We have seen that change in direction is accounted by radial acceleration (centripetal acceleration), which is given by following relation,

a R = v 2 r a R = v 2 r

The change in speed have implications on radial (centripetal) acceleration. There are two possibilities :

1: The radius of circle is constant (like in the motion along a circular rail or motor track)

A change in “v” shall change the magnitude of radial acceleration. This means that the centripetal acceleration is not constant as in the case of uniform circular motion. Greater the speed, greater is the radial acceleration. It can be easily visualized that a particle moving at higher speed will need a greater radial force to change direction and vice-versa, when radius of circular path is constant.

2: The radial (centripetal) force is constant (like a satellite rotating about the earth under the influence of constant force of gravity)

The circular motion adjusts its radius in response to change in speed. This means that the radius of the circular path is variable as against that in the case of uniform circular motion.

In any eventuality, the equation of centripetal acceleration in terms of “speed” and “radius” must be satisfied. The important thing to note here is that though change in speed of the particle affects radial acceleration, but the change in speed is not affected by radial or centripetal force. We need a tangential force to affect the change in the magnitude of a tangential velocity. The corresponding acceleration is called tangential acceleration.

The angular velocity in non-uniform circular motion is not constant as ω = v/r and “v” is varying.

We construct a data set here to have an understanding of what is actually happening to angular speed with the passage of time. Let us consider a non-uniform circular motion of a particle in a centrifuge, whose linear speed, starting with zero, is incremented by 1 m/s at the end of every second. Let the radius of the circle be 10 m.

---------------------------
 t         v           ω
(s)      (m/s)      (rad/s)
---------------------------
 0         0          0.0
 1         1          0.1
 2         2          0.2    
 3         3          0.3
---------------------------

The above data set describes just a simplified situation for the purpose of highlighting variation in angular speed. We can visualize this change in terms of angular velocity vector with increasing magnitudes as shown in the figure here :

Figure 2: Angular velocity changes with time in non-uniform circular motion.

Angular velocity

Note that magnitude of angular velocity i.e. speed changes, but not its direction in the illustrated case. However, it has been pointed out earlier that there are actually two directional possibilities i.e. clockwise and anti-clockwise rotation. Thus, a circular motion may also involve change of direction besides a change in its magnitude.

The non-uniform circular motion involves a change in speed. This change is accounted by the tangential acceleration, which results due to a tangential force and which acts along the direction of velocity circumferentially as shown in the figure. It is easy to realize that tangential velocity and acceleration are tangential to the path of motion and keeps changing their direction as motion progresses.

Figure 3: Velocity, tangential acceleration and tangential force all act along the same direction.

Tangential acceleration

We note that velocity, tangential acceleration and tangential force all act along the same direction. It must, however, be recognized that force (and hence acceleration) may also act in the opposite direction to the velocity. In that case, the speed of the particle will decrease with time.

The magnitude of tangential acceleration is equal to the time rate of change in the speed of the particle.

The magnitude of angular acceleration is the ratio of angular speed and time interval.

If the ratio is evaluated for finite time interval, then the ratio is called average angular acceleration and If the ratio is is evaluated for infinitesimally small period (Δ t →0), then the ratio is called instantaneous angular acceleration. Mathematically, the instantaneous angular acceleration is :

The angular acceleration is measured in “ rad / s 2 rad / s 2 ”. It is important to emphasize here that this angular acceleration is associated with the change in angular speed (ω) i.e. change in the linear speed of the particle (v = ωr) - not associated with the change in the direction of the linear velocity (v). In the case of uniform circular motion, ω = constant, hence angular acceleration is zero.

We can relate angular acceleration (α) with tangential acceleration ( a T a T ) in non – uniform circular motion as :

a T = đ v đ t = đ 2 s đ t 2 = đ 2 đ t 2 ( r θ ) = r đ 2 θ đ t 2 ⇒ a T = α r a T = đ v đ t = đ 2 s đ t 2 = đ 2 đ t 2 ( r θ ) = r đ 2 θ đ t 2 ⇒ a T = α r

We see here that angular acceleration and tangential acceleration are representation of the same aspect of motion, which is related to the change in angular speed or the equivalent linear speed. It is only the difference in the manner in which change of the magnitude of motion is described.

The existence of angular or tangential acceleration indicates the presence of a tangential force on the particle.

Note : All relations between angular quantities and their linear counterparts involve multiplication of angular quantity by the radius of circular path “r” to yield to corresponding linear equivalents. Let us revisit the relations so far arrived to appreciate this aspect of relationship :

We can represent the relation between angular acceleration and tangential acceleration in terms of vector cross product :

a T = α X r a T = α X r

Figure 4: Angular acceleration is an axial vector.

Tangential and angular acceleration

The order of quantities in vector product is important. A change in the order of cross product like ( r X α r X α ) represents the product vector in opposite direction. The directional relationship between thee vector quantities are shown in the figure. The vectors “ a T a T ” and “r” are in “xz” plane i.e. in the plane of motion, whereas angular acceleration (α) is in y-direction i.e. perpendicular to the plane of motion. We can know about tangential acceleration completely by analyzing the right hand side of vector equation. The spatial relationship among the vectors is automatically conveyed by the vector relation.

We can evaluate magnitude of tangential acceleration as :

a T = α X r ⇒ a T = | a T | = α r sin θ a T = α X r ⇒ a T = | a T | = α r sin θ

where θ is the angle between two vectors α and r. In the case of circular motion, θ = 90°, Hence,

a T = | α X r | = α r a T = | α X r | = α r

In the light of new quantities and new relationships, we can attempt analysis of the general circular motion (including both uniform and non-uniform), using vector relations. We have seen that :

v = ω X r v = ω X r

A close scrutiny of the quantities on the right hand of the expression of velocity indicate two possible changes :

The angular velocity ( ω ) can change either in its direction (clockwise or anti-clockwise) or can change in its magnitude. There is no change in the direction of axis of rotation, however, which is fixed. As far as position vector ( r) is concerned, there is no change in its magnitude i.e. | r | or r is constant, but its direction keeps changing with time. So there is only change of direction involved with vector “ r”.

Now differentiating the vector equation, we have

đ v đ t = đ ω đ t X r + ω X đ r đ t đ v đ t = đ ω đ t X r + ω X đ r đ t

We must understand the meaning of each of the acceleration defined by the differentials in the above equation :

⇒ a = α X r + ω X v ⇒ a = α X r + ω X v

⇒ a = a T + a R ⇒ a = a T + a R

where, a T = α X r a T = α X r is tangential acceleration and is measure of the time rate change of the magnitude of the velocity of the particle in the tangential direction and a R = ω X v a R = ω X v is the radial acceleration also known as centripetal acceleration, which is measure of time rate change of the velocity of the particle in radial direction.

Various vector quantities involved in the equation are shown graphically with respect to the plane of motion (xz plane) :

Figure 5

Vector quantities

The magnitude of total acceleration in general circular motion is given by :

Figure 6

Resultant acceleration

a = | a | = ( a T 2 + a R 2 ) a = | a | = ( a T 2 + a R 2 )