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Chapter 6,9,10

Chapter 6,9,10. Circular Motion, Gravitation, Rotation, Bodies in Equilibrium. Circular Motion. Ball at the end of a string revolving Planets around Sun Moon around Earth. The Radian. The radian is a unit of angular measure

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Chapter 6,9,10

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  1. Chapter 6,9,10 Circular Motion, Gravitation, Rotation, Bodies in Equilibrium

  2. Circular Motion • Ball at the end of a string revolving • Planets around Sun • Moon around Earth

  3. The Radian • 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 57.3°

  4. More About Radians • Comparing degrees and radians • Converting from degrees to radians

  5. Angular Displacement • Axis of rotation is the center of the disk • Need a fixed reference line • During time t, the reference line moves through angle θ

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

  7. Average Angular Speed • The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

  8. Angular Speed, cont. • The instantaneous angular speed • Units of angular speed are radians/sec • rad/s • Speed will be positive if θ is increasing (counterclockwise) • Speed will be negative if θ is decreasing (clockwise)

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

  10. Angular Acceleration, cont • Units of angular acceleration are rad/s² • 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 speed and the same angular acceleration

  11. Angular Acceleration, final • The sign of the acceleration does not have to be the same as the sign of the angular speed • The instantaneous angular acceleration

  12. Analogies Between Linear and Rotational Motion Linear Motion with constant acc. (x,v,a) Rotational Motion with fixed axis and constant a (q,,a)

  13. Examples • 78 rev/min=? • A fan turns at a rate of 900 rpm • Tangential speed of tips of 20cm long blades? • Now the fan is uniformly accelerated to 1200 rpm in 20 s

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

  15. Centripetal 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

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

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

  18. Centripetal Acceleration and Angular Velocity • The angular velocity and the linear velocity are related (v = ωR) • The centripetal acceleration can also be related to the angular velocity

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

  20. Examples • Ball at the end of revolving string • Fast car rounding a curve

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

  22. Example • 200 grams mass revolving in uniform circular motion on an horizontal frictionless surface at 2 revolutions/s. What is the force on the mass by the string (R=20cm)?

  23. 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.

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

  25. Universal Gravitation, 3 • The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1 • The forces form a Newton’s third law action-reaction

  26. Universal Gravitation, 4 • The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center

  27. Gravitation Constant • Determined experimentally • Henry Cavendish • 1798 • The light beam and mirror serve to amplify the motion

  28. Applications of Universal Gravitation • Weighing the Earth

  29. Applications of Universal Gravitation • “g” will vary with altitude

  30. Escape Speed • The escape speed is the speed needed for an object to soar off into space and not return • For the earth, vesc is about 11.2 km/s • Note, v is independent of the mass of the object

  31. Various Escape Speeds • The escape speeds for various members of the solar system • Escape speed is one factor that determines a planet’s atmosphere

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

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

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

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

  36. Force vs. Torque • Forces cause accelerations • Torques cause angular accelerations • Force and torque are related

  37. Torque • The door is free to rotate about an axis through O • There are three factors that determine the effectiveness of the force in opening the door: • The magnitude of the force • The position of the application of the force • The angle at which the force is applied

  38. Torque, cont • Torque, t, is the tendency of a force to rotate an object about some axis • t is the torque • F is the force • symbol is the Greek tau • l is the length of lever arm • SI unit is N.m • Work done by torque W=

  39. Direction of Torque • 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

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

  41. Torque and Equilibrium • First Condition of Equilibrium • The net external force must be zero • This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium • This is a statement of translational equilibrium

  42. Torque and Equilibrium, cont • To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational • The Second Condition of Equilibrium states • The net external torque must be zero

  43. Equilibrium Example • The woman, mass m, sits on the left end of the see-saw • The man, mass M, sits where the see-saw will be balanced • Apply the Second Condition of Equilibrium and solve for the unknown distance, x

  44. Moment of Inertia • 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 • SI units are kg m2

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

  46. 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. • The moment of inertia also depends upon the location of the axis of rotation

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