Chapter 6: Systems in Motion
The celestial order and the beauty of the universe compel me to admit that there is some excellent and eternal Being, who deserves the respect and homage of men Cicero (106 BC 43 BC) 2
6.1 Projectile Motion Define projectile. Recognize the independence of a projectile s horizontal and vertical velocities. Describe the path of a projectile. Calculate a projectile s horizontal or vertical distance or speed. Explain how a projectile s launch angle affects its range. 3
Projectile Motion Any object moving through air and affected only by gravity is called a projectile. The horizontal (x) and vertical (y) motions of the projectile are independent of each other! There is no acceleration in the horizontal (x) direction so the projectile keeps moving at a constant speed horizontally. Gravity acts in the vertical (y) direction causing a downward acceleration (increase in speed). 4
Two Types of Problems Cliff problems where the projectile is shot horizontally from a height. Cannonball problems where the projectile is shot from ground level at an angle. The path a projectile follows is called its trajectory and follows a specially shaped curve called a parabola. The range of a projectile is the horizontal distance it travels in the air before touching the ground. Cliff Problem Cannon ball Problem 5
Projectile Equations Projectile motion is two dimensional motion. Therefore we have to keep track of two velocities, a x and a y velocity! The x velocity, v x, never changes since there is no force to change it! 6
Projectile Equations Projectile motion is two dimensional motion. Therefore we have to keep track of two distances, a x (range) and a y distance (height)! (5,2) 5t 2 7
Class Problem 1: Cliff Problem If you know the height of the cliff you can calculate the time it will take the projectile to hit the ground. If you know the initial horizontal velocity, v x, you can calculate the range of the projectile, d x. d y 5t 2 d x v x t 8
Review of Free Fall Motion Shoot a ball off the ground with some initial upward velocity, let s say 40 m/s, how much time does it take to reach its highest point? What do you suppose is the total time the ball is in the air? Remember each second the ball is going up it is losing 10 m/s of speed, so if the initial velocity was 40 m/s, it would take 4 seconds to reach its highest point. It will be in the air a total of 8 seconds! It takes the same amount of time to return to the ground. What if the initial velocity was 50 m/s? 9
The Velocity Vector Remember velocity is a vector and has both a size and a direction you need two bits of information to uniquely describe a velocity. We will break the velocity vector into its components. 10
Cannon Ball Problem There are two component velocities to keep track of now. One in the x direction, and one in the y direction. The x velocity never changes. The y velocity changes because of gravity. 11
Class Problem 2: Cannon Ball Problem A cannon ball fired off the ground so that it is initially moving 8 m/s in the x direction and 40 m/s in the y direction. a) You know the acceleration due to gravity is 10 m/s 2 that means is loses 10 m/s of speed every second on the way up. How many seconds does it take to get to the maximum height, point A? b) It will take just as long to reach the ground again, what is the total time in the air when the ball hits the ground at point B? c) Knowing the x velocity and the time, how far away did it land? d) Can you use the height equation to find the maximum height, d y? d y 5t 2 d x v 4 sec, 8 sec, 64 meters, 80 meters high at point A x t 12
Class Problem 3: Cannon Ball Problem Assume a ball was thrown with an initial overall speed was 50 m/s and it was projected at a 37 o angle. The x velocity is 40 m/s The y velocity is 30 m/s How many seconds did it take to reach the highest point? What was the total time in the air? How far away did it land? 3 sec, 6 sec, 240 meters! You can see how the velocity changes over time. The x velocity remains constant while the y velocity decreases on the way up, is zero at the top, and increases on the way down. 13
The Range of a Projectile The range of a projectile depends on both the speed and the angle of launch. For any given speed, what angle produces the greatest range? 45 o 60 o and 30 o are +/- 15 o from 45 o. What is special about these angles? They both land at the same spot. 14
6.2 Circular Motion Distinguish between rotation and revolution Calculate angular speed Explain how angular speed, linear speed, and distance are related 15
Rotation & Revolution An object rotates about its own internal axis. An object revolves about some external object/axis. The basketball rotates The boy revolves 16
Rotation & Revolution 17
Angular or Rotational Speed Circular motion is described by angular speed. The angular speed is the rate at which something turns. RPM s or rotations per minute, is commonly used for angular speed. If a basketball turns 15 times in three seconds, its angular speed is five rotations per second. 18
Rotations (Revolutions) Per Minute You can also think of RPM s is a frequency besides an angular speed. When thinking of frequency think how frequent does something happen. Frequency & Period are inverses: f = 1/t For example: Find the period of a spinning 45 record. 45rev 1min f 0.75rev /sec min 60sec t 1 1sec 1.33sec/ rev f 0.75rev 19
Relationship between Angular speed Each point on a rotating object has the same angular speed. But not the same linear speed! This depends also depends on the distance from the axis of rotation. Which kids has the greatest linear speed? Dwayne and Linear speed 20
Rolling Rolling is a combination of linear motion and rotational motion. 21
Class Problem: Circus Ride Let s say you are on the ride shown below. Your seat is 16 meters from the center of the ride. You spin around 6 times per minute. What is your RPMs? What is your angular speed? What is the period of revolution (in seconds)? What is your linear speed that distance away? What if your friend is on a shorter cable of only 8 meters, what is her linear speed? 22
6.3 Centripetal Force, Gravitation, and Satellites Explain how a centripetal force causes circular motion List the factors that affect centripetal force Describe the relationship between gravitational force, mass, and distance Relate centripetal force to orbital motion 23
Centripetal Force Recall Newton s 1 st Law: An object will move in a straight line at constant speed as long as there is no outside force on it. For something to move in a circle and continually change directions, there must be a force acting on it. There must be a centripetal or center seeking force acting! 24
Centripetal Force The centripetal force is a type of force, not the cause itself. There must always be some physical cause for an object to move in a circle. What is it in this case? The string continually changing the direction of the green ball. 25
Centripetal Acceleration The centripetal force continually pulls the object moving in a circle toward the center. Therefore the acceleration must also act in the same direction, toward the center! v 2 a c r mv 2 F c r 26
Centripetal Forces What causes the tether ball to move in a circle? The String! What causes a communication satellite to orbit the Earth? The Earth s gravity! What causes a car to keep moving in around a curve? Friction with the road! 27
Fictitious Centrifugal Forces What seems to push you to the outside door of your car as you go around a curve? What ultimately does cause the box in the back of the truck in the diagram to change its motion? 28
Newton's Law Of Universal Gravitation At the Earth's surface we have been using the equation w = mg for the force of gravity. Over longer range distances we have to use Newton's Universal Law of Gravity. There is a mutual (equal & opposite) gravitational force between any two masses, given by the equation: G is the universal gravitational constant and its value is measured to be 6.67 x10-11 Nm 2 /kg 2. 29
The Falling Apple The apple falls towards the earth. The apple accelerates towards the earth. It accelerates because of the gravitational force of the earth. 30
The Falling Moon Does earth s gravity affect the moon? Why doesn t the moon fall to the earth just like the apple does? What path would the moon take if there was no gravity? 31
Newton s Law of Universal Gravitation It seems all things (i.e., material objects) in the universe are attracted to each other by a gravitational force. You are attracted to the earth, but also the fire extinguisher, even Pluto and distant stars. 32
Measuring Big G G is called the universal gravitational constant and its value is measured to be 6.67 x10-11 Nm 2 /kg 2. F G G m m 1 d 2 2 33
Inverse Square Law m1m 2 Remember the formula is F G G 2 d So as the distance increases the force of gravity decreases much more! Doubling the distance does what to the force of gravity? 34
Is there zero gravity out in space? Astronauts in training weightlessness is simulated for a couple minutes only by putting an airplane in a nose dive! 35
Real and Apparent Weight Even though we are continually under the influence of the earth s gravity, we sometimes feel weightless the example to the left is not one you want to be in. Can you think of others? 36
Astronauts are in? Astronauts in the international space station appear to be weightless, this is because: They are in zero gravity They are in free fall 37
Class Problem What is the force of gravity between the Earth and the moon? Use the data below: Earth s mass is ~ 6.0 x10 24 kg Moon s mass is ~ 7.3 x10 22 kg The orbital radius of the moon is ~ 3.8 x10 5 m 38
6.4 Center of Mass and Rotational Inertia Define center of mass and center of gravity Explain how to locate an object s center of mass and center of gravity Use the concept of center of gravity to explain toppling 39
Finding the Center of Mass The center of mass is the average position of all the particles that make up the object s mass. Sometimes that is at a point where there is no mass at all! 40
Center of Mass 41
Stars, Planets, and Moons Two bodies of similar mass orbiting around a common center of mass. Two bodies with a major difference in mass orbiting around a common center of mass. This could be the Earth-Moon system! 42
Stars, Planets, and Moons 43
The Trajectory of a Projectile 44
Center of Gravity The center of gravity is the average position of an object s weight. In a uniform gravitational field, the center of mass and the center of gravity will be the same point. 45
Stability 46