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Angular quantities for general motion

Module by: Sunil Kumar Singh. E-mail the author

Summary: Angular quantities are not limited to rotation about fixed axis only.

Note: You are viewing an old version of this document. The latest version is available here.

Note:

This module is under major revision. Readers are requested not to read this module for the time being. Inconvenience is regretted.

Angular quantities like angular displacement, velocity, acceleration and torque etc. have been discussed in earlier modules. We were, however, restricted in interpreting and applying these quantities to circular motion or pure rotational motion. Essentially, these physical quantities have been visualized in reference to an axis of rotation and a circular path.

In this module, we shall expand the meaning and application of angular quantities in very general terms, capable of representing pure as well as impure rotation and translation. We shall find, in this module, that pure translation and rotation are, as a matter of fact, special cases.

General interpretation of angular quantities

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, which is universally valid unlike Newton's law (for translational or rotational motion).

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
 Angular velocity  (aq3.gif)

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.

Angular velocity

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.

ω avg = Δ θ Δ t ω avg = Δ θ Δ t
(1)

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 :

ω = d θ d t ω = d θ d t
(2)

Figure 2: The angular displacement in a small time interval as measured with respect to a point.
Angular velocity
 Angular velocity  (aq4.gif)

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.

Angular and linear velocity

The relation of angular velocity of a particle about a point has the same vector form as for circular motion about an axis, but here again its interpretation is different. The angular velocity is related to linear velocity by the following vector product :

v = ω x r v = ω x r
(3)

The vector "r" is measured from the point about which angular velocity is being calculated. In case, the point coincides with the origin of the reference system (we normally plan so), the vector "r" becomes the position vector of the particle in the given reference.

It is important to realize that angular quantities like angular velocity are defined in terms of vector product. This is required to associate direction with angular quantities. The evaluation techniques for its magnitude and direction follow certain steps.

Evaluation of angular velocity

Evaluation of a vector quantity involves magnitude and direction :

(i) Magnitude of angular velocity

We can find the magnitude of angular velocity, using any of the following two alternatives :

(a) The magnitude of linear velocity as product angular velocity is given as :

v = ω r sin θ v = ω r sin θ

Rearranging for angular velocity, we have :

ω = v r sin θ ω = v r sin θ
(4)

In order to evaluate this expression, we need to find angle between position and velocity vector. The angle involved in the expression is shown in the left hand side figure.

Figure 3
Magnitude of angular velocity
(a) Magnitude in terms of enclosed angle (b) Magnitude in terms of moment arm
Figure 3(a) (aq2.gif)Figure 3(b) (aq5.gif)

(b) The expression for the magnitude of vector product relating linear velocity with angular velocity can also be interpreted in terms of moment arm as :

v = ω ( r sin θ ) = ω r v = ω ( r sin θ ) = ω r

ω = v r ω = v r
(5)

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

(ii) Direction of angular velocity

The direction of angular velocity is such that the plane formed by it and position vector is perpendicular to linear velocity. The angular velocity is also individually perpendicular to linear velocity vector. We must here understand that angular velocity is an operand of the vector product - not the vector product itself. As such, we can not directly apply right hand vector multiplication rule to obtain the direction of angular velocity.

The easiest directional visualization of angular velocity is gathered from the sense of angular displacement. If we curl fingers of the right hand along the motion about the point, then the direction of thumb points towards the direction of angular velocity. This estimate of direction together with fact that angular velocity is perpendicular to the plane of angular motion, completely determines the direction of angular velocity, provided motion is taking place in a plane.

Example 1

Problem : The velocity of a particle confined to xy - plane is (8i – 6j) at an instant when its position is (3 m, 4m). Find the angular velocity of the particle at that instant.

Solution : Angular velocity is related to linear velocity by the following relation,

v = ω x r v = ω x r

An inspection of the equation reveals that angular velocity is perpendicular to the direction of velocity vector. Now, velocity vector lies in xy-plane. It, then, follows that angular velocity is directed along z-axis. Let angular velocity be represented as :

ω = a k ω = a k

Putting the values, we have :

8 i - 6 j = a k x ( 3 i + 4 j ) 8 i - 6 j = a k x ( 3 i + 4 j )

8 i - 6 j = 3 a j - 4 a i ) 8 i - 6 j = 3 a j - 4 a i )

Comparing the coefficients of unit vectors on either side of the equation, we have :

3 a = - 6 a = - 2 3 a = - 6 a = - 2

and

- 4 a = 8 a = - 2 - 4 a = 8 a = - 2

Thus,

ω = - 2 k ω = - 2 k

The angular velocity, therefore, is 2 rad/s in the negative z – direction.

Summary

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.

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
 Motion of a rod  (aq6.gif)

Solution : In this case, the point about which angular velocity is to be determined is itself moving with a velocity of 10 m/s. However, we are required to find angular velocity for a particular instant i.e. instantaneous angular velocity for which the point can be considered at rest.

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 and equate the same with the given value.

Figure 5: The rod is constrained to move between a vertical wall and a horizontal surface.
Motion of a rod
 Motion of a rod  (aq7.gif)

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.

x t = AB t ( cos θ ) = - AB sin θ x θ t x t = AB t ( cos θ ) = - AB sin θ x θ 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 signifies that the angle “θ” decreases with time ultimately becoming equal to 0° as measured in anticlockwise direction from x-axis. We can, therefore, conclude that instantaneous angular velocity of point “A” is clockwise having a magnitude of 2 rad/s.

Figure 6: The rod is constrained to move between a vertical wall and a horizontal surface.
Motion of a rod
 Motion of a rod  (aq8.gif)

Interpretation of angular velocity

In this sub-section, we shall interpret angular velocity for general motion with a reference to circular motion (pure rotation), highlighting "where and how they are different or same".

We can see that the expressions of angular velocity are same as in the case of pure rotation. In the case of pure rotation, the angular velocity is perpendicular to the plane of angular displacement and is aligned either in the positive (clockwise) or negative (anticlockwise) direction of the axis. In the general case, the angular velocity is similarly perpendicular to plane of angular displacement and its sense of direction is clockwise or anticlockwise like in the case of rotation, but with respect to a point - not with respect to an axis. This makes the difference to the measurement of angular displacement and hence to the angular velocity as shown in the figure below :

Figure 7: The angular velocity with respect to point "O" is different to that with respect to y-axis in both magnitude and direction.
Comparison of angular velocities measurements
 Comparison of angular velocities measurements  (aq1.gif)

In the above figure, we have considered rotation of a particle about y-axis. For visualization, we have considered the initial position of the particle in the yz - plane. The particle moves from "A" to "B" in anticlockwise direction as seen from the top. We note following important differences for general consideration for rotation of a particle :

  • The linear distances of the particle at two instants are not same (OA # OB) from the point "O" as against from the axis, which are equal (O'A # O'B).
  • The angle measured from "O" (general case) and "O'" (rotational case) are different (θ # θ').
  • The moment arm from "O" is OO' not O'A as in the case of rotation.
  • The direction of angular velocity vector is not in the direction of y-axis as in the case of rotation.

For the sake of comparison, here, we have analyzed a pure rotation. In general, however, the plane of angular displacement, unlike pure rotation, can change and so the direction of angular velocity. We must understand that particle is free to move along any path.

As a matter of fact, angular velocity in rotation is just a special case of angular velocity defined in general. This can be visualized easily, if we make (i) the origin "O" to coincide with "O'" (ii) ensure that the magnitude of position vector (r) of the particle does not change during motion and (iii) the plane of motion is perpendicular to the axis of rotation. In that case, the particle will be constrained to rotate about an axis. Clearly, the angular velocity as defined for rotational motion is a special case of general definition with certain restrictions.

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