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Angular acceleration indicates the time rate of change of angular velocity and is usually denoted by α and is expressed in radians per second. Moreover, the angular acceleration is constant and does not depend on the time variable as it varies linearly with time. Angular Acceleration is also called Rotational Acceleration.
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Key Terms: Angular Acceleration, Angular Velocity, Speed, Displacement, Rotational Motion, Circular Motion, Average Angular Acceleration
What is Angular Velocity?
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Angular velocity is a vector quantity that is defined as the rate of change in displacement of a body that is revolving around a fixed point in a circular fashion in a given time period. Angular velocity is denoted by the Greek alphabet omega, i.e. ω.
Hence, it is calculated by the formula:
| ω = dθ / dt |
Where,
- ω = angular velocity
- θ = angle
- t = time
- dθ = change of angle
- dt = change in time
Solved Example on Angular VelocityQues. A tyre of radius 20 inches rotates at eight revolutions per second. Determine its angular velocity. (2 marks) Ans. In order to determine the Angular velocity, we have to consider the equation, ⇒ \(\omega = {{\theta} \over t}\) Now, since the tyre has completed eight revolutions per second, we have to multiply it by 2 because a full rotation of 360 degrees is equal to 2. Replacing the values, we get, ⇒\(\omega = {{16 \pi radians} \over 1s}\) = 16 rads-1 Thus, its angular velocity is 16 rads-1. |
Units of Angular Velocity
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The SI unit of angle is radian and the SI unit of time is second, hence we see that the SI unit of angular velocity is radian/second or rad/s. Angular velocity can also be calculated through linear velocity if we know the radius of the circle around which the object is spinning, with the formula:
| ω = v / r |
Where,
- ω = angular velocity
- v = linear velocity
- r = radius of the circle
What is Angular Acceleration?
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Angular acceleration is derived from angular velocity and is defined as the rate of change of angular velocity over a time period. Angular acceleration can also be called rotational acceleration. Angular acceleration is denoted by the Greek alphabet alpha, i.e. α.
This means that angular acceleration can be calculated using the following formula:
| α = dω / dt |
Where,
- α = Angular acceleration
- ω = Angular velocity
- t = Time
- dω = Change in angular velocity
- dt = Change in time
Since we know that ω = dθ/dt, by putting it into the above formula we get another formula for angular acceleration: α = d2θ / dt2
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Units of Angular Acceleration
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Since we already know that the SI unit of angular velocity is radian per second (rad/s) and that the SI unit of time is second(s), this means using the above formula (α = d2θ / dt2), we know that the SI unit of angular acceleration is radian per square second, which is denoted as rad/s2.
It is similar to linear acceleration, which we already know. It is basically just linear acceleration at a given angle.
Solved Examples Related to Angular AccelerationQues. State the occurrence of angular acceleration when the angular velocity is constant. (1 mark)
Ans. The correct answer is a, Zero. Ques. What relationship do angular acceleration and angular velocity have? (1 mark) Ans. Angular acceleration “α” is can be expressed as the rate of change of angular velocity. This is how both angular acceleration and angular velocity are related. |
Read More: Kinematics of Rotational Motion around a Fixed Axis
Formula of Angular Acceleration
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The formula that we used earlier is the one we use in case the change in angular velocity is constant. However, in case the change in angular velocity is not constant and is itself changing from time to time, then a new formula is used. It is often called average angular acceleration.
This formula is as follows:
| αavg = ω2 – ω1 / d2 – d1 |
Where,
- αavg = average angular acceleration
- ω1 = initial angular velocity
- ω2 = terminal angular velocity
- d1 = initial time
- d2 = terminal time
Since angular acceleration is a vector, it means that it has a direction. To find the direction of the angular velocity we must see in which direction the object is rotating. If the object is rotating in a clockwise direction, then the direction of the angular velocity is moving away from you.
However, if the object is rotating or moving in a counter-clockwise direction, then the direction of the angular velocity is moving towards you.
Read More: Derivation of Kinetic Energy
Formula of Average Angular Velocity
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Average angular velocity refers to the ratio of change in angular coordinate to change in time. The formula of average angular velocity is:
| ∏avg = ∏θ /Δt = (θ2 - θ1)/ (t2 - t1) |
Where,
- ∏avg = the average angular velocity in radian s-1
- Δθ = change in angular coordinate in radians
- θ1 = the initial angular coordinate in radians
- θ2 = the final angular coordinate in radians
- Δt = the change in time in s
- t1 = the initial time in s
- t2 = the final time in s
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Previous Year Questions
- The moment of inertia of a uniform cylinder of length … [JEE Mains 2017]
- A circular disc D1 of mass M and radius R has two identical discs … [JEE Mains 2019]
- A long cylindrical vessel is half filled with a liquid … [JEE Mains 2019]
- A metal coin of mass 5g and radius 1cm is fixed … [JEE Mains 2019]
- A particle of mass m is fixed to one end of a light spring … [JEE Mains 2020]
- A rectangular solid box of length … [JEE Mains 2019]
- A rigid massless rod of length … [JEE Mains 2019]
- A roller is made by joining together … [JEE Mains 2016]
- A slender uniform rod of mass … [JEE Mains 2017]
- A solid sphere of mass M and radius R … [JEE Mains 2019]
- A string is wound around a hollow cylinder of mass … [JEE Mains 2019]
- A tennis ball (treated as hollow spherical shell) … [JEE Mains 2013]
- A thin bar of length L has a mass per unit length … [JEE Mains 2014]
- A thin circular plate of mass M and radius R … [JEE Mains 2019]
- A thin disc of mass MM and radius RR has mass per … [JEE Mains 2019]
- A uniform rectangular thin sheet … [JEE Mains 2019]
- A uniform solid cylindrical roller of mass … [JEE Mains 2015]
- A thin ring of mass 2 kg and radius 0.5 m is rolling without slipping on a … [JEE Advanced 2011]
- The total torque about pivot A provided by the forces … [AMUEEE 2012]
- A wheel turning with angular speed of … [AMUEEE 2012]
- A particle moves in a circular path with decreasing speed … [JEE Advanced 2005]
- A sphere is rolling without slipping on a fixed horizontal plane surface … [JEE Advanced 2009]
- A block of base l0cm×10cm and height … [JEE Advanced 2009]
- Two point masses of 0.3 kg and 0.7 kg … [JEE Advanced 1995]
- Two solid cylinders P and Q of same mass and same radius … [JEE Advanced 2012]
Things to Remember
- Angular acceleration implies the time rate of change of angular velocity.
- Angular velocity is denoted by the Greek alphabet omega, i.e. ω.
- The SI unit of angle is radian and the SI unit of time is second, hence we see that the SI unit of angular velocity is radian/second or rad/s.
- Angular acceleration can also be called rotational acceleration.
- Angular acceleration is denoted by the Greek alphabet alpha, i.e. α.
- Average angular velocity refers to the ratio of change in angular coordinate to change in time.
- SI unit of angular acceleration is radian per square second, which is denoted as rad/s2.
Sample questions
Ques: Find out the components along the x, y, z-axes of the angular momentum I of a particle whose position vector is r with components x, y, z and momentum is p with components px, py and pz. Show that the particle moves only in the x-y plane and the angular momentum has only a z-component. (5 marks)
Ans: As per the question,
Ques: A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. Find out the angular acceleration of the cylinder if the rope is pulled with a force of 30 N. What will be the linear acceleration of the rope while assuming that there is no slipping. (3 marks)
Ans: Given,
M = 3 kg, R = 40 m = 0.4 m
Moment of inertia of the hollow cylinder about its axis.
I = MR2 = 3 (0.4)2 = 0.48 kg m2
Ques: From a uniform disk of radius R, a circular hole of radius R/2 is cut out and the centre of the hole is at R/2 from the centre of the original disc. Find the centre of gravity of the resulting flat body. (3 marks)
The centre of gravity of the resulting flat body
Ans: Let us assume that a bigger uniform disc of radius R and with centre O is a smaller circular hole of radius R/2 with its centre at O1 which is cut out. Let the centre of gravity or the centre of mass of the resulting flat body be at O2, where OO2 = x. If σ be mass per unit area, then mass of the whole disc is M1 = πR2 σ and the mass of the cut out part is:
Ques: What is angular acceleration? What is the SI unit of angular acceleration? (2 marks)
Ans: Angular acceleration is derived from angular velocity and is defined as the rate of change of angular velocity over a time period. Angular acceleration can also be called rotational acceleration. Angular acceleration is denoted by the Greek alphabet alpha, i.e. α.
SI unit of angular acceleration is radian per square second, which is denoted as rad/s2.
It is similar to linear acceleration, which we already know. It is basically just linear acceleration at a given angle.
Ques: a) A child stands at the centre of a turntable with his arms outstretched. The table is set rotating with an angular speed of 40 rev/min. Find the angular speed of the child if he folds his hands back and reduces his moment of inertia to ? times the initial value. Assume that the turntable rotates with friction. b) show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How will you account for such an increase in kinetic energy? (3 marks)
Ans: Let us assume the initial inertia of the child be I1, then the final moment of inertia will be:
Ques: An ant has been sitting on the edge of a rotating circular disc. Assuming that its angular velocity changes at the rate of 60 rad/s for 10 seconds, determine its angular acceleration. (2 marks)
Ans: As per the given question, the change in angular velocity equals, dω = 60 rad/s.
Time taken for change equals, t = 10s.
Hence, by substituting the above values,
⇒ \(\alpha = {{d \omega} \over dt} = {{30} \over 5} =\) 6 rad/s2
Ques: What is Angular Velocity? (1 mark)
Ans: Angular velocity is a vector quantity which is expressed as the rate of change in displacement of a body that is revolving around a fixed point in a circular fashion in a given time period.
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