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Circular motion and rotational kinematics

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

Summary: Each part of a rigid body under pure rotational motion describes a circular motion about a fixed axis.

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

In pure rotational motion, the constituent particles of a rigid body rotate about a fixed axis in a circular trajectory. The particles, composing the rigid body, are always at a constant perpendicular distance from the axis of rotation as their internal distances within the rigid body is locked. Farther the particle from the axis of rotation, greater is the speed of rotation of the particle.

Figure 1: Each particle constituting the body executes an uniform circular motion about the fixed axis.
Rotation of a rigid body about a fixed axis
 Rotation of a rigid body about a fixed axis  (am1.gif)

We shall study these and other details about the rotational motion of rigid bodies at a later stage. For now we confine ourselves to the aspects of rotational motion, which are connected to the circular motion as executed by a particle. In this background, we can say that uniform circular motion represents the basic form of circular motion and hence that of rotational motion.

The description of rotational motion is best suited to corresponding angular quantities as against linear quantities that we have so far used to describe translational motion. In this module, we shall introduce these angular quantities and prepare the ground work to enable us apply the concepts of angular motion to “circular motion” in general and “uniform circulation motion” in particular.

Most important aspect of angular description as against linear description is that there exists one to one correspondence of quantities describing motion : angular displacement (linear displacement), angular velocity (linear velocity) and angular acceleration (linear acceleration).

Angular quantities

In this section, we discuss some of the defining quantities, which are used to describe uniform circular motion. These quantities are angular position, angular displacement and angular velocity.

Notably, we shall not discuss angular acceleration. It will be discussed as a part of non-uniform circular motion in a separate module.

Angular position (θ)

We need two straight lines to measure an angle. In rotational motion, one of them represents fixed direction, while another represents the rotating arm containing the particle. Both these lines are perpendicular to the rotating axis and passing through the position of the particle.

Figure 2: Angular position is the angle between reference direction and rotating arm.
Angular position (θ)
 Angular position (θ)   (am2.gif)

For convenience, the reference direction like x – axis of the coordinate system serves to represent fixed direction. The angle between reference direction and rotating arm (OP) at any instant is the angular position of the particle (θ).

It must be clearly understood that angular position (θ) is an angle and does not represent the position of the particle by itself. It requires to be paired with radius of the circle (r) along which particle moves in order to specify the position of the particle. Thus, a specification of a position in the reference system will require both “r” and “θ” to be specified.

Figure 3
Relation between distance (s) and angle (θ)
 Relation between distance (s) and angle (θ)   (am3.gif)

By geometry,

θ = s r s = θ r θ = s r s = θ r

where s is the length of the arc subtending angle “θ” and “r” is the radius of the circle containing the position of the particle. The angular position is measured in “radian”, which has no dimension, being ratio of two lengths. One revolution contains 2π radians. The unit of radian is related to other angle measuring units as :

1 revolution = 360° = 2π radian

Note:

The quantities related to angular motion are expressed in terms of angular position. It must be ensured that values of angular position wherever it appears in the expression be substituted in radians only. If the given value is in some other unit, then first we need to change the value into radian. It is so because, radian is a unit derived from the definition of the angle. The defining relation θ = s/r will not hold unless “θ” is in radian.

Angular displacement (Δθ)

Angular displacement is equal to the difference of angular positions at two instants of rotational motion.

Figure 4: Angular displacement is equal to the difference of angular positions at two positions.
Angular displacement (Δθ)
 Angular displacement (Δθ)  (am4.gif)

Δ θ = θ 2 - θ 1 Δ θ = θ 2 - θ 1

The angular displacement is also measured in “radian” like angular position.

Angular velocity (ω)

Angular speed is the ratio of the magnitude of angular displacement and time interval.

ω = Δ θ Δ t ω = Δ θ Δ t

This ratio is called average angular velocity, when evaluated for finite time interval and instantaneous angular velocity, when evaluated for infinitesimally small period (Δ→0).

ω = θ t ω = θ t

The angular position is measured in “rad/s”.

Description of circular motion

Circular motion is completely described when angular position of a particle is given as a function of time like :

θ = f ( t ) θ = f ( t )

For example, θ = 2 t 2 - 3 t + 1 θ = 2 t 2 - 3 t + 1 tells us the position of the particle with the progress of time. The attributes of circular motion such as angular velocity and acceleration are derivatives of this function.

Similarity to pure translational motion is quite obvious here. In pure translational motion, each particle constituting a rigid body follows parallel linear paths. The position of a particle is a function of time, whereby :

x = f ( t ) x = f ( t )

Example 1

Problem : The angular position (in radian) of a particle under circular motion about a perpendicular axis with respect to reference direction is given by the function in time (seconds) as :

θ = t 2 - 0.2 t + 1 θ = t 2 - 0.2 t + 1

Find (i) angular position when angular velocity is zero and (ii) determine whether rotation is clock-wise or anti-clockwise.

Solution : The angular velocity is equal to first derivative of angular position,

ω = θ t = t ( t 2 - 0.2 t + 1 ) = 2 t - 0.2 ω = θ t = t ( t 2 - 0.2 t + 1 ) = 2 t - 0.2

For ω= 0, we have :

2 t - 0.2 = 0 t = 0.1 s 2 t - 0.2 = 0 t = 0.1 s

The angular position at t = 0.1 s,

θ = ( 0.1 ) 2 - 0.2 x 0.1 + 1 = 0.99 rad θ = 0.99 x 360 2 π = 0.99 x 360 x 7 2 x 22 = 56.7 0 θ = ( 0.1 ) 2 - 0.2 x 0.1 + 1 = 0.99 rad θ = 0.99 x 360 2 π = 0.99 x 360 x 7 2 x 22 = 56.7 0

As the particle is at a positive angle with respect to reference direction, we conclude that the particle is moving in anti-clockwise direction.

Relationship between linear (v) and angular speed (ω)

Let us consider two concentric uniform circular motions with equal time period (T) along two circular trajectories of radii r1 and r2. It is evident that particle along the outer circle is moving at a greater speed as it has to cover greater perimeter or distance. On the other hand angular speeds of the two particles are equal as they transverse equal angles in a given time.

This observation is key to understand the relation between linear and angular speed. In words, linear velocity (v) is proportional to the radius of circle (r) for a constant angular velocity (ω).

Now, we know that :

s = r θ s = r θ

Differentiating with respect to time, we have :

s t = r t θ + r θ t s t = r t θ + r θ t

Since, “r” is constant for a given circular motion, r t = 0 r t = 0 .

s t = θ t r = ω r s t = θ t r = ω r

Now, s t s t is equal to linear speed, v. Hence,

v = ω r v = ω r

This is the relation between angular and linear speeds. Though it is apparent, but it is emphasized here for clarity that angular and linear speeds do not represent two separate individual speeds. Remember that a particle can have only one speed at a particular point of time. They are, as a matter of fact, equivalent representation of the same change of position with respect to time. They represent same speed – but in different language or notation.

Example 2

Problem : The angular position (in radian) of a particle along a circle of radius 0.5 m about a perpendicular axis with respect to reference direction is given by the function in time (seconds) as :

θ = t 2 - 0.2 t θ = t 2 - 0.2 t

Find linear velocity of the particle at t = 0 second.

Solution : The angular velocity is given by :

ω = θ t = t ( t 2 - 0.2 t ) = 2 t - 0.2 ω = θ t = t ( t 2 - 0.2 t ) = 2 t - 0.2

For t = 0, the angular velocity is :

ω = 2 x 0 - 0.2 = 0.2 rad ω = 2 x 0 - 0.2 = 0.2 rad

The linear velocity at this instant is :

v = ω r = 0.2 x 0.5 = 0.1 m / s v = ω r = 0.2 x 0.5 = 0.1 m / s

Vector representation of angular quantities

The angular quantities (displacement, velocity and acceleration) are also vector quantities like their linear counter parts and follow vector rules of addition and multiplication, with the notable exception of angular displacement. Angular displacement does not follow the rule of vector addition, which states that the result of vector addition does not depend on the order in which vectors are added. We intend here to skip the details of this exception to focus on the subject matter in hand.

The vector angular quantities like angular velocity ( ω) is represented by a vector, whose direction is obtained by applying “Right hand rule”. We just hold the axis of rotation with right hand in such a manner that the direction of fingers is along the direction of the rotation. The direction of extended thumb (along y-axis in the figure below) then represents the direction of angular velocity ( ω).

Figure 5: Right Hand Rule (RHR)
Vector cross product
 Vector cross product  (am5.gif)

The important aspect of angular vector representation is that the angular vector is essentially a straight line of certain magnitude represented on certain scale with an arrow showing direction (shown in the figure as a red line with arrow) – not a curl as some may have expected.

Further, the angular vector quantities are axial in nature. This means that they apply along the axis of rotation. Now, there are only two possible directions along the axis of rotation. Thus, we can work with sign (positive or negative convention) to indicate directional attribute of angular quantities.

The angular quantities measured in counter clockwise direction is considered positive, whereas quantities measured in clockwise direction is considered negative.

This simplicity resulting from fixed axis of rotation is very useful. We can take the liberty to represent angular vector quantities in terms of signed scalar quantities as done in the case of linear quantities. The sign of the angular quantity represents the relative direction of the angular quantity with respect to a reference direction.

Adapting to angular vector representation is tedious. For this reason, vector analysis of angular motion is not always emphasized. However, the vector equations showing relationship of various angular and linear quantities are exact in interpretation and are useful. Therefore, vector notations of angular quantities will be attempted only in relation with the final result of any derivation in this module.

It is strongly recommended to remember vector relations among angular quantities. The vector representation provides the most details in the most accurate manner with the least of expression requirement (we shall demonstrate this aspect wherever applicable). In the beginning, vector relations of angular quantities may be slightly difficult to comprehend or interpret, but a bit of insistence resolves to a much improved understanding of the concept. This bargain to study final result in vector form is worth the efforts as we shall soon find out.

Note:

The treatment of the rotation motion of rigid bodies is built upon the concepts of rotational kinematics. Besides, the topics such as magnetic effect of current, torque on rotating coil and particle physics involve angular motion. All these topics are heavily dependent on the directional features of interacting angular vectors and hence vector interpretation of angular equations are very important.

Linear and angular velocity relation in vector form

If we want to write the relation for velocities (as against the one derived for speed, v = ω r), then we need to write the relation as vector cross product :

v = ω x r v = ω x r

The order of quantities in vector product is important. A change in the order of cross product like ( r x ω r x ω ) represents the product vector in opposite direction. The directional relationship between thee vector quantities are shown in the figure. The vectors “v” and “r” are in the plane of “xz” plane, whereas angular velocity ( ω ), is in y-direction.

Figure 6: Directional relation between linear and angular velocity
Linear and angular velocity
 Linear and angular velocity  (am6.gif)

Here, we shall demonstrate the usefulness of vector notation. Let us do a bit of interpretation here to establish the directional relationship among the quantities from the vector notation. It is expected from the equation ( v = ω x r v = ω x r ) that the vector product of angular velocity ( ω ) and radius vector (r) should yield the direction of velocity (v).

Remember that a vector cross product is evaluated by yet another Right Hand Rule (RHR). We move from first vector ( ω ) to the second vector (r) of the vector product in an arc as shown in the figure.

Figure 7: Determining direction of vector cross product
Vector cross product
 Vector cross product  (am7.gif)

We place our right hand such that the curl of fingers follows the direction of arc. The extended thumb, then, represents the direction of cross product (v), which is perpendicular (this fact lets us draw the exact direction) to the vectors or the plane containing two vectors ( ω and r) whose products is being evaluated. In the case of circular motion, vectors ω and r are perpendicular to each other and vector v is perpendicular to the plane defined by vectors ω and r.

Figure 8: Determining direction of vector cross product
Vector cross product
 Vector cross product  (am8.gif)

Thus, we see that the interpretation of cross products completely defines the directions of quantities involved at the expense of developing skill to interpret vector product (we may require to do a bit of practice).

Also, we can evaluate magnitude (speed) as :

v = | v | = ω r sin θ v = | v | = ω r sin θ

where θ is the angle between two vectors ω and r. In the case of circular motion, θ = 900, Hence,

v = | v | = ω r v = | v | = ω r

Thus, we have every detail of directional quantities involved in the equation by remembering vector form of equation.

Uniform circular motion revisited

In the case of the uniform circular motion, the speed (v) of the particle is constant (by definition). This implies that angular velocity (ω = v/r) in uniform circular motion is also constant.

ω = v r = constant ω = v r = constant

Also, the time period of the uniform circular motion is :

T = 2 π r v = 2 π ω T = 2 π r v = 2 π ω

Linear .vs. angular quantity

The description of circular motion is described better in terms of angular quantity than its linear counter part.

The reasons are easy to understand. For example, consider a case of uniform circular motion. Here, the velocity of particle is changing - though the motion is “uniform”. The two concepts do not go together. The general connotation of the term “uniform” indicates “constant”, but the velocity is actually changing.

When we describe the same uniform circular motion in terms of angular velocity, there is no contradiction. The velocity (i.e. angular velocity) is indeed constant. This is the first advantage of describing uniform circular motion in terms of angular velocity.

In other words, the vector manipulation or analysis of linear velocity along the circular path is complicated as its direction is specific to a particular point on the circular path and is basically multi-directional. On the other hand, direction of angular velocity is limited to be bi-directional along the axis of rotation at the most.

Figure 9
Linear and angular velocity
 Linear and angular velocity  (am9.gif)

Second advantage is that angular velocity conveys the physical sense of the rotation of the particle as against linear velocity, which indicates translational motion. Alternatively, angular description emphasizes the distinction between two types of motion (translational and rotational).

Finally, angular quantities allow to write equations of motion as available for translational motion with constant acceleration. For illustration purpose, we can refer to equation of motion connecting initial and final angular velocities for a motion with constant angular acceleration “α” as :

ω 2 = ω 1 + α t ω 2 = ω 1 + α t

We shall study detailed aspect of circular motion under constant angular acceleration in a separate module.

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