Tangential velocity is the linear component of the speed of any object which is moving along a circular path. When an object moves in a circular path at a distance r from the center, then the body’s velocity is directed tangentially at any instant. This is termed as tangential velocity. Also, we may say that the linear velocity is its tangential velocity at any instant. This topic will explain the tangential velocity formula with examples. Let us learn it!

**What is Tangential Velocity?**

A tangent is simply a line that touches a non-linear curve (like a circle) at only a single point. It represents an equation with the relationship between the coordinates “x” and “y” on a two-dimensional graph.

The tangential velocity is the measurement of the speed at any point tangent to a rotating wheel in a circular motion. Thus angular velocity, \(\omega\), is related to the tangential velocity, \(V_{t}\), through the formula. Tangential velocity is the component of the motion along the edge of a circle measured at any arbitrary point of time. As per its name, tangential velocity describes the motion of an object along the edge of the circle, whose direction at any given point on the circle is always along the tangent to that point.

**The Formula for Tangential Velocity**

First, we have to calculate the angular displacement \(\theta\), which is the ratio of the length of the arc ‘s’ that an object traces on this circle to its radius ‘r’. We measure it in radians.

The rate of change of the object’s angular displacement is its angular velocity. It is denoted by \(\omega\) and its standard unit is radians per second. It is different from the linear velocity, as it only deals with objects moving in a circular motion. Therefore, it measures the rate at which angular displacement is swept.

- Tangential Velocity Formula is: \(V_{t} = r \times \frac {d\theta}{dt}\)
- Tangential Velocity Formula can also be: \(V_{t} = r \times \omega\)
- Another formula for Tangential Velocity is: \(V_{t} = \frac{ 2 \pi r}{t}\)

Where,

\(V_{t}\) | Tangential Velocity |

r | The radius of the circular path |

\(\omega\) | Angular Velocity |

t | Time |

\(\theta\) | Angular Displacement |

Tangential velocity formula is applicable in calculating the tangential velocity of any object moving in a circular path. Its unit is meter per second.

**Solved Examples for Tangential Velocity Formula**

Q.1: If the angular velocity of a wheel is 40 \frac{rad}{s}, and the wheel diameter is 60 cm. Determine the tangential velocity of the wheel.

Solution: Given parameters are,

Radius, \(r = \frac {1}{2} \times diameter\)

\(r = \frac{1}{2} \times 60\)

r= 30 cm

r = 0.30 m.

Angular velocity, \(\omega = 40 rad per s.\)

Tangential velocity formula is as given:

\(V_{t} = r \times \omega\)

\(= 0.30 \times 40\)

= 12 m/s

Thus tangential velocity will be 12 meters per sec.

Q.2: If a wheel is turning with a speed of 12 m per sec, and its angular velocity is 6 radians per sec. Then find out its radius.

Solution: The tangential velocity is as follows,

\(V_{t} = 12 m per sec.\)

The angular velocity, \(\omega\), is 6 radians/sec.

Now the formula for tangential velocity is:

\(V_{t} = r \times \omega\)

Rearranging it,

\(r = \frac {V_{t}}{\omega}\)

\(= \frac {12}{6}\)

= 2 m

Therefore, the radius is 2 meters.

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