Here, we shall define and interpret angular quantities very generally with respect to a "point" in the reference system - rather than an axis. This change in reference of measurement allows us to extend application of angular quantities beyond the context of rotational motion. We can actually associate all angular quantities even with a straight line motion i.e. pure translational motion. For example, we can calculate torque on a particle, which is moving along a straight line. Similarly, we can determine angular displacement and velocity for a projectile motion, which we have studied strictly from the point of view of translation. We shall work out appropriate examples to illustrate extension of angular concepts to these motion.

We must understand here that the broadening the concept of angular quantities is not without purpose. We shall find out in the subsequent modules that the de-linking of angular concepts like torque and angular momentum from an axis, lets us derive very powerful law known as conservation of angular momentum.

The example given below calculates average angular velocity of a projectile to highlight the generality of angular quantity.

Example

Problem 1 : A particle is projected with velocity "v" at an angle of "θ" with the horizontal. Find the average angular speed of the particle between point of projection and point of impact.

Solution : The average angular speed is given by :

Figure 1: Average angular speed during the flight of a projectile.

Angular velocity

ω avg = Δ θ Δ t ω avg = Δ θ Δ t

From the figure, magnitude of the total angular displacement is :

Δ θ = 2 θ Δ θ = 2 θ

On the other hand, time of flight is given by :

Δ t = 2 v sin θ g Δ t = 2 v sin θ g

Putting these values in the expression of angular velocity, we have :

ω avg = Δ θ Δ t = 2 θ g 2 v sin θ ω avg = θ g v sin θ rad / s ω avg = Δ θ Δ t = 2 θ g 2 v sin θ ω avg = θ g v sin θ rad / s

From this example, we see that we can indeed associate angular quantity like angular speed with motion like that of projectile, which is not strictly rotational.

As a particle moves, the line joining a fixed point and particle moves through angular displacement. Important thing to note here is that the particle may not follow a circular path - it can describe any curve even a straight line. We, then, define average angular velocity as :

Definition 1: Average angular velocity

Average angular velocity of a particle is equal to the ratio of change in angular displacement and time.

Instantaneous angular velocity is obtained by taking the limit when time interval tends to become zero. In other words, the instantaneous angular velocity (simply referred as angular velocity) is equal to the first differential of angular displacement with respect to time :

Figure 2: The angular displacement in a small time interval as measured with respect to a point.

Angular velocity

Since angle is measured with respect to a point, the measurement of angular velocity, in turn, will depend on the choice of origin. Further, the linear distance of the particle from a given point is not constant like in circular motion.

In the case of circular motion about an axis, we have seen that the vector relation of linear velocity of a particle is related to angular velocity by the relation given by :

In the case of rotation about an axis, the plane of velocity and position vector is same. This is not necessary for motion of a particle about any arbitrary point. The position vector and velocity can be oriented in any direction. It means that plane of motion can be any plane in three dimensional coordinate system and is not constrained to be in the plane containing position vector.

Interpretation of the vector equation given above is difficult. We observe that angular velocity is an operand of the cross product. We can say that linear velocity is perpendicular to the plane formed by angular velocity and position vector. The difficulty is interpreting direction of angular velocity in terms of the directions of velocity and position vector. From the point of view of angular velocity, we can only say that it is perpendicular to linear velocity, but it can have any orientation with respect to position vector.

Here, we consider a situation as shown in the figure. We see that velocity of the particle and the position vector form an angle, "θ" between them. For convenience, we have considered the point about which angular velocity is defined as the same as origin of coordinate system.

Figure 3: The angular displacement in a small time interval as measured with respect to a point.

Angular velocity

To analyze the situation, let us consider components of velocity in the radial and tangential directions.

The radial component of velocity is not a component for rotation. Its direction is along the line passing through the origin. The rotation is accounted by the component of velocity perpendicular to position vector. Now, tangential velocity is perpendicular to position vector. Thus, we can use the fact that magnitude of tangential velocity is equal to the product of angular velocity and linear distance from the origin. Hence, the magnitude of velocity is given as :

v sin θ = ω r v sin θ = ω r

⇒ ω = v sin θ r = v ⊥ r ⇒ ω = v sin θ r = v ⊥ r

We can use the expression as defined above to calculate the magnitude of angular velocity. The evaluation of the first expression will require us to determine angle between two vectors. The second expression, which uses tangential velocity, gives easier option, when this component is known.

Instantaneous axis of rotation

The notion of rotation with the motion of a particle is visualized in terms of moving axis. We have discussed that the direction of angular velocity is perpendicular to the plane formed by tangential velocity and the point about which angular velocity is measured. We can infer this direction to be the axial direction for the angular velocity at that instant.

In case, these instantaneous axes are parallel to each other, then it is possible to assign unique angular velocity of the particle about any one of these instantaneous axes. We shall work out an example here to illustrate determining angular velocity about an instantaneous axis.

Example 3

Problem 2 : A rod of length 10 m lying against a vertical wall starts moving. At an instant, the rod makes an angle of 30° as shown in the figure. If the velocity of end “A” at that instant, is 10 m/s, then find the angular velocity of the rod about “A”.

Figure 4: The rod is constrained to move between a vertical wall and a horizontal surface.

Motion of a rod

Solution : In this case, the point about which angular velocity is to be determined is itself moving towards right with a velocity of 10 m/s. However, the instantaneous axis of rotation is parallel to each other. As such, it is possible to assign unique angular velocity for the motion under consideration.

We shall examine the situation with the geometric relation of the length of rod with respect to the position of its end “A”. This is a logical approach as the relation for the position of end “A” shall let us determine its velocity.

Figure 5: The rod is constrained to move between a vertical wall and a horizontal surface.

Motion of a rod

x = AB cos θ x = AB cos θ

An inspection of the equation reveals that if we differentiate the equation with respect to time, then we shall be able to relate linear velocity with angular velocity.

ⅆ x ⅆ t = AB ⅆ ⅆ t ( cos θ ) = - AB sin θ x ⅆ θ ⅆ t ⅆ x ⅆ t = AB ⅆ ⅆ t ( cos θ ) = - AB sin θ x ⅆ θ ⅆ t

⇒ v = - AB ω sin θ ⇒ ω = - v AB sin θ ⇒ v = - AB ω sin θ ⇒ ω = - v AB sin θ

Putting values,

⇒ ω = - 10 10 sin 30 ° = - 2 rad / s ⇒ ω = - 10 10 sin 30 ° = - 2 rad / s

Negative sign here needs some clarification. We can visualize that rod is rotating about instantaneous axis in anticlockwise direction. As such, the sign of angular velocity should have been positive - not negative. But, note that we are measuring angle from the negative x-direction as against positive x-direction. Therefore, we conclude that rod rotates about the instantaneous axis in anticlockwise direction.

Figure 6: The rod is constrained to move between a vertical wall and a horizontal surface.

Motion of a rod

1: Angular quantities are general quantities, which can be defined and interpreted for any motion types – pure/impure translation or rotation.

2: Angular velocity is defined as the time rate of change of angular displacement with respect to a point :

ω = d θ d t ω = d θ d t

3: If the point, about which angular velocity is determined, is the origin of the reference system, then the position of the particle is represented by position vector.

4: The measurement and physical interpretation of angular velocity are different than that in the case of rotational motion.

5: Angular velocity in rotation is a sub-set of angular velocity in general case. In the case of rotation, moment arm is radius vector, which is perpendicular to the axis of rotation. Also, the plane of motion in rotation is perpendicular to the axis of rotation.

6: The relation between linear and angular velocity has the following form,

v = ω x r v = ω x r

where “r” determines the position of the particle with respect to the point. The magnitude of angular velocity can be determined, using any of the following two relations :

ω = v r sin θ ω = v r sin θ

and

ω = v r ⊥ ω = v r ⊥

where " r ⊥ r ⊥ " represents perpendicular distance of the velocity vector from the point of reference

7: Direction of angular velocity is perpendicular to the plane of motion. The plane of motion, however, may not be fixed like in the case of rotation. As such, direction of angular velocity may not be limited to only two directions as in the case of rotation. For this reason, we need to treat angular velocity as a general vector quantity, which can not be represented by a scalar with positive and negative signs as in the case of rotation.