Rotational motion

Rotational motion is the motion of an object around its own axis.

1. Rotational Motion 1 Rotational Motion- By Aditya Abeysinghe

2. Definitions of some special terms Angular position ( Φ ) - The angular position of a particle is the angle ɸ made between the line connecting the particle to the original and the positive direction of the x-axis, measured in a counterclockwise direction 2. Angular displacement ( θ ) - The radian value of the angle displaced by an object on the center of its path in circular motion from the initial position to the final position is called the angular displacement. 2 Rotational Motion- By Aditya Abeysinghe ɸ = l / r

3. 3. Angular Velocity ( ω ) - Angular velocity of an object in circular motion is the rate of change of angular displacement 3 Rotational Motion- By Aditya Abeysinghe θ = θ f - θ i ω = θ / t Unit - rads -1 Vector direction by Right hand rule

4. 4. Angular acceleration- Angular acceleration of an object in circular motion is the rate of change of angular velocity 4 Rotational Motion- By Aditya Abeysinghe θ ω ω r t = 0 t = t α = ( ω – ω ) / t Unit- rads -2 Direction- By right hand rule

5. Angular equations of movement α = ( ω – ω ) / t ω = ω + α t ( ω + ω )/2 = θ / t θ = ( ω + ω )t/2 ( ω + ω )/2 = θ / t , ω t = 2 θ + ω t , ( ω + α t)t = 2 θ + ω t θ = ω t + ½ α t 2 θ = ( ω + ω ) t/2, θ = ( ω + ω )( ω - ω ) 2 α 5 Rotational Motion- By Aditya Abeysinghe ω 2 = ω 2 + 2 αθ

6. Therefore the four equations of angular movement are- ω = ω + α t θ = ( ω + ω )t/2 θ = ω t + ½ α t 2 ω 2 = ω 2 + 2 αθ It should be noted that these four equations are analogous to the four linear equations of motion: V = U + at S = (V + U)t/2 S = Ut + ½ at 2 V 2 = U 2 + 2as 6 Rotational Motion- By Aditya Abeysinghe

7. Right hand rule Take your right hand and curl your fingers along the direction of the rotation. Your thumb directs along the specific vector you need. ( angular velocity, angular acceleration, angular momentum etc.) 7 Rotational Motion- By Aditya Abeysinghe Direction of rotation Axis of rotation Thus, right hand rule is used whenever, in rotational motion, to measure the direction of a particular vector

8. Relationship between physical quantities measured in angular motion and that in linear motion Linear displacement- Angular displacement Linear velocity- Angular velocity S / t = r θ / t V = r ω 8 Rotational Motion- By Aditya Abeysinghe θ r S S= (2 π r / 2 π ) × θ = r θ S = r θ

9. 3 . Linear acceleration- Angular acceleration α = ( ω – ω ) / t α r = ( ω – ω )r / t α r = ( ω r – ω r) / t α r = (V - V ) / t α r = a a = r α 9 Rotational Motion- By Aditya Abeysinghe Displacement Velocity Acceleratiom Translational motion S V a Rotational motion θ ω α Relationship S = r θ V = r ω a = r α

10. The period and the frequency of an object in rotation motion Period (T) is the time taken by an object in rotational motion to complete one complete circle. Frequency (f) is the no. of cycles an object rotates around its axis of rotation Thus, f = 1/ T . However, ω = θ / t Therefore, ω = 2 π / T Therefore, ω = 2 π / (1/f) Thus, ω = 2 π f 10 Rotational Motion- By Aditya Abeysinghe

11. Moment of Inertia Unlike in the case of linear movement’s inertia (reluctance to move or stop) , inertia of circular/rotational motion depends both upon the mass of the object and the distribution of mass (how the mass is spread across the object) Moment of Inertia of a single object- I = mr 2 Moment of Inertia is a scalar quantity 11 Rotational Motion- By Aditya Abeysinghe m r Axis of rotation Path of the object

12. Radius of gyration Suppose a body of mass M has moment of Inertia I about an axis. The radius of gyration, k, of the body about the axis is defined as I= Mk 2 That is k is the distance of a point mass M from the axis of rotation such that this point mass has the same moment of inertia about the axis as the given body. 12 Rotational Motion- By Aditya Abeysinghe

13. Moment of Inertia of some common shapes Rotational Motion- By Aditya Abeysinghe 13 Body Axis Figure I k Ring (RadiusR ) Perpendicular to the plane at the center MR 2 R Disc (Radius R) Perpendicular to the plane at the center ½ MR 2 R / √2 Solid Cylinder (Radius R) Axis of cylinder ½ MR 2 R / √2 Solid Sphere (Radius R) Diameter ⅖ MR 2 R√(⅖)

14. Angular Momentum Angular momentum is the product of the moment of inertia and the angular velocity of the object. 14 Rotational Motion- By Aditya Abeysinghe θ r P y x z L = r sin θ × P L = r P Sin θ However, P= mv , ( Linear moment= mass × linear velocity ) Therefore , L= m r v Sin θ = m r ( ω r ) Sin θ (as V= r ω ) = mr 2 ω Sin θ Therefore, L = I ω Sin θ (as I = mr 2 ) L= I ω Sin θ

15. But in most cases the radius of rotation is perpendicular to the momentum. Thus, θ = 90° , L = I ω 15 Rotational Motion- By Aditya Abeysinghe P r Axis of rotation

16. Torque The rate of change of angular momentum of an object in rotational motion is proportional to the external unbalanced torque. The direction of the torque also lies in the direction of the angular momentum. Torque is called the moment of force and is a measure of the turning effect of the force about a given axis. A torque is needed to rotate an object at rest or to change the rotational mode of an object. 16 Rotational Motion- By Aditya Abeysinghe

17. 17 Rotational Motion- By Aditya Abeysinghe τ = (I ω – I ω )/t τ = I [( ω - ω )/ t] Therefore, τ = I α By Newton’s second law of motion F = ma Fr = mra = mr (r α ) ( as a=r α ) Fr= mr 2 α Fr = I α ( as I= mr 2 ) However, from the above derivation, I α = τ . Therefore, Fr = τ Thus , τ = Fr

18. This theory can be expressed as: 18 Rotational Motion- By Aditya Abeysinghe θ d r F y z x τ = Fd τ = Fr Sin θ O τ = F × r

19. Relationship between Torque and Angular Momentum- τ = dL / dt τ – Net torque L- Angular momentum This result is the rotational analogue of Newton’s second law: F = dP / dt 19 Rotational Motion- By Aditya Abeysinghe

20. Applying Newton’s laws to rotational motion Newton’s First law and equilibrium- If the net torque acting on a rigid object is zero, it will rotate with a constant angular velocity. Concept of equilibrium- 20 Rotational Motion- By Aditya Abeysinghe M m d 3d If the system is at equilibrium, total torque around O should be zero O Therefore, τ M + τ m = 0 Mgd + [ -(mg(3d))] = 0 Therefore, m = M/3

21. 2 . Newton’s second law of motion The rate of change of angular momentum of an object in rotational motion is proportional to the external unbalanced torque. The direction of the torque also lies in the direction of the angular momentum . τ α α τ = I α This is analogous to the linear equivalent, which states, The rate of change of momentum is directly proportional to the external unbalanced force applied on an object and that force lies in the direction of the net momentum. Thus, torque could be treated as the rotational analogue of the force applied on an object. 21 Rotational Motion- By Aditya Abeysinghe

22. Theorem of Parallel Axes Let I CM be the moment of inertia of a body of mass M about an axis passing through the center of mass and let I be the moment of inertia about a parallel axis at a distance d from the first axis. Then, I = I CM + Md 2 22 Rotational Motion- By Aditya Abeysinghe Thus the minimum moment of inertia for any object is at the center of mass, as x in the above expression is zero.

23. Theorem of perpendicular axis The moment of inertia of a body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two mutually perpendicular axis in its own plane and crossing through the point through which the perpendicular axis passes 23 Rotational Motion- By Aditya Abeysinghe I z = I x + I y

24. Rotational Kinetic Energy E k = Σ ½ m i v i 2 But , v i = r i ω So , E k = Σ ½ m i r i 2 ω 2 = ½ ( Σ m i r i 2 ) ω 2 Therefore , E k = ½ I ω 2 24 Rotational Motion- By Aditya Abeysinghe r 1 m 1 r 2 m 2 r n m n ω E k = ½ I ω 2 Rotational kinetic energy is the rotational analogue of the translational kinetic energy, which is E K = ½ mv 2 . In fact, rotational kinetic energy equation can be deduced by substituting v =r ω and viceversa.

25. Law of Conservation of Angular Momentum Law- If the resultant external torque on a system is zero, its total angular momentum remains constant. That is, if τ = 0, dL / dt = 0 , which means that L is a constant This is the rotational analogue of the law of conservation of linear momentum. 25 Rotational Motion- By Aditya Abeysinghe

26. Work done by a torque dW = τ d θ Therefore, the total work done in rotating the body from an angular displacement of θ 1 to an angle displacement θ 2 is W= ∫ τ d θ W = τ ∫ 1. d θ Therefore, W = τ [ θ 2 - θ 1 ] 26 Rotational Motion- By Aditya Abeysinghe θ 1 θ 2 θ 1 θ 2

27. Power The rate at which work is done by a torque is called Power P = dw/dt = τ d θ /dt = τ ω Therefore, P = τ ω 27 Rotational Motion- By Aditya Abeysinghe

28. Work - Energy Principle From ω 2 = ω 2 + 2 αθ and W= τθ , it is clear that the work done by the net torque is equal to the change in rotational kinetic energy. τ = I α and ω 2 = ω 2 + 2 αθ . Therefore, ω 2 = ω 2 + 2( τ /I) θ . Thus, ω 2 = ω 2 + 2[( τθ )/I] . Thus, ω 2 = ω 2 + 2W/I OR W= ½ I ( ω 2 - ω 2 ). This is called the work-energy principle. 28 Rotational Motion- By Aditya Abeysinghe

29. Relationship between Angular momentum and Angular velocity τ = I α = I d ω /dt = d(I ω )/dt But, τ = dL/dt Thus, dL/dt = d(I ω )/ dt By integrating both sides, It can be shown that , L = I ω This is the rotational analogue of P = mv 29 Rotational Motion- By Aditya Abeysinghe

30. Rolling Body Rolling is a combination of rotational and transitional(linear) motions. Suppose a sphere is rolling on a plane surface, the velocity distribution can be expressed as, 30 Rotational Motion- By Aditya Abeysinghe

31. V 2V V V √2 V = 0 This two systems (rotational and transitional) can be combined together to understand how actually the sphere above moves in the plane. The final distribution shows clearly that in reality the ball always instantly experiences a zero velocity at the point of contact with the surface and the maximum velocity is at the top 31 Rotational Motion- By Aditya Abeysinghe

32. Thus, E total = E transitional + E rotational E = ½ mv 2 + ½ I ω 2 E = ½ mR 2 ω 2 + ½ mk 2 ω 2 (Where k- radius of gyration) E = ½m ω 2 (R 2 + k 2 ) 32 Rotational Motion- By Aditya Abeysinghe

33. A body rolling down an inclined plane From the conservation of energy, Mgh = ½mv 2 + ½I ω 2 33 Rotational Motion- By Aditya Abeysinghe θ h s mg

34. Equilibrium of a rigid body Transitional Equilibrium For a body to be in transitional equilibrium, the vector sum of all the external forces on the body must be zero. F ext = M a cm and a cm must be zero for transitional equilibrium. 2 . Rotational Equilibrium For a body to be in rotational equilibrium, the vector sum of all the external torques on the body about any axis must be zero. τ ext = I α and α must be zero for rotational equilibrium. 34 Rotational Motion- By Aditya Abeysinghe

35. Summarizing it up!! 35 Rotational Motion- By Aditya Abeysinghe Term Definition Angular position The angular position of a particle is the angle ɸ made between the line connecting the particle to the original and the positive direction of the x-axis, measured in a counterclockwise direction Angular displacement The radian value of the angle displaced by an object on the center of its path in circular motion from the initial position to the final position is called the angular displacement. Angular velocity Angular velocity of an object in circular motion is the rate of change of angular displacement Angular acceleration Angular acceleration of an object in circular motion is the rate of change of angular velocity Angular equations of motion ω = ω + α t , θ = ( ω + ω )t/2 , θ = ω t + ½ α t 2 , ω 2 = ω 2 + 2 αθ

36. Rotational Motion- By Aditya Abeysinghe 36 Right hand rule Take your right hand and curl your fingers along the direction of the rotation.Your thumb directs along the specific vector you need Relationship between linear and angular qualities S = rθ , V = rω , a = rα (s- translational displacement , v- translational velocity, a-translational acceleration ) Period and frequency Period (T) is the time taken by an object in circular motion to complete one complete circle. Frequency (f) is the no. of cycles an object rotates around its axis of rotation Thus, f = 1/ T Moment of inertia Inertia of circular/rotational motion depends both upon the mass of the object and the distribution of mass. Radius of gyration The radius of gyration, k, of the body about the axis is defined as I= Mk 2 Angular momentum Angular momentum is the product of the moment of inertia and the angular velocity of the object.

37. Torque Torque is called the moment of force and is a measure of the turning effect of the force about a given axis. Theorem of parallel axes Let I CM be the moment of inertia of a body of mass M about an axis passing through the center of mass and let I be the moment of inertia about a parallel axis at a distance d from the first axis. Then, I = I CM + Md 2 Theorem of perpendicular axes The moment of inertia of a body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two mutually perpendicular axis in its own plane and crossing through the point through which the perpendicular axis passes I z = I x + I y Rotational kinetic energy E k = ½ I ω 2 Law of conservation of angular momentum If the resultant external torque on a system is zero, its total angular momentum remains constant. Work done W = τ [θ 2 - θ 1 ]

38. Power The rate at which work is done by a torque is called Power P = τ ω Work-Energy principle W= ½ I (ω 2 - ω 2 ). This is called the work-energy principle. Rolling body Rolling is a combination of rotational and transitional(linear) motions. Equilibrium of a rigid body 1. Transitional Equilibrium For a body to be in transitional equilibrium, the vector sum of all the external forces on the body must be zero. F ext = Ma cm and a cm must be zero for transitional equilibrium. 2. Rotational Equilibrium For a body to be in rotational equilibrium, the vector sum of all the external torques on the body about any axis must be zero. τ = Iα and α must be zero for rotational equilibrium.