Rotational motion

1. Rotational motion Chapter 9

2. Rigid objects • A rigid object has a perfectly definite and unchanging shape and size. • In this class, we will approximate everything as a rigid object

3. Radians • In describing rotational motion, we will use angles in radians, not degrees.

4. q in radians

5. q in radians • An angle in radians is the ratio of two lengths, so it has no units. • We will often write “rad” as the units on such an angle to make it clear that it’s not in degrees • But in calculations, “rad” doesn’t factor into unit analysis.

6. Angular velocity • Rate of change of q • w (omega) is the symbol for angular velocity

7. Angular velocity • At any instant, all points on a rigid object have the same angular velocity. • The units of angular velocity are rad/s. • Sometimes angular velocity is given in rev/s or rpm. • 1 rev is 2p radians • Angular speed is the magnitude of angular velocity

8. Angular acceleration • Rate of change of angular velocity • a (alpha) is the symbol for angular acceleration

9. Angular acceleration • The units for angular acceleration are rad/s2.

10. x is linear position v is linear velocity a is linear acceleration q is angular position w is angular velocity a is angular acceleration Comparison

11. Rotation with constant angular acceleration

12. Example • A CD rotates from rest to 500 rev/min in 5.5 s. • What is its angular acceleration, assuming it is constant? • How many revolutions does the disk make in 5.5 s? • 9.52 rad/s 22.9 rev

13. Relating linear and angular kinematics • We might want to know the linear speed and acceleration of a point on a rotating rigid object. • So we need relationships between • v and w • a and a

14. Speed relationship • Note: these are speeds, not velocities

15. Acceleration relationship Change in direction

16. Example • Find the required angular speed, in rev/min, of an ultracentrifuge for the radial acceleration of a point 2.50 cm from the axis to equal 400,000 times the acceleration due to gravity. • 1.25 x 104 rad/s = 1.19 x 105 rev/min

17. Moment of Inertia • Rotating objects have inertia, but is more than just their mass. • It depends on how that mass is distributed.

18. Moment of inertia • The moment of inertia, I, of an object is found by taking the sum of the mass of each particle in the object times the square of it’s perpendicular distance from the axis of rotation.

19. Moment of Inertia • For continuous distributions of particles, i.e. large objects, the sum becomes an integral. • The moments of inertia for several familiar shapes with uniform densities are given on page 215 of your book. • Moments of inertia are given in terms of masses and dimensions.

20. Kinetic energy of rotating objects

21. Gravitational potential energy of rotating objects • Same as for other objects, but use total mass and position of the center of mass.

22. Example • A uniform thin rod of length L and mass M, pivoted at one end, is held horizontal and then released from rest. Assuming the pivot is frictionless, find • The angular velocity of the rod when it reaches its vertical position • Sqrt(3g/L)

23. On your own • An airplane propeller (I=(1/12)ML2) is 2.08 m in length (from tip to tip) with mass 117 kg. The propeller is rotating at 2400 rev/min about an axis through it’s center. • What is its rotational kinetic energy? • If it were not rotating, how far would it have to drop in free fall to acquire the same kinetic energy? • 1.33 x 106 J 1.16 km

24. Torque • The measure of the tendency of a force to change the rotational motion of a object. • Torque depends on the perpendicular distance between the force and the axis of rotation

25. Torque magnitude • Where t (tau) [your book uses G (gamma)] is the magnitude of the torque • Also called moment • F is the magnitude of the force • l is the perpendicular distance between the force and the axis of rotation. • Also called lever arm or moment arm

26. Torque magnitude

27. Torque magnitude

28. Torque Magnitude

29. + Torque sign • Counterclockwise rotation is caused by positive torques and clockwise rotation is caused by negative torques. • We can use this symbol to indicate which direction is positive torque.

30. Torque Units • The SI-unit of torque is the Newton-meter. • Torque is not work or energy, so it should not be expressed as Joules.

31. Torque Vector Direction

32. Visual aid for torque direction • Torque is perpendicular to both r and F. • Think of a normal, right-handed screw. • The torque vector points in the direction the screw moves.

33. Discussion Question • Why are doorknobs located far from the hinges?

34. Example • Forces F1 = 8.60 N and F2 = 2.40 N are applied tangentially to a wheel with a radius of 1.50 m, as shown on the next slide. What is the net torque on the wheel if it rotates on an axis perpendicular to the wheel and passing through its center?

35. F1 F2

36. F q = 60° 2 m O You try • Calculate the torque (magnitude and direction) about point O due to the force shown below. The bar has a length of 4.00 m and the force is 30.0 N.

37. Torque and angular acceleration • Only valid for rigid objects • a must be in rad/s2 for units to work

38. Example • A torque of 32.0 N-m on a certain wheel causes an angular acceleration of 25.0 rad/s2. What is the wheel’s moment of inertia?

39. On your own • A solid sphere has a radius of 1.90 m. An applied torque of 960 N-m gives the sphere an angular acceleration of 6.20 rad/s2 about an axis through its center. Find • The moment of inertia of the sphere • The mass of the sphere

40. Example • An object of mass m is tied to a light string wound around a wheel that has a moment of inertia I and radius R. The wheel is frictionless, and the string does not slip on the rim. Find the tension in the string and the acceleration of the object. • T=(I/(I+mR2)*mg a=(mR2/(I+mR2))g

41. On your own a

42. On your own • Two blocks are connected by a string that passes over a pulley of radius R and moment of inertia I. The block of mass m1 slides on a frictionless, horizontal surface; the block of mass m2 is suspended from the string. Find the acceleration a of the blocks and the tensions T1 and T2 assuming that the string does not slip on the pulley. • a=(m2/(m1+m2+I/R2))m2g T1=(m1/(m1+m2+I/R2))m2g T2=((m1+I/R2)/(m1+m2+I/R2))m2g

43. Rigid object rotation about a moving axis • Combined translation and rotation. • Translation of center of mass • Rotation about the center of mass • There is friction, but only static friction to keep the object from slipping

44. Kinetic Energy • The kinetic energy is the sum of translational and rotational kinetic energies.

45. Rolling without slipping • When something is rolling without slipping,

46. On your own • A hollow cylindrical shell with mass M and radius R rolls without slipping with speed V on a flat surface. What is its kinetic energy? • MV2

47. Example • A solid disk and a hoop with the same mass and radius roll down an incline of height h without slipping. • Which one reaches the bottom first? • The disk • What if they had different masses? • Different radii?

48. Dynamics of translating and rotating objects • We can use both Newton’s 2nd law and its rotational counterpart

49. Example • A uniform solid ball of mass m and radius R rolls without slipping down a plane inclined at an angle q. A frictional force f is exerted on the ball by the incline. Find the acceleration of the center of mass. • (5/7) gsinq

50. Work and Power • Work done by a constant torque