Forces, Momentum, & Gravity (Chapter 3) Force and Motion Cause and Effect In chapter 2 we studied motion but not its cause. In this chapter we will look at both force and motion the cause and effect. We will consider Newton s: Newton s three laws of motion Newton s law of universal gravitation Buoyancy and momentum Intro Student Learning Objectives Recall and apply each of Newton s Laws Relate momentum to impact force Use conservation of momentum to analyze motion Describe gravity and its applications. 1
Sir Isaac Newton (1642 1727) Only 25 when he formulated most of his discoveries in math and physics His book Mathematical Principles of Natural Philosophy is considered to be the most important publication in the history of Physics. Intro What is a force & when are forces balanced? A force is the amount of push or pull on an object. Forces can cause a change in motion; a net force results in acceleration. An object in mechanical equilibrium maintains its motion. There is no change. F net = 0 Forces are balanced F net = F Balanced (equal) forces, therefore no motion. Equal in magnitude but in opposite directions. Section 3.1 2
Unbalanced forces result in motion Net force to the right Section 3.1 Practice 1) A 150 lb person is standing still on the floor. What is the net force? 2) If a car engine provides 700 Newtons of push, and there is 200 Newtons of opposing friction, what is the net force on the car? 3) When the car engine continues to provide 700 Newtons of push, but the friction force is increased to 700 Newtons, what is the net force on the car? Would this cause the car to stop? 4) Is a car with a constant velocity of 30 mph in mechanical equilibrium? Why must you keep pressure on the accelerator? What are Newton s Laws of Motion & how do they apply? All objects have the same motion within a specific reference frame. Newton s 1 st Law of Motion: Inertia An object will remain at rest or maintain a constant velocity unless an unbalanced force causes the object s motion to change. Inertia is the tendency of an object to maintain its motion. Quick stops Cornering Something sliding across your dash as you turn The tablecloth trick 3
A spacecraft keeps going because no forces act to stop it Photo Source: Copyright Bobby H. Bammel. All rights reserved. Section 3.2 A large rock in Yellowstone N.P. stays put until a large enough force acts on it. Photo Source: Copyright Bobby H. Bammel. All rights reserved. Section 3.2 http://www.physicsclassroom.com/mmedia/newtlaws/cci.cfm https://www.youtube.com/watch?v=d7iyzpp2zyy Mass Inertia depends on mass. more mass more inertia harder to change motion Mass is the amount of material contained in an object. Empty 747 Jet 160,000 kg Average Man 73 kg 5 coin 0.0052 kg Mass is a fundamental quantity. 4
Mass and Inertia The large man has more inertia more force is necessary to start him swinging and also to stop him due to his greater inertia Section 3.2 Mass and Inertia Quickly pull the paper and the stack of quarters tend to stay in place due to inertia. Section 3.2 Practice: Mass is often defined in elementary school as the amount of space an object takes up. Why is this not correct? Newton s 2 nd Law of Motion: F = ma An unbalanced force acting on a mass gives the mass an acceleration in the same direction as the unbalanced force. Example: Soccer ball F = ma http://www.grc.nasa.gov/www/k-12/airplane/socforce.html https://www.youtube.com/watch?v=8hgns1yqab0 5
Force, Mass, Acceleration a) Original situation a F m b) If we double the force we double the acceleration. c) If we double the mass we half the acceleration. Section 3.3 When an object is in motion, friction is always in the opposite direction of the motion. F f Big Box Motion Practice 1) If the force on a 5 kg mass is tripled, what will happen to the rate of acceleration? 2) A 2000 kg car engine provides 9800 N of push southward. The opposing frictional force is 1200 N. What is the average acceleration? Weight is a force; it is the gravitational force acting on a mass. W = mg 1 kg of mass weighs 9.8 Newtons or 2.2 pounds on Earth. Practice 1) Does weight depend on volume? 2) Would 1 kg of mass weigh 2.2 pounds on the Moon? 3) If you were instantly transported to Mars, what would change? a. Mass? b. Weight? c. Inertia? 4) What would a person weigh on Mars if the person weighs150 lb (667 N) on Earth? The acceleration due to gravity on Mars is 3.72 m/s 2. 6
Vectors Forces are vectors, and vector addition is the addition of both the size and direction of each quantity. http://www.physicsclassroom.com/mmedia/vectors/rb.cfm http://www.physicsclassroom.com/mmedia/vectors/plane.cfm The resultant vector shows the result of two or more vectors acting simultaneously. Practice An airplane s speedometer reads 500 mph North. What is the net velocity of the airplane in each case? a) Wind is blowing North at 50 mph. b) Wind is blowing South at 50 mph. c) Wind is blowing East at 50 mph. Newton s 3 rd Law of Motion: Action-Reaction When two objects interact, they create equal and opposite forces on each other. F 1 = F 2 To every action force there is an equal (in magnitude) and opposite (in direction) reaction force when two objects are in contact. Examples: Pushing on a wall, Bat & Ball, Rockets 7
Newton's Laws in Action Friction on the tires provides necessary centripetal acceleration. Passengers continue straight ahead in original direction and as car turns the door comes toward passenger 1 st Law As car turns you push against door and the door equally pushes against you 3 rd Law Section 3.4 http://www.wimp.com/golfball/ http://www.cnn.com/videos/us/2015/02/11/sot-spacex-rocket-launch.nasa-tv Practice If I give the chair a good push, it goes from rest to having a velocity, and then stops. How do each of Newton s laws apply to this system? What is momentum? Linear momentum is the combination of mass (inertia) and velocity. p= mv The greater the momentum, the harder it is to stop an object! 8
Momentum If we have a system of masses, the linear momentum is the sum of all individual momentum vectors. P f = P i (final = initial) P = p 1 + p 2 + p 3 + (sum of the individual momentum vectors) Law of Conservation of Linear Momentum Law of Conservation of Linear Momentum - the total linear momentum of an isolated system remains the same if there is no external, unbalanced force acting on the system Linear Momentum is conserved as long as there are no external unbalance forces. It does not change with time. Conservation of Linear Momentum P i = P f = 0 (for man and boat) When the man jumps out of the boat he has momentum in one direction and, therefore, so does the boat. Their momentums must cancel out! (= 0) 9
Applying the Conservation of Linear Momentum Two masses at rest on a frictionless surface. When the string (weightless) is burned the two masses fly apart due to the release of the compressed (internal) spring (v 1 = 1.8 m/s). Applying the Conservation of Linear Momentum Two masses at rest on a frictionless surface. When the string (weightless) is burned the two masses fly apart due to the release of the compressed (internal) spring (v 1 = 1.8 m/s). GIVEN: P f = P i = 0 m 1 = 1.0 kg P f =p 1 + p 2 = 0 m 2 = 2.0 kg p 1 = -p 2 v 1 = 1.8 m/s, v 2 =? m 1 v 1 = -m 2 v 2 Applying the Conservation of Linear Momentum m 1 v 1 = -m 2 v 2 v 2 = - m 1v 1 = - (1.0 kg) (1.8 m/s) = -0.90 m/s m 2 2.0 kg 10
Jet Propulsion Jet Propulsion can be explained in terms of both Newton s 3 rd Law & Linear Momentum p 1 = -p 2 m 1 v 1 =-m 2 v 2 The exhaust gas molecules have small m and large v. The rocket has large m and smaller v. BUT m 1 v 1 = -m 2 v 2 (momentum is conserved) Practice 1) What is an example of a moving object that could have a large momentum because it has a large mass? 2) What is an example of a small object that could have a large momentum because it has a large velocity? 3) Which has the most momentum? a) 10,000 lb (4535 kg) 18-wheeler parked at the curb b) 300 lb (136 kg) football player running 10 mph (4.46 m/s) c) 150 lb (68 kg) sprinter running 22 mph (9.83 m/s) d) 1200 kg car moving at 1 m/s Torque Torque the twisting effect caused by one or more forces As we have learned, the linear momentum of a system can be changed by the introduction of an external unbalanced force. Similarly, angular momentum can be changed by an external unbalanced torque. https://www.youtube.com/watch?v=h4jutnkigpi 11
Torque Torque is a twisting action that produces rotational motion or a change in rotational motion. Torque and Lever Arm Torque varies with the length of the lever arm. As the length of the lever arm is doubled, the torque is doubled, for a given force. Law of Conservation of Angular Momentum Law of Conservation of Angular Momentum - the angular momentum of an object remains constant if there is no external, unbalanced torque (a force about an axis) acting on it Concerns objects that go in paths around a fixed point, for example a satellite orbiting the earth https://www.youtube.com/watch?v=rmwj7wa09ge 12
Angular Momentum L = mvr L = angular momentum, m = mass, v = velocity, and r = distance to center of motion L 1 = L 2 m 1 v 1 r 1 = m 2 v 2 r 2 Angular Momentum - Example Mass (m) is constant. As r changes so must v. When r decreases, v must increase so that m 1 v 1 r 1 = m 2 v 2 r 2 Angular Momentum in our Solar System In our solar system the planet s orbit paths are slightly elliptical, therefore both r and v will slightly vary during a complete orbit. 13
Conservation of Angular Momentum Example A comet at its farthest point from the Sun is 900 million miles, traveling at 6000 mi/h. What is its speed at its closest point of 30 million miles away? EQUATION: m 1 v 1 r 1 = m 2 v 2 r 2 GIVEN: v 2, r 2, vr 1, and m 1 2 r 2 (6.0 = m 2 x 10 3 mi/h) (900 x 10 6 mi) FIND: v 1 = = 30 x 10 6 mi r 1 1.8 x 10 5 mi/h or 180,000 mi/h Conservation of Angular Momentum Rotors on large helicopters rotate in the opposite direction Conservation of Angular Momentum Figure Skater she/he starts the spin with arms out at one angular velocity. Simply by pulling the arms in the skater spins faster, since the average radial distance of the mass decreases. m 1 v 1 r 1 = m 2 v 2 r 2 m is constant; r decreases; Therefore v increases 14
Angular momentum is momentum in a circular path. L = mvr The angular momentum vector is perpendicular to the plane of the circular path. L http://hyperphysics.phy-astr.gsu.edu/hbase/bike.html https://www.youtube.com/watch?v=8h98bgrzpom v How does momentum affect the force of impact? https://www.youtube.com/watch?v=qlp6qh1izxw https://www.youtube.com/watch?v=r8e5dunlmh4 During an impact, the force of impact depends on how quickly the momentum is changed. F = p t Examples: Water barrels at the start of a divided highway Carpet vs. concrete Practice 1) You (75 kg) are riding in your 2000 kg car at 30 m/s (67 mph) when suddenly a squirrel runs in front of you; you swerve, and hit a tree. If the duration of the impact is 1/2 of a second, what is the impact force? 2) Two 2000 kg cars, each with a 75 kg person, are traveling toward each other with a speed of 30 m/s (67 mph); suddenly one swerves into the wrong lane and there is a head-on collision. If the duration of the impact is 1/2 of a second, what is the impact force? 3) What are some features of car design that decrease force of impact? 15
How is conservation of momentum applied? The total momentum of an isolated system remains constant. p f = p i L f = L i Examples: Collisions & Ice Skaters http://www.physicsclassroom.com/mmedia/momentum https://www.youtube.com/watch?v=wsg28pzvxjs https://www.youtube.com/watch?v=aqltceag9v0 Practice Always assume momentum is conserved. 1)Two cars of equal mass collide. One is traveling West at 30 m/s, the other is at rest. Then there is an inelastic collision between the two cars. If linear momentum is conserved, what is the final velocity of each car? 2) An ice skater spins 5 m/s with outstretched arms. The radius of the circular path traced by her arms is 1 meter. Then she pulls her arms in, changing the radius of the circular path to 1/3 m. If angular momentum is conserved, what is her new spinning speed? Newton s Law of Gravitation Gravity is a fundamental force of nature We do not know what causes it We can only describe it Law of Universal Gravitation Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them Section 3.5 16
What is Newton s description of gravity? Newton s Universal Law of Gravitation F g = GMm Mutual force of attraction d 2 All masses pull the same on each other! Causes acceleration due to gravity, g = 9.81 m/s 2 G is the universal gravitational constant G = 6.67 x 10-11 N. m 2 /kg 2 G: is a very small quantity thought to be valid throughout the universe was measured by Cavendish 70 years after Newton s death not equal to g and not a force http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html Newton s Law of Gravitation The forces that attract particles together are equal and opposite F 1 = -F 2 or m 1 a 1 = -m 2 a 2 Section 3.5 Newton's Law of Gravitation F = Gm 1 m 2 r 2 For a homogeneous sphere the gravitational force acts as if all the mass of the sphere were at its center Section 3.5 17
Applying Newton s Law of Gravitation Two objects with masses of 1.0 kg and 2.0 kg are 1.0 m apart. What is the magnitude of the gravitational force between the masses? Section 3.5 Applying Newton s Law of Gravitation Example Two objects with masses of 1.0 kg and 2.0 kg are 1.0 m apart. What is the magnitude of the gravitational force between the masses? Gm 1 m 2 F = r 2 F = (6.67 x 10-11 N-m 2/kg 2)(1.0 kg)(2.0 kg) F = 1.3 x 10-10 N (1.0 m) 2 Section 3.5 Practice 1) Would the acceleration due to gravity (9.81 m/s 2 ) be different for an object dropped from a high mountain top than it is at sea level? 2) If Earth had twice as much mass, would this change our weight? Would it change our mass? 3) What is the gravitational attraction between Earth and a 75 kg person standing on the surface, at sea level (M E = 6 x 10 24 kg, r E = 6.4 x 10 6 m)? What do we normally call this? 4) How would the gravitational force change if the distance doubled? 5) Is the gravitational force zero in space? 18
What are some effects of gravity? https://www.youtube.com/watch?v=wmk36dphikg The feeling of weightlessness occurs when an object and its reference frame accelerate at the same rate. Airplane drops Large, rolling dip in the road Freefall ride If there is no support force, then objects will fall together. https://www.youtube.com/watch?v=bv7tpvk ke Practice If you were standing on a scale in the elevator that measured your weight, how would the scale reading change as the elevator a) Accelerates up b) Accelerates down c) Free falls Gravity Effects Orbits Tides Atmospheres Changing Earth-Moon System Changing Earth-Moon system: Earth s rotation is slowing (0.0015 seconds/century) Our Moon is drifting away (3.8 cm/year) The synchronous orbit of the Moon (same face) 19
Weightlessness in space is the result of both the astronaut and the spacecraft falling to Earth at the same rate Photo Source: Standard HMCO copyright line Section 3.5 https://www.youtube.com/watch?v=l37ofe9hamu Practice: The Sun's tidal affects are weak compared to the Moon. Why? What is Einstein s description of Gravity? Every object with mass creates a curvature of space-time. https://www.youtube.com/watch?v=loaohvy5aca https://www.youtube.com/watch?v=jzwyavn970c According to Einstein, mass does not create a force, but rather a warping of space which other objects follow. More Mass = More Curvature Objects fall independent of their mass because they all follow the same path in curved space-time. 20