured at any arbitrary instant.

A tangent is a line that only touches one point of a non-linear curve (such as a circle). On a two-dimensional graph, it represents an equation with the relationship between the coordinates x and y. 

In a circular motion, the tangential velocity is the measurement of the speed at any point tangent to the revolving wheel. Through the formula, angular velocity ω is connected to tangential velocity, Vt. Tangential velocity is the component of motion along the circle’s edge that may be measured at any time. 

Formula for Tangential Velocity

First, we must determine the angular displacement θ, which is defined as the ratio of the length of the arc’s traced by an item on this circle to its radius ‘r’.

The angular velocity of an object is the rate at which its angular displacement changes. Its standard unit is radians per second, and it is represented by ω. It differs from linear velocity in that it only considers objects that move in a circular motion. As a result, it is used to calculate the rate at which angular displacement is swept.

Mathematically, the tangential velocity Vt is given as:

Vt = r × ω

where, 

  • r is the radius of the circular path, and
  • ω is the angular velocity
  • But, the angular velocity is defined as,

ω = dθ/dt = 2π/t

where,

  • dθ/dt is the time rate change of angular displacement θ, and
  • t is the time taken.

Therefore, the tangential velocity becomes:

Vt = r × dθ/dt

or 

Vt = r × 2π/t

The tangential velocity of any object moving in a circular direction can be calculated using the tangential velocity formula. 

  • Unit of Tangential Velocity: Metre per second or m/s.
  • Dimension formula for Tangential Velocity: [M0LT-1].

Sample Problems

Problem 1: The angular velocity of a circular ring is 20 rad/s, and its diameter is 20 cm. Find its tangential velocity.

Solution:

Given that:

The angular velocity, ω = 20 rad/s,

The diameter of the ring, d = 20 cm.

Therefore, the Radius, r = 20 cm/2 = 10 cm = 0.1 m

The formula for Tangential velocity is as given:

Vt = r × ω

    = 0.1 m × 20 rad/s

    = 2 m/s

Problem 2: Determine the tangential velocity of a disc that has an angular velocity of 10 rad/s, and a radius of 5 m.

Solution:

Given that:

The angular velocity, ω = 10 rad/s,

The radius of the disc, d = 5 m.

The formula for Tangential velocity is as given:

Vt = r × ω

    = 5 m × 10 rad/s

    = 50 m/s

Problem 3: What is the radius of the wheel which turns with a speed of 10 m/s, and its angular velocity is 5 rad/s?

Solution:

Given that:

The tangential velocity, Vt = 10 m/s,

The angular velocity, ω = 5 rad/s.

The formula for Tangential velocity is as given:

Vt = r × ω

10 m/s = r × 5 rad/s

r = 2 m

Problem 4: What is the radius of the ring which has a tangential velocity of 50 m/s, and its angular velocity is 5 rad/s?

Solution:

Given that:

The tangential velocity, Vt = 50 m/s,

The angular velocity, ω = 5 rad/s.

The formula for Tangential velocity is as given:

Vt = r × ω

50 m/s = r × 5 rad/s

r = 10 m

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