Is Angular Velocity Negative: 7 Facts(Read This First!)

The angular velocity is a radial displacement of an object rotating about its fixed rotational axis. Let us discuss whether is angular velocity negative.

The angular velocity is perpendicular to the angular acceleration of the object. The angular velocity is negative if the variation in the angular displacement decreases with time or the direction of angular displacement is clockwise, that is if the object is in a reverse motion.

Why angular velocity is negative?

If the angular velocity of the object decreases or if the direction of angular displacement of an object is in a clockwise direction, then the angular velocity will be negative.

The angular velocity is a change in the angular displacement with respect to time, and also it is the ratio of the tangential velocity of a rotating object and the radius of a circular path traced by the rotating body.

The direction of the tangential/linear velocity of a rotating object is perpendicular to the radius and the angular velocity of a body. The relation between the angular velocity and linear velocity is given as,

ω = v/r

Here, ω is an angular velocity, v is a linear velocity and r is a radius of a circular path traced by the rotating object from the axis of rotation.

The frequency of the rotating object is the time required to make one complete rotation of 3600 angle. If the direction of the linear velocity vector is negative, then the angular velocity will also be negative.

When angular velocity is negative?

The angular velocity is negative when the direction of rotation of a body is in a clockwise direction or the direction of linear velocity is along the negative axis.

The variation in the angular displacement with time determines the angular velocity of an object. If the change in the angular displacement is the maximum is a small amount of time, then the angular velocity will be positive and high. While the angular displacement decreases, then the angular velocity will be negative.

The change in the angular displacement with time gives the angular velocity. The following expression gives the angular velocity:-

ω = dθ/dt

Here, ω is an angular velocity, dθ is the change in the angular displacement of the object. It is equal to the difference in the final position of an object to its initial position of an object. It is given by the relation dθ= θ2 – θ1.

dt is a change in time from the initial angular position of an object to the final displacement.

Suppose the final angular displacement of an object is less than the initial position. In that case, if θ2 < θ1, the change in the angular displacement of an object will be negative, and hence the angular velocity will be negative.

How angular velocity is negative?

The angular velocity is positive if the rotation of a body is anti-clockwise, and the angular velocity is negative if the direction of rotation is clockwise.

The angular velocity of an object can also be negative if the frequency of rotation of a body decreases with time. If an object’s velocity or rotational speed decreases, then the angular velocity of an object during that time interval will become negative.

The negative angular velocity with the deceleration of the rotating object is obvious because the total angular displacement with every time interval reduces frequently. Hence, any object coming to a rest position can be believed to have a negative angular velocity.

On the contrary, it is also true that if there is no variation in an object’s angular displacement rate, then the angular velocity is constant and can have either positive or negative values.

What causes negative angular velocity?

The negative angular velocity can be related to the deceleration of an object.

The negative angular velocity is mainly observed in an object, which decelerates with time. The deceleration of an object is due to the imposition of the external force that opposes the speed of a rotating object.

The negative angular velocity of an object is when the angular displacement of an object decreases with every time interval. The angular velocity depends upon the rate at which there is a variation seen in the angular displacement of an object.

Examples of Negative Angular Velocity

• A cart slowing down: A cart slowing down will decrease its angular displacement gradually till it comes to rest. Hence, the change in the angular displacement becomes negative, and therefore the angular velocity is negative.
• Reducing the speed of vehicles: Upon reducing the speed of a vehicle, the angular frequency of tires decreases. Hence the angular velocity of the tires during that time interval is negative.
• Taking a car in the reverse direction: While driving a car in a reverse direction, the rotation of the tires is in a clockwise direction, and therefore the angular velocity is negative. Moreover, the displacement of cars is in the negative direction.
• A Cricket ball kicked high in the air: When the cricket ball collides with the bat, the angular acceleration of a ball is high. The acceleration of a ball becomes nearly zero when a ball rises high in the air as it gains the maximum potential energy.
• A gravitron coming to rest: Initially, a graviton’s frequency is gradually increased. Thus there is a positive angular velocity and acceleration. While bringing the gravitron to the rest, the angular velocity of the graviton and its acceleration becomes negative.
• Spinning Top: The angular displacement of the rotating spinning top decreases as the center of gravity of the spinning-top points more and more outside the point of contact between a tip of a spinning top and the ground. Thus, this also gives the negative angular velocity while decelerating.

When angular velocity is positive?

The angular velocity is positive when the angular displacement of an object increases with every time interval.

The angular velocity of a rotating body with continuously increasing speed or an object rotating in a counterclockwise direction has a positive angular velocity.

What is the angular velocity if the angular displacement of a rotating disk was measured 5π rad at time t1 = 20 sec and 3π at time t2 = 25 sec?

Given: The initial angular displacement is, θ1 = 5π rad.

The final angular displacement is, θ2 = 3π rad.

The initial time is, t1 = 20 sec.

The final time is, t2 = 25 sec.

The formula to calculate the angular velocity with the variable angular displacement is,

ω = dθ/dt

ω = ( θ2 – θ1)/( t2 – t1)

ω = (3π- 5π) rad /( 25- 20) sec

ω = – 2π rad/ 5 sec

ω = – 2π rad/ 5 sec

Hence, the angular velocity of a rotating disk between the time interval t1 = 20 sec and t2 = 25 sec is – 0.4π rad/sec.