1. Chapter 7 7.1 Measuring rotational motion

2. Rotational Quantities • Rotational motion: motion of a body that spins about an axis • Axis of rotation: the line about which the rotation occurs • Circular motion: motion of a point on a rotating object

3. Rotational Quantities • Circular Motion • Direction is constantly changing • Described as an angle • All points (except points on the axis) move through the same angle during any time interval

4. Circular Motion • Useful to set a reference line • Angles are measured in radians • s= arc length • r = radius

5. Angular Motion • 360o = 2rad • 180o = rad

6. Angular displacement • Angular dispacement: the angle through which a point line, or body is rotated in a specified direction and about a specified axis • Practice: • Earth has an equatorial radius of approximately 6380km and rotates 360o every 24 h. • What is the angular displacement (in degrees) of a person standing at the equator for 1.0 h? • Convert this angular displacement to radians • What is the arc length traveled by this person?

7. Angular speed and acceleration • Angular speed: The rate at which a body rotates about an axis, usually expressed in radians per second • Angular acceleration: The time rate of change of angular speed, expressed in radians per second per second

8. Angular speed and acceleration ALL POINTS ON A ROTATING RIGID OBJECT HAVE THE SAME ANGULAR SPEED AND ANGULAR ACCELERATION

9. Rotational kinematic equations

10. Angular kinematics • Practice • A barrel is given a downhill rolling start of 1.5 rad/s at the top of a hill. Assume a constant angular acceleration of 2.9 rad/s • If the barrel takes 11.5 s to get to the bottom of the hill, what is the final angular speed of the barrel? • What angular displacement does the barrel experience during the 11.5 s ride?

11. Homework Assignment • Page 269: 5 - 12

12. Chapter 7 7.2 Tangential and Centripetal Acceleration

13. Tangential Speed • Let us look at the relationship between angular and linear quantities. • The instantaneous linear speed of an object directed along the tangent to the object’s circular path • Tangent: the line that touches the circle at one and only one point.

14. Tangential Speed • In order for two points at different distances to have the same angular displacement, they must travel different distances • The object with the larger radius must have a greater tangential speed

15. Tangential Speed

16. Tangential Acceleration • The instantaneous linear acceleration of an object directed along the tangent to the object’s circular path

17. Lets do a problem • A yo-yo has a tangential acceleration of 0.98m/s2 when it is released. The string is wound around a central shaft of radius 0.35cm. What is the angular acceleration of the yo-yo?

18. Centripetal Acceleration • Acceleration directed toward the center of a circular path • Although an object is moving at a constant speed, it can still have an acceleration. • Velocity is a vector, which has both magnitude and DIRECTION. • In circular motion, velocity is constantly changing direction.

19. Centripetal Acceleration • vi and vf in the figure to the right differ only in direction, not magnitude • When the time interval is very small, vf and vi will be almost parallel to each other and acceleration is directed towards the center

20. Centripetal Acceleration

21. Tangential and centripetal accelerations • Summary: • The tangential component of acceleration is due to changing speed; the centripetal component of acceleration is due to changing direction • Pythagorean theorem can be used to find total acceleration and the inverse tangent function can be used to find direction

22. What’s coming up • HW: Pg 270, problems 21 - 26 • Monday: Section 7.3 • Wednesday: Review • Friday: TEST over Chapter 7

23. Chapter 7 7.3: Causes of Circular Motion

24. Causes of circular motion • When an object is in motion, the inertia of the object tends to maintain the object’s motion in a straight-line path. • In circular motion (I.e. a weight attached to a string), the string counteracts this tendency by exerting a force • This force is directed along the length of the string towards the center of the circle

25. Force that maintains circular motion • According to Newton’s second law or:

26. Force that maintains circular motion • REMEMBER: The force that maintains circular motion acts at right angles to the motion. • What happens to a person in a car(in terms of forces) when the car makes a sharp turn.

27. Chapter 9 9.2 - Fluid pressure and temperature

28. Pressure • What happens to your ears when you ride in an airplane? • What happens if a submarine goes too deep into the ocean?

29. What is Pressure? • Pressure is defined as the measure of how much force is applied over a given area • The SI unit of pressure is the pascal (PA), which is equal to N/m2 • 105Pa is equal to 1 atm

30. Some Pressures

31. Pressure applied to a fluid • When you inflate a balloon/tire etc, pressure increases • Pascal’s Principle • Pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of a container

32. Lets do a problem • In a hydraulic lift, a 620 N force is exerted on a 0.20 m2 piston in order to support a weight that is placed on a 2.0 m2 piston. • How much pressure is exerted on the narrow piston? • How much weight can the wide piston lift?

33. Pressure varies with depth in a fluid • Water pressure increases with depth. WHY? • At a given depth, the water must support the weight of the water above it • The deeper you are, the more water there is to support • A submarine can only go so deep an withstand the increased pressure

34. The example of a submarine • Lets take a small area on the hull of the submarine • The weight of the entire column of water above that area exerts a force on that area

35. Fluid Pressure • Gauge Pressure • does not take the pressure of the atmosphere into consideration • Fluid Pressure as a function of depth • Absolute pressure = atmospheric pressure + (density x free-fall acceleration x depth)

36. Point to remember These equations are valid ONLY if the density is the same throughout the fluid

37. The Relationship between Fluid pressure and buoyant forces • Buoyant forces arise from the differences in fluid pressure between the top and bottom of an immersed object

38. Atmospheric Pressure • Pressure from the air above • The force it exerts on our body is 200 000N (40 000 lb) • Why are we still alive?? • Our body cavities are permeated with fluids and gases that are pushing outward with a pressure equal to that of the atmosphere -> Our bodies are in equilibrium

39. Atmospheric • A mercury barometer is commonly used to measure atmospheric pressure

40. Kinetic Theory of Gases • Gas contains particles that constantly collide with each other and surfaces • When they collide with surfaces, they transfer momentum • The rate of transfer is equal to the force exerted by the gas on the surface • Force per unit time is the gas pressure

41. Lets do a Problem • Find the atmospheric pressure at an altitude of 1.0 x 103 m if the air density is constant. Assume that the air density is uniformly 1.29 kg/m3 and P0=1.01 x 105 Pa

42. Temperature in a gas • Temperature is the a measure of the average kinetic energy of the particles in a substance • The higher the temperature, the faster the particles move • The faster the particles move, the higher the rate of collisions against a given surface • This results in increased pressure

43. HW Assignment • Page 330: Practice 9C, page 331: Section Review

44. Chapter 9 9.3 - Fluids in Motion

45. Fluid Flow • Fluid in motion can be characterized in two ways: • Laminar: Every particle passes a particular point along the same smooth path (streamline) traveled by the particles that passed that point earlier • Turbulent: Abrupt changes in velocity • Eddy currents: Irregular motion of the fluid

46. Ideal Fluid • A fluid that has no internal friction or viscosity and is incompressible • Viscosity: The amount of internal friction within a fluid • Viscous fluids loose kinetic energy because it is transformed into internal energy because of internal friction.

47. Ideal Fluid • Characterized by Steady flow • Velocity, density and pressure are constant at each point in the fluid • Nonturbulent • There is no such thing as a perfectly ideal fluid, but the concept does allow us to understand fluid flow better • In this class, we will assume that fluids are ideal fluids unless otherwise stated

48. Principles of Fluid Flow • If a fluid is flowing through a pipe, the mass flowing into the pipe is equal to the mass flowing out of the pipe

49. Pressure and Speed of Flow • In the Pipe shown to the right, water will move faster through the narrow part • There will be an acceleration • This acceleration is due to an unbalanced force • The water pressure will be lower, where the velocity is higher

50. Bernoulli’s Principle • The pressure in a fluid decreases as the fluid’s velocity increases