Angular Velocity Formula: Explanation and Solved Examples

Angular Velocity Formula is used to calculate the distance traveled by a body in terms of rotation and revolutions per unit of time.

• Angular velocity is the time rate at which an object rotates or circles around an axis, or the rate at which the angular displacement between two bodies varies.
• The angular velocity of an object moving on a circular path applies to the entire object.
• The angle traced by a particle during circular motion is known as angular displacement.
• Angular displacement is an axial vector since its direction is along the axis.
• The SI unit of angular velocity is radian per second (rad/s).

Read Also: Angular Acceleration

Key Terms: Angular Speed, SI Unit, Linear Velocity, Angular Displacement, Radius, Angular acceleration, Instantaneous angular velocity, Average angular velocity

---

Angular Velocity

[Click Here for Sample Questions]

Angular velocity is a vector number that indicates the angular speed or rotational speed of an object and the axis around which the object is revolving.

• The amount of change in the particle's angular displacement over time is known as angular velocity.
• The angular velocity vector's course is vertical to the plane of rotation, as indicated by the right-hand rule.
• An example of angular velocity is a roulette ball on a roulette wheel, a race car on a circular circuit, or a Ferris wheel.
• Furthermore, the object's angular velocity is its angular displacement with respect to time.
• Additionally, as an object moves along a circular path, the central angle that corresponds to the object's position on the circle shifts.
• Furthermore, the angular velocity, denoted by the symbol ω, is the rate at which this angle changes with respect to time.

Also Read:

SI Unit of Angular Velocity

The SI unit of angular velocity is radians per second, with the radian having a dimensionless value of one, hence the SI units of angular velocity are 1/s or s^-1. The symbol omega (ω, sometimes Ω). is commonly used to signify angular velocity.

---

Angular Velocity Formula

[Click Here for Sample Questions]

Let θ be the angular displacement of a body in time t, then the formula of angular velocity is given by

\(\omega=\frac{\theta}{t}\)

Where

• The change in angular displacement is denoted by θ.
• The shift in time t is represented by t.

Positive angular velocity indicates counterclockwise rotation, whereas negative indicates clockwise rotation.

Average Angular Velocity Formula

The ratio of the angular displacement to the time interval represents the average angular velocity of a rotating rigid body. It is given by the formula

\(\omega_{avg}=\frac{\Delta \theta}{\Delta t}=\frac{\theta_2-\theta_1}{t_2-t_1}\)

Instantaneous Angular Velocity Formula

The limit of the average angular velocity as the time interval approaches to zero is known as the instantaneous angular velocity.

\(\omega_{ins}=\lim_{\Delta x \to 0}\frac{\Delta \theta}{\Delta t}=\frac{d\theta}{dt}\)

Also Check:

---

Linear Velocity

[Click Here for Sample Questions]

Linear velocity describes the motion of an object or particle in a straight line.

• It also refers to the rate at which an object's location changes over time.
• Furthermore, the most typical example is the speed with which you drive along the road.
• Furthermore, the speedometer displays your linear velocity, which is measured in kilometers per hour (km/h).

Read More: Moment of Torque

---

Relation Between Linear and Angular Velocity

[Click Here for Sample Questions]

The relation between linear and angular velocity for a body moving in a uniform circular motion is:

where,

• The linear velocity is denoted by the letter v.
• The radius of the circular path is denoted by r.

The linear velocity (v) is proportional to the particle's distance from the circular path's center and its angular velocity, according to this equation.

• The linear velocity varies depending on where you are in the circle.
• At the center, it's zero.
• The distance between the center and any point on the circumference is the shortest.
• It reaches its apex at the circle's circumference. The angular velocity, on the other hand, is constant along the circular journey.

Also Read:

---

Things to Remember

• S = rθ simplifies the computation by putting the value of in radians in the equation.
• The angular velocity, like the displacement, is an axial vector quantity. Using the Right-hand Thumb Rule, we can determine its direction. The following rule applies:
• Curl your fingers counterclockwise, and the direction of angular motion is the thumb pointing outwards (along the axis). Similarly, curling your fingers clockwise gives you the direction of your thumb pointing inwards.
• This equation holds for any particle with a rigid body at any given time.
• The speed (velocity isn't really right here; that's a vector) will increase if the radius is changed by gently reducing the string. Again, when the radius decreases, velocity increases, but in order to stay in orbit, one must fire engines to accelerate the satellite!

Also Read:

---