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1800-102-2727You might have observed that when you switch ON the fan, it starts from rest and continuously increases its velocity till it comes to the particular speed you chose. This means, similar to the acceleration that occurs in linear motion, rotational and circular motion also has some acceleration. This acceleration is known as angular acceleration. Let’s see more about this here!
Table of contents
It is the rate of change of angular velocity.
We know that,
Where,
_{} is the linear velocity
_{} is the angular velocity
_{} is the position vector of the point under consideration from the axis of rotation
In scalar form we can write,
Differentiating both sides with respect to t,
It is to be noted that, in a non-uniform circular motion, both angular and tangential acceleration exist.
If _{}and _{} are the instantaneous angular velocities at times _{} and _{} , respectively, then the average angular acceleration is,
SI unit of angular acceleration is _{}
It is the instantaneous rate of change of angular velocity with time.
SI unit: _{}
Let _{} be the angular acceleration and _{} be the angular velocity of a particle in the circular motion at any instant of time _{}.
Let _{} be the angular position of the particle at time t and it covers an angular displacement _{} in time .
We know that,
Consider a particle moving in a circular motion.
At any instant _{}, let the velocity of the particle be _{}and the net acceleration be _{}
We know,
But
So,
Thus, total acceleration is the vector sum of tangential acceleration and centripetal acceleration.
Where _{} is the angle made by the resultant acceleration with the centripetal acceleration.
Parameter | UCM | NUCM |
Angular speed | Constant | Variable |
Angular velocity | Constant | Variable |
Angular acceleration | Zero | Non-zero |
Tangential acceleration | Zero | Non-zero |
Centripetal acceleration | Non-zero | Non-zero |
Parameter | Linear kinematics | Angular kinematics |
Position/Angle | Position (_{$\overrightarrow{X}$}) | Angle (_{$\overrightarrow{\theta}$}) |
Velocity | $\overrightarrow{v}=\frac{d\overrightarrow{x}}{dt}$ | $\overrightarrow{\omega}=\frac{d\overrightarrow{\theta}}{dt}$ |
Acceleration | $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=\frac{{d}^{2}\overrightarrow{x}}{d{t}^{2}}$ | $\overrightarrow{\alpha}=\frac{d\overrightarrow{\omega}}{dt}=\frac{{d}^{2}\overrightarrow{\theta}}{d{t}^{2}}$ |
For applying equations of motion in non-uniform circular motion, we can do it only for a constant angular and tangential acceleration. _{} _{}
The equations of motion in angular kinematics are:
In scalar and simple form,
Angular kinematics | Linear kinematics |
$\omega ={\omega}_{o}+\alpha t$ | $v=u+at$ |
$\Delta \theta ={\omega}_{o}t+\frac{1}{2}\alpha {t}^{2}$ | $s=ut+\frac{1}{2}a{t}^{2}$ |
${\omega}^{2}={{\omega}_{o}}^{2}+2\alpha \Delta \theta $ | ${v}^{2}={u}^{2}+2as$ |
Q1. A particle moves in a circle of radius of _{} at a speed that increases uniformly. Find the angular acceleration of the particle, if its speed changes from _{} to _{} in _{}.
Ans. Given,
Radius = _{}
Final speed =
Initial speed = _{}
Time taken = 4 s
We know that,
Also tangential acceleration, _{} (where _{} Angular acceleration)
Q2. A pulley wheel of diameter 8 cm has a 5m long cord wrapped around its periphery. Starting from rest, the wheel is given an angular acceleration of_{}.
Ans. Given,
Radius of wheel (_{}) = _{}
Initial angular velocity (_{}) _{}
Angular acceleration (_{})_{}
This is a case of constant angular acceleration, so we can use the equations of motion.
(a) Since the cord is to unwind completely, the equivalent linear displacement of the wheel is equal to the length of the string.
Now, angular displacement of wheel, _{}
(b) For the time taken, we use the equation,
_{} (we can neglect negative value of t)
Q3. A solid body rotates about a stationary axis with an angular retardation _{} , where 𝜔 is the angular velocity of the body. Find the time after which the body comes to rest if at _{}, the angular velocity of body was _{}.
Ans. Given,
Initial angular velocity = _{}
Angular retardation, _{}
Final angular velocity (_{}) = 0
We need to find time (t) after which the body comes to rest.
We know that,
_{} (Negative sign due to retardation)
Integrating both sides with limits,
Q4. A solid body rotates about a stationary axis with an angular retardation _{} , where _{} is the angular velocity of the body at an instant. Find the average angular speed of the body, averaged over the whole time of rotation if at the initial moment of time, its angular velocity was equal to _{}.
Ans. Given,
Angular retardation, _{} where _{} is the angular velocity
Initial angular velocity = _{}
We have,
Average angular velocity, _{}
Now,
Integrating both sides with limits,
Also angular acceleration, _{}
Integrating both sides with limits,
Substituting _{} and _{} equation in _{} equation,
FAQs
Q1. What are the dimensions of angular acceleration?
Ans. The angular acceleration is the rate of change of angular velocity with time.
So it is the double derivative with respect to time over the angular displacement. We know that the angle expressed in terms of radians has no dimension as it is a ratio of arc length to the length of radius. So the dimension of angular acceleration is _{}.
Q2. What happens to the angular acceleration, if angular velocity becomes constant?
Ans. The angular acceleration becomes zero when angular velocity remains constant. Since _{} , if the angular velocity remains constant, the rate of change will become zero.
Q3. What causes constant angular acceleration?
Ans. Constant angular acceleration is achieved in a uniform circular motion. This happens when there is no torque acting on the body under consideration. We know, torque _{}.
So, angular acceleration becomes zero only when the torque acting is also zero and thus the constant angular acceleration is achieved.
Q4. Why does angular acceleration not change with radius?
Ans. Angular acceleration is the rate of change of angular velocity with time. Angular velocity is the rate of change of angular displacement with time. Angular displacement is measured as the angle subtended by the radius. It does not change with the radial distance from the axis. So the angular velocity and angular acceleration also doesn't change with the radial distance from the axis.