Chapter 7 7.1 Measuring rotational motion
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
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
Circular Motion • Useful to set a reference line • Angles are measured in radians • s= arc length • r = radius
Angular Motion • 360o = 2rad • 180o = rad
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?
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
Angular speed and acceleration ALL POINTS ON A ROTATING RIGID OBJECT HAVE THE SAME ANGULAR SPEED AND ANGULAR ACCELERATION
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?
Homework Assignment • Page 269: 5 - 12
Chapter 7 7.2 Tangential and Centripetal Acceleration
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.
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
Tangential Acceleration • The instantaneous linear acceleration of an object directed along the tangent to the object’s circular path
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?
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.
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
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
What’s coming up • HW: Pg 270, problems 21 - 26 • Monday: Section 7.3 • Wednesday: Review • Friday: TEST over Chapter 7
Chapter 7 7.3: Causes of Circular Motion
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
Force that maintains circular motion • According to Newton’s second law or:
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.
Chapter 9 9.2 - Fluid pressure and temperature
Pressure • What happens to your ears when you ride in an airplane? • What happens if a submarine goes too deep into the ocean?
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
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
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?
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
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
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)
Point to remember These equations are valid ONLY if the density is the same throughout the fluid
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
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
Atmospheric • A mercury barometer is commonly used to measure atmospheric pressure
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
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
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
HW Assignment • Page 330: Practice 9C, page 331: Section Review
Chapter 9 9.3 - Fluids in Motion
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
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.
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
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
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
Bernoulli’s Principle • The pressure in a fluid decreases as the fluid’s velocity increases