10/16 do now • A racing car is moving around the circular track of radius 300 meters shown. At the instant when the car’s velocity is directed due east, its acceleration is directed due north and has a magnitude of 3 m/s2. When viewed from above, the car is moving • Clockwise at 30 m/s • Clockwise at 10 m/s • Counterclockwise at 30 m/s • Counterclockwise at 10 m/s • With constant velocity
Chapter 4 Newton’s Laws of Motion
Goals for Chapter 4 • To visualize force as a vector • To find the net force acting on a body and apply Newton’s First Law • To study mass, acceleration, and their application to Newton’s Second Law • To calculate weight and compare/contrast it with mass • To see action–reaction pairs and study Newton’s Third Law
4-1 force and interactions • What are the properties of force(s)?
There are four common types of forces • The normal force—When an object rests or pushes on a surface, the surface pushes back. • Frictional forces—In addition to the normal force, surfaces can resist motion along the surface.
There are four common types of forces II • Tension forces—When a force is exerted through a rope or cable, the force is transmitted through that rope or cable as a tension. • Weight—Gravity’s pull on an object. This force can act from large distances.
The unit of force • Force is a quantity which is measured using the standard metric unit known as the Newton. • A Newton is abbreviated by a "N." To say "10.0 N" means 10.0 Newton of force. • One Newton is the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s. Thus, the following unit equivalency can be stated:
Force is a vector quantity • Use a vector arrow to indicate magnitude and direction of the force.
Use the net (overall) force—Figure 4.4 • Several forces acting on a point have the same effect as their vector sum acting on the same point.
Decomposing a force into components • Fx and Fy are the parallel and perpendicular components of a force to a sloping surface. • Use F*Cosθ and F*Sinθ operations to find force components.
Notation and method for the vector sum—Figure 4.7 • We refer to the vector sum or resultant as the “sum of forces” R = F1 + F2 + F3 … Fn = ΣF. • Use Tanθ = Ry/Rx and R = (Rx2 + Ry2)1/2.
It’s more convenient to describe a force F in terms of its x and y components Fxand Fy • Our coordinate axes does not have to be vertical and horizontal. Note: we draw a wiggly line through the force vector F to show that we have replaced it by its x and y components.
Example 4.1 superposition of forces • Three professional wrestlers are fighting over the same champion's belt. As viewed from above, they apply the three horizontal forces the belt that are shown in the figure. The magnitudes of the three forces are F1 = 250 N, F2= 50 N, F3 = 120 N. Find the x and y components of the net force on the belt, and find the magnitude and direction of the net force.
Test your understanding 4.1 • The figure shows a force F acting on a crate. With the x- and y-axes shown in the figure, which statement about the components of the gravitational force that the earth exerts on the crate (the crate’s weight) is correct? • The x- and y-components are both positive • The x-components is zero and the y-component is positive • The x-component is negative and the y-component is positive • The x- and y-components are both negative • The x-component is zero and the y-component is negative • The x-component is positive and the y-component is negative.
10/17 Do now • A 50 Newton weight is suspended by two cords as shown in the figure above. Then tension in the slanted cord is • 50 N • 100 N • 150 N • 200 N • 250 N
4.2 Newton’s First Law – a body in equilibrium • Simply stated—“objects at rest tend to stay at rest, objects in motion stay in motion.” • More properly, “A body acted on by no net force moves with constant velocity and zero acceleration.”
Newton’s First Law II—Figure 4.10 • Figure 4.10 shows an unbalanced force causing an acceleration and balanced forces resulting in no motion.
Example 4.2 – zero net force means constant velocity • In the classic 1950 science fiction film X-M, a spaceship is moving in the vacuum of outer space, far from any planet, when its engine dies. As a result, the spaceship slows down and stops, what does Newton's first law say about this event? According the Newton’s 1st Law, an object in motion will remain in motion. The spaceship will not slow down, it will travel at constant velocity forever.
Example 4.3: constant velocity means zero net force You are driving a Porsche Carrera GT on a straight testing track at a constant speed of 150 km/h. you pass a 1971 Volkswagen beetle doing a constant 75 km/h. For which car is the net force greater? Since both cars are in equilibrium because their velocities are constant, therefore the net force on each car is zero.
Suppose you are in a bus that is traveling on a straight road and speeding up. If you could stand in the aisle on roller skates, you would start moving backward relative to the bus as the bus gains speed. If instead the bus was slowing to a stop, you would start moving forward down the aisle. In either case, it looks as though Newton’s first law is not obeyed; there is no net force acting on you, yet your velocity change. What wrong?
Inertial frames of reference—Figure 4.11 • When a car turns and a rider continues to move, the rider perceives a force.
Inertial frame of reference • The bus is accelerating with respect to the earth and in not a suitable frame of reference for Newton’s 1st law. A frame of reference in which Newton's 1st law is valid is called an inertial frame of reference. The earth is at least approximately an inertial frame of reference, but the bus is not. • Because Newton’s first law is used to determine what we mean by an inertial frame of reference, it is sometimes called the law of inertia.
Test your understanding 4.2 • In which of the following situations is there zero net force on the body? • An airplane flying due north at a steady 120 m/s and at a constant altitude. • A car driving straight up a hill with a 3o slope at a constant 90 km/h • A hawk circling at a constant 20 km/h at a constant height of 15 m above an open field • A box with slick, frictionless surfaces in the back of a truck as the truck accelerates forward on a level road at 5 m/s2.
Newton’s Second Law—Figure 4.13 • An unbalanced force (or sum of forces) will cause a mass to accelerate.
An object undergoing uniform circular motion • Centripetal force is the net force. We have already seen the centripetal acceleration. But, if we measure the mass in motion, Newton’s Second Law allows us to calculate the centripetal force.
The relationship of F, m, and a • Acceleration is directly proportional to the applied force and always in the same directions as the net force.
The relationship of F, m, and a redux • acceleration is inversely proportional to the object’s mass.
Suppose we apply a constant net force ∑F to a body having a known mass m1 and we find an acceleration of magnitude a1. We then apply the same force to another body having an unknown mass m2, and we find an acceleration of magnitude a2. For the same net force, the ratio of the masses of two bodies is the inverse of the ratio of their acceleration.
Applying Newton's second law • Since the equation is a vector equation, we will use it in component form, with a separate equation for each component of force and the corresponding acceleration: Note: 1. ∑F means the sum of all external forces. 2. the equation is only valid if mass is constant. 3. the equation is only valid in the inertial frame of reference
caution • ma is not a force. • The vector ma is equal to the vector sum of all the forces acting on the body (∑F)
Example 4.4 determining acceleration from force • A worker applies a constant horizontal force with magnitude 20 N to a box with mass 40 kg resting on a level floor with negligible friction. What is the accelerations of the box?
Example 4.5 determining force from acceleration • A waitress shoves a ketchup bottle with mass 0.45 kg to the right along a smooth, level lunch counter. The bottle leaves her hand moving at 2.8 m/s, then slows down as it slides because of the constant horizontal friction force exerted on it by the counter top. It slides a distance of 1.0 m before coming to rest. What are the magnitude and direction of the friction force acting on the bottle?
Newtons, kilograms, pounds, and slugs—Table 4.2 • Table 4.2 rightly points out that the pound is a force. The popular culture refers to it as a weight (which is actually a slug). • The Dyne is actually a cgs version of the Newton (sometimes used with fine work on tiny objects).
Test your understanding • Rank the following situations in order of the magnitude of the object’s acceleration, from lowest to highest. Are there any cases that have the same magnitude of acceleration? • A 2.0 kg object acted on by a 2.0 N net force; • A 2.0 kg object acted on by an 8.0 N net force. • An 8.0 kg object acted on by a 2.0 N net force. • An 8.0 kg object acted on by a 8.0 N net force. c, d = a, b
10/25 do now • A small box is on ramp tilted at an angle θ above the horizontal. The box may be subject to the following forces: friction (f), gravity (mg) and normal force (N). Draw a free body diagram best represents the box if it is at rest on the ramp. N f mg 54%
4.4 mass and weight • Mass characterizes the inertial properties of a body. The greater the mass, the greater the force needed to cause a given acceleration; ∑F = ma • Weight is a force exerted on a body the pull of the earth. Bodies having large mass also have large weight.
The force that makes the body accelerate downward at 9.8 m/s2 is its weight. • A body with mass m has weight of magnitude of w w = mg • The magnitude w of a body’s weight is directly proportional to its mass m. • The weight of a body is a force, a vector quantity, w = mg
Caution: A body’s weight acts at all times
Example 4.6 Net force and acceleration in free fall • A one-euro coin was dropped from rest from the Leaning Tower of Pisa. If the coin falls freely, so that the effects of the air are negligible, how does the net force on the coin vary as it falls?
g, and hence weight, is only constant on earth, at sea level • On Earth, g depends on your altitude. • On other planets, gravity will likely have an entirely new value. • Mass doesn’t change, g and weight changes with location.
Measuring mass and weight • The easiest way to measure the mass of a body is to measure its weight, often by comparing with a standard. • Two bodies that have the same weight at a particular location also have the same mass. • The equal-arm balance can determine with great precision when the weights of two bodies are equal and hence their masses are equal. • In outer space, we can compute the mass as the ratio of force to acceleration.
Test your understanding of section 4.4 • Suppose an astronaut landed on a planet where g = 19.6 m/s2. Compare to Earth, would it be easier, harder, or just as earth for her to walk around? • Would it be easier, harder, or just as easy for her to catch a ball that is moving horizontally at 12 m/s (assume that the astronaut’s spacesuit is a light-weight model that doesn’t impede her movements in any way.)
10/18 Do now (ap 2009 exam) • The velocity v of an elevator moving upward between adjacent floors is shown as a function of time t in the graph. At which of the following times is the force exerted by the elevator floor on a passenger the least? Fnorm v (m/s) A. 1 s B. 3 s C. 4 s D. 5 s E. 6 s t (s) Fgrav 1 2 3 4 5 6 7 8 29% correct
4.5 Newton’s Third Law • A force acting on a body is always the result of its interaction with another body, so forces always comes in pairs. • Exerting a force on a body results in a force back upon you.
CAUTION: the two forces in an action-reaction pair act on different bodies. Unlike in Newton's 1st or 2nd Law, which involve the forces act on one body. The action and reaction forces can be contact forces or long-range forces. When you drop a ball, both the ball and the earth accelerate toward each other. The net force on each body ahs the same magnitude, but the earth’s acceleration is microscopically small because its mass is so great. Nevertheless it does move!
Example 4.8 which force is greater? • After your sports car breaks down, you start to push it to the nearest repair shop. While the car is starting to move, how does the force you exert on the car compare to the force the car exerts on you? How do these forces compare when your are pushing the car along at a constant speed? In both cases, the force you exert on the car is equal in magnitude and opposite in direction to the force the car exerts on you.
Example 4.9 applying Newton's 3rd law – object at rest An apple sits on a table in equilibrium. What forces act on it? What is the reaction force to each of the forces acting on the apple? What are the action-reaction pairs?
Example 4.10 applying Newton's 3rd law – object in motion • A stonemason drags a marble block across a floor by pulling on a rope attached to the block. The block may or may not be in equilibrium. How are the various force related? What are the action-reaction pairs?