2. Chapter 9 Rotational Motion Rotational Motion

3. Rotational Motion Many interesting physical phenomena are not “linear” Many interesting physical phenomena are not “linear” Ball at the end of a string revolving Ball at the end of a string revolving Planets around Sun Planets around Sun Moon around Earth Moon around Earth New Physical Quantities: Angular Displacement Angular velocity Angular acceleration

4. The Radian The radian is a unit of angular measure The radian is a unit of angular measure The radian can be defined as the arc length s along a circle divided by the radius r The radian can be defined as the arc length s along a circle divided by the radius r 57.3°

5. More About Radians Whole circle Whole circle Comparing degrees and radians Comparing degrees and radians Converting from degrees to radians Converting from degrees to radians

6. Example Express it in radians, revolutions.

7. Angular Displacement Axis of rotation is the center of the disk Axis of rotation is the center of the disk Need a fixed reference line Need a fixed reference line During time t, the line on rotating object moves through angle θ During time t, the line on rotating object moves through angle θ

8. Angular Displacement, cont. The angular displacement is defined as the angle the object rotates through during some time interval The angular displacement is defined as the angle the object rotates through during some time interval The unit of angular displacement is the radian The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement Each point on the object undergoes the same angular displacement

9. Average Angular Velocity The average angular velocity, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval The average angular velocity, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

10. Angular velocity, cont. The instantaneous angular velocity The instantaneous angular velocity Units of angular velocity are radians/sec Units of angular velocity are radians/sec rad/srad/s Velocity will be positive if θ is increasing (counterclockwise) Velocity will be positive if θ is increasing (counterclockwise) Velocity will be negative if θ is decreasing (clockwise) Velocity will be negative if θ is decreasing (clockwise)

11. Average Angular Acceleration The average angular acceleration of an object is defined as the ratio of the change in the angular velocity to the time it takes for the object to undergo the change: The average angular acceleration of an object is defined as the ratio of the change in the angular velocity to the time it takes for the object to undergo the change:

12. Angular Acceleration, cont Units of angular acceleration are rad/s² Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular velocity and the same angular acceleration When a rigid object rotates about a fixed axis, every portion of the object has the same angular velocity and the same angular acceleration Angular speed Angular speed

13. Angular Acceleration, final The sign of the acceleration does not have to be the same as the sign of the angular velocity The sign of the acceleration does not have to be the same as the sign of the angular velocity The instantaneous angular acceleration The instantaneous angular acceleration Only consider rotations with uniform angular acceleration Only consider rotations with uniform angular acceleration

14. Example A bicycle wheel makes 50 revolutions in 0.5 min. Compute the angular speed in rev/s, rad/s, deg/s.

15. Example The bicycle wheel traveling at 1.67 rev/s decelerates uniformly to a stop in 3 seconds. Compute the angular deceleration in rev/s², rad/s², deg/s²

16. Analogies Between Linear and Rotational Motion Linear Motion with constant acc. (x,v,a) Rotational Motion with fixed axis and constant  

17. Example A car coasts to a stop with a uniform deceleration of 1.2 rad/s² for its wheel. If its initial angular speed was 1000 rev/min, how many revolutions does the wheel make before coming to a stop? 1000 rev/min, how many revolutions does the wheel make before coming to a stop?

18. Relationship Between Angular and Linear Quantities Displacements Displacements Speeds Speeds Accelerations Accelerations Every point on the rotating object has the same angular motion Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion Every point on the rotating object does not have the same linear motion

19. Examples Pulley Pulley Rolling wheel Rolling wheel A belt runs on a pulley with diameter 5 cm which revolves at a speed of 1000 rev/min. What length of belt passes over the pulley each second? A belt runs on a pulley with diameter 5 cm which revolves at a speed of 1000 rev/min. What length of belt passes over the pulley each second?

20. Example A 60 cm diameter wheel is rotating @ speed of 3.0 rev/s. How fast is this car going?

21. Example If the masses of Atwood’s machine are accelerating at 4.8 m/s² and radius of 0.3 m, what is 

22. Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration The centripetal acceleration is due to the change in the direction of the velocity The centripetal acceleration is due to the change in the direction of the velocity

23. Centripetal Acceleration, cont. Centripetal refers to “center-seeking” Centripetal refers to “center-seeking” The direction of the velocity changes The direction of the velocity changes The acceleration is directed toward the center of the circle of motion The acceleration is directed toward the center of the circle of motion

24. Centripetal Acceleration, final The magnitude of the centripetal acceleration is given by The magnitude of the centripetal acceleration is given by This direction is toward the center of the circleThis direction is toward the center of the circle

25. Centripetal Acceleration and Angular Velocity The angular velocity and the linear velocity are related (v = rω) The angular velocity and the linear velocity are related (v = rω) The centripetal acceleration can also be related to the angular velocity The centripetal acceleration can also be related to the angular velocity  must be in rad/s  must be in rad/s

26. Forces Causing Centripetal Acceleration Newton’s Second Law says that the centripetal acceleration is accompanied by a force Newton’s Second Law says that the centripetal acceleration is accompanied by a force F = ma F = ma  F stands for any force that keeps an object following a circular pathF stands for any force that keeps an object following a circular path Tension in a string Tension in a string Gravity Gravity Force of friction Force of friction

27. Examples Ball at the end of revolving string (m=500 grams, r=120 cm,  rev/s). Tension? Ball at the end of revolving string (m=500 grams, r=120 cm,  rev/s). Tension? Fast car rounding a curve Fast car rounding a curve

28. More on circular Motion Length of circumference = 2R Length of circumference = 2R Period T (time for one complete circle) Period T (time for one complete circle)

29. Newton’s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

30. Universal Gravitation, 2 G is the constant of universal gravitational G is the constant of universal gravitational G = 6.673 x 10 -11 N m² /kg² G = 6.673 x 10 -11 N m² /kg² This is an example of an inverse square law This is an example of an inverse square law

32. Motion of Satellites Consider only circular orbit Consider only circular orbit Radius of orbit r: Radius of orbit r: Gravitational force is the centripetal force. Gravitational force is the centripetal force.

33. Motion of Satellites Period  Period  Synchronous Orbits Synchronous Orbits Kepler’s 3 rd Law

34. Communications Satellite A geosynchronous orbit A geosynchronous orbit Remains above the same place on the earthRemains above the same place on the earth The period of the satellite will be 24 hrThe period of the satellite will be 24 hr r = h + R E r = h + R E Still independent of the mass of the satellite Still independent of the mass of the satellite

35. Satellites and Weightlessness weighting an object in an elevator weighting an object in an elevator Elevator at rest: mg Elevator at rest: mg Elevator accelerates upward: m(g+a) Elevator accelerates upward: m(g+a) Elevator accelerates downward: m(g+a) with a<0 Elevator accelerates downward: m(g+a) with a<0 Satellite: a=-g!! Satellite: a=-g!!

37. Chapter 10 Dynamics of Rotation Dynamics of Rotation

38. Force vs. Torque Forces cause accelerations Forces cause accelerations Torques cause angular accelerations Torques cause angular accelerations rotationrotation Force and torque are related Force and torque are related

39. Torque, cont Torque, , is the tendency of a force to rotate an object about some axis Torque, , is the tendency of a force to rotate an object about some axis  is the torque  is the torque F is the force F is the force symbol is the Greek tausymbol is the Greek tau l is the lever arm l is the lever arm SI unit is N. m SI unit is N. m Lever Arm: Perpendicular distance from the pivot point to the line of force Lever Arm: Perpendicular distance from the pivot point to the line of force

40. Direction of Torque Torque is a vector quantity Torque is a vector quantity We will treat only 2-d torque so no need for vector notion.We will treat only 2-d torque so no need for vector notion. If the turning tendency of the force is counterclockwise, the torque will be positive (+)If the turning tendency of the force is counterclockwise, the torque will be positive (+) If the turning tendency is clockwise, the torque will be negative (-)If the turning tendency is clockwise, the torque will be negative (-)

41. Multiple Torques When two or more torques are acting on an object, the torques are added When two or more torques are acting on an object, the torques are added with the signswith the signs If the net torque is zero, the object’s rate of rotation doesn’t change If the net torque is zero, the object’s rate of rotation doesn’t change

42. General Definition of Torque The applied force is not always perpendicular to the position vector The applied force is not always perpendicular to the position vector The component of the force perpendicular to the object will cause it to rotate The component of the force perpendicular to the object will cause it to rotate F is the force F is the force L is distance between pivot and point of action L is distance between pivot and point of action  is the angle  is the angle

43. Moment of Inertia The angular acceleration is inversely proportional to the analogy of the mass in a rotating system The angular acceleration is inversely proportional to the analogy of the mass in a rotating system This mass analog is called the moment of inertia, I, of the object This mass analog is called the moment of inertia, I, of the object SI units are kg m 2SI units are kg m 2

44. Newton’s Second Law for a Rotating Object The angular acceleration is directly proportional to the net torque The angular acceleration is directly proportional to the net torque The angular acceleration is inversely proportional to the moment of inertia of the object The angular acceleration is inversely proportional to the moment of inertia of the object

45. More About Moment of Inertia There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. The moment of inertia also depends upon the location of the axis of rotation The moment of inertia also depends upon the location of the axis of rotation

46. Moment of Inertia of a Uniform Ring Image the hoop is divided into a number of small segments, m 1 … Image the hoop is divided into a number of small segments, m 1 … These segments are equidistant from the axis These segments are equidistant from the axis

47. Other Moments of Inertia

48. Example Wheel of radius R=20 cm and I=30kg·m^2. Force F=40N acts along the edge of the wheel. 1. Angular acceleration? 2. Angular speed 4s after starting from rest? 3. Number of revolutions for the 4s?

49. Rotational Kinetic Energy An object rotating about some axis with an angular speed, ω, has rotational kinetic energy E kr = ½Iω 2 An object rotating about some axis with an angular speed, ω, has rotational kinetic energy E kr = ½Iω 2 Energy concepts can be useful for simplifying the analysis of rotational motion Energy concepts can be useful for simplifying the analysis of rotational motion

50. Total Energy of a System Conservation of Mechanical Energy Conservation of Mechanical Energy Remember, this is for conservative forces, no dissipative forces such as friction can be presentRemember, this is for conservative forces, no dissipative forces such as friction can be present Potential energies of any other conservative forces could be addedPotential energies of any other conservative forces could be added

51. Rolling down incline Energy conservation Energy conservation Linear velocity and angular speed are related v=R Linear velocity and angular speed are related v=R Smaller I, bigger v, faster!! Smaller I, bigger v, faster!!

52. Work-Energy in a Rotating System In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy (E kt +E kr +E p ) i + W=(E kt +E kt +E p ) f (E kt +E kr +E p ) i + W=(E kt +E kt +E p ) f

53. Angular Momentum Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum Angular momentum is defined as Angular momentum is defined as L = I ωL = I ω andand

54. Angular Momentum, cont If the net torque is zero, the angular momentum remains constant If the net torque is zero, the angular momentum remains constant Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. That is, whenThat is, when

55. Conservation Rules, Summary In an isolated system, the following quantities are conserved: In an isolated system, the following quantities are conserved: Mechanical energyMechanical energy Linear momentumLinear momentum Angular momentumAngular momentum

56. Conservation of Angular Momentum, Example With hands and feet drawn closer to the body, the skater’s angular speed increases With hands and feet drawn closer to the body, the skater’s angular speed increases L is conserved, I decreases,  increasesL is conserved, I decreases,  increases

59. Example A 500 grams uniform sphere of 7.0 cm radius spins at 30 rev/s on an axis through its center. Moment of inertia Moment of inertia Rotational kinetic energy Rotational kinetic energy Angular momentum Angular momentum

60. Example A turntable is a uniform disk of metal of mass 1.5 kg and radius 13 cm. What torque is required to drive the turntable so that it accelerates at a constant rate from 0 to 33.3 rpm in 2 seconds?