C 5. Chapter 5 Newton s Laws: Force and Motion. Motion and Force in One Dimension. Objectives: Key Questions: Vocabulary. Unit.

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1 U2 Unit Motion and Force in One Dimension Chapter 5 Newton s Laws: Force and Motion Objectives: C 5 Chapter By the end of this chapter you should be able to: 4 Describe how the law of inertia affects the motion of an object. 4 Give an example of a system or invention designed to overcome inertia. 4 Measure and describe force in newtons (N) and pounds (lb). 4 Calculate the net force for two or more forces acting along the same line. 4 Calculate the acceleration of an object from the net force acting on it. 4 Determine whether an object is in equilibrium by analyzing the forces acting on it. 4 Draw a diagram showing an action-reaction pair of forces. 4 Determine the reaction force when given an action force. Key Questions: When do you encounter Newton s laws of motion in daily life? How are force, mass, and acceleration related? What are some common action-reaction force pairs? Vocabulary action force net force Newton s second law reaction dynamics law of inertia newton (N) Newton s third law statics equilibrium locomotion Newton s first law 99

2 Chapter 5 NEWTON S LAWS: FORCE AND MOTION 5.1 The First Law: Force and Inertia Sir Isaac Newton ( ), an English physicist and mathematician, is one of the most famous scientists who have ever lived. Before the age of 30, he made several important discoveries in physics and invented a whole new kind of mathematics calculus. The three laws of motion discovered by Newton are probably the most widely-used natural laws in all of science. Together, Newton s laws are the model which connects the forces acting on an object, its mass, and its resulting motion. This chapter is about Newton s laws, and the first section is about the first law, the law of inertia. Force Changing an object s motion Force is an action that can change motion Creating force Changes in motion only occur through force Suppose you want to move a box from one side of the room to the other. What would you do? Would you yell at it until it moved? Hey, box, get going! Move to the other side of the room! Of course not! You would push or pull it across the room. In physics terms, you would apply a force to the box. A force is what we call a push or a pull, or any action that has the ability to change an object s motion. Forces can be used to increase the speed of an object, decrease the speed of an object, or change the direction in which an object is moving. For something to be considered a force, it does not necessarily have to change the motion, but it must have the ability to do so. For example, if you push down on a table, it will probably not move. But if the legs were to break, the table could move. Therefore, your push qualifies as a force. Forces can be created by many different processes. For example, gravity creates force. Muscles can create force. The movement of air, water, sand, or other matter can create force. Electricity and magnetism can create force. Even light can create force. No matter how force is created, its effect on motion is always described by Newton s three laws. Forces create changes in motion, and there can be no change in motion without also having a force (Figure 5.1). Anytime there is a change in motion, a force must exist, even if you cannot immediately recognize the force. For example, when a rolling ball stops by hitting a wall, its motion changes rapidly. That change in motion is caused by the wall exerting a force that stops the ball. Figure 5.1: Force is the action which has the ability to change motion. Without force, the motion of an object cannot be started or changed. 100 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

3 NEWTON S LAWS: FORCE AND MOTION Chapter 5 Inertia Objects tend to keep doing what they are doing Newton s first law Consider that box you wish to move across the room. What if the box had been moving and you wanted to stop it? Again, yelling a command will not make it stop. The only way to stop the box is to apply enough force in a direction opposite to its motion. In general, objects tend to continue doing what they are already doing. If they are moving, they tend to keep moving, in the same direction, at the same speed. If they are at rest, they tend to stay at rest. This idea is known as Newton s first law of motion. Newton s first law states that an object will continue indefinitely in its current state of motion, speed, and direction, unless acted upon by a net force. Intuitively, you know that the motion of a massive object is harder to change than the motion of a lighter object. Inertia is a term used to measure the ability of an object to resist a change in its state of motion. An object with a lot of inertia takes a lot of force to start or stop; an object with a small amount of inertia requires a small amount of force to start or stop. Because inertia is a key idea in Newton s first law, the first law is sometimes referred to as the law of inertia. Which systems in a car overcome the law of inertia? The engine supplies force that allows you to change motion by pressing the gas pedal. The brake system is designed to help you change your motion by slowing down. Inertia is a property of mass Origin of the word inertia The amount of inertia an object has depends on its mass. More massive objects have more inertia than less massive objects. Recall that mass is a measure of the amount of matter in an object. Big trucks are made of more matter than small cars; thus, they have greater mass and a greater amount of inertia. It takes more force to stop a moving truck because it has more inertia than a small car. This is a common-sense application of the first law. The word inertia comes from the Latin word inertus, which can be translated to mean lazy. It can be helpful to think of things that have a lot of inertia as being very lazy when it comes to change. In other words, they want to maintain the status quo and keep doing whatever they are currently doing. The steering wheel and steering system is designed to help you change your motion by changing your direction. Can you think of three parts of a bicycle that are designed to overcome the law of inertia? 5.1 THE FIRST LAW: FORCE AND INERTIA 101

4 Chapter 5 NEWTON S LAWS: FORCE AND MOTION Applications of Newton s first law Seat belts and air bags Cup holders The tablecloth trick Two very important safety features of automobiles are designed with Newton s first law in mind: seat belts and air bags. Suppose you are driving down the highway in your car at 55 miles per hour when the driver in front of you slams on the brakes. You also slam on your brakes to avoid an accident. Your car slows down but the inertia of your body resists the change in motion. Your body tries to continue doing what it was doing traveling at 55 miles per hour. Luckily, your seat belt or air bag or both supplies a restraining force to counteract your inertia and slow your body down with the car (Figure 5.2). A cup holder does almost the same thing for a cup. Consider what happens if you have a can of soda on the dashboard. What happens to the soda can when you turn sharply to the left? Remember, the soda can was not at rest to begin with. It was moving at the same speed as the car. When your car goes left, the soda can s inertia causes it to keep moving forward (Figure 5.3). The result is quite a mess. Automobile cup holders are designed to keep the first law from making messes. Have you ever wondered how a magician is able to pull a tablecloth out from underneath dishes set on a table? It s not a trick of magic at all, but just physics. The dishes have inertia and therefore tend to resist changes in motion. Before the magician pulls on the cloth, the dishes are at rest. So when the tablecloth is whisked away, the inertia of the dishes keeps them at rest. This trick works best when the tablecloth is pulled very rapidly and the table is small. It would be quite difficult to perform this trick with the long table in the diagram. Can you think why the long table would make the trick hard to do? Figure 5.2: Because of its inertia, your body tends to keep moving when your car stops suddenly. This can cause serious injury if you are not wearing a seat belt. Figure 5.3: Because of its inertia, a soda can on the dashboard will tend to keep moving forward when the car turns left. 102 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

5 5.2 The Second Law: Force, Mass, and Acceleration Newton s discovery of the connection among force, mass, and acceleration was a milestone in our understanding of science. The second law is the most widely used equation in physics because it is so practical. This section shows you how to apply Newton s second law to practical situations. Newton s second law of motion Force is related to acceleration The acceleration of an object is equal to the force you apply divided by the mass of the object. This is Newton s second law, and it states precisely what you already know intuitively. If you apply more force to an object, it accelerates at a higher rate. If the same force is applied to an object with greater mass, the object accelerates at a lower rate because mass adds inertia. The rate of acceleration is the ratio of force divided by mass. NEWTON S LAWS: FORCE AND MOTION Chapter 5 Figure 5.4: An ice-skater can coast for quite a long time because motion at constant speed does not require force. If there was no friction, a skater could coast at constant speed forever. Force is required only to speed up, turn, or stop. Motion at constant speed Force is not necessary to keep an object in motion at constant speed. An ice-skater will coast for a long time without any outside force (Figure 5.4). However, the ice-skater does need force to speed up, slow down, turn, or stop. Recall that changes in speed or direction always involve acceleration. Force causes acceleration, and mass resists acceleration. 5.2 THE SECOND LAW: FORCE, MASS, AND ACCELERATION 103

6 Chapter 5 NEWTON S LAWS: FORCE AND MOTION The definition of force Pounds Newtons Converting newtons and pounds In the English system, the unit of force, the pound, was originally defined by gravity. One pound is the force of gravity pulling on a mass of kilograms. When you measure your weight in pounds on a bathroom scale, you are measuring the force of gravity acting on your mass. The metric definition of force depends on the acceleration per unit of mass. A force of one newton is exactly the amount of force needed to cause a mass of one kilogram to accelerate at one m/s 2. We call the unit of force the newton (N) because force in the metric system is defined by Newton s second law. The newton is a useful way to measure force because it connects force directly to its effect on matter and motion. A net force of one newton will always accelerate a 1-kilogram mass at 1 m/s 2, no matter where you are in the universe. The newton is a smaller unit of force than the pound. A force of one pound is equal to about newtons. This means a pound of force can accelerate a 1-kilogram mass at m/s 2. Pounds are fine for everyday use here on Earth but inconvenient for physics because of the conversion factor of What is force? The simplest concept of force is a push or a pull. On a deeper level, force is the action that has the ability to create or change motion. Pushes or pulls do not always change motion. But they could. The unit of force is derived from fundamental quantities of length, mass, and time. Using the second law, the units of force work out to be kg m/s 2. A force of 1 N causes a 1 kg mass to accelerate at 1 m/s 2. We could always write forces in terms of kg m/s 2. This would remind us what force is. But, writing kg m/s 2 everywhere would be a nuisance. Instead we use newtons. One newton (1 N) is 1 kg m/s UNIT 2 MOTION AND FORCE IN ONE DIMENSION

7 NEWTON S LAWS: FORCE AND MOTION Chapter 5 Using the second law of motion Net force The force (F) that appears in the second law is the net force. There are often many forces acting on the same object. Acceleration results from the combined action of all the forces that act on an object. When used this way, the word net means total. To solve problems with multiple forces, you have to add up all the forces to get a single net force before you can calculate any resulting acceleration. Calculating the acceleration of a cart on a ramp Three forms of the second law Units for the second law The second law can be rearranged three ways. Choose the form that is most convenient for calculating what you want to know. The three ways to write the law are summarized below. Table 5.1: Three forms of the second law Use... if you want to find... and you know... F a= M F=ma F M= a The acceleration (a) The net force (F) The mass (m) The net force (F) and the mass (m) The acceleration (a) and the mass (m) The acceleration (a) and the net force (F) To use Newton s second law in physics calculations, you must be sure to have units of m/s 2 for acceleration, newtons for force, and kilograms for mass. Many problems will require you to convert forces from pounds to newtons. Other problems may require you to convert weight in pounds to mass in kilograms. Remember also that m stands for mass in the formula for the second law. Do not confuse the variable m with the abbreviation m that stands for meters. A cart rolls down a ramp. Using a spring scale, you measure a net force of 2 newtons pulling the car down. The cart has a mass of 500 grams (0.5 kg). Calculate the acceleration of the cart. 1. You are asked for the acceleration (a). 2. You are given mass (m) and force (F). 3. Newton s second law applies. a = F m 4. Plug in numbers. Remember that 1 N = 1 kg m/s 2. a = (2 N) / (0.5 kg) = (2 kg m/s 2 ) / (0.5 kg) = 4 m/s THE SECOND LAW: FORCE, MASS, AND ACCELERATION 105

8 Chapter 5 NEWTON S LAWS: FORCE AND MOTION Finding the acceleration of moving objects Dynamics Direction of acceleration Positive and negative The sign of acceleration The word dynamics refers to problems involving motion. In dynamics problems, the second law is often used to calculate the acceleration of an object when you know the force and mass. For example, the second law is used to calculate the acceleration of a rocket from the force of the engines and the mass of the rocket. The acceleration is in the same direction as the net force. Common sense tells you this is true, and so does Newton s second law. Speed increases when the net force is in the same direction as the motion. Speed decreases when the net force is in the opposite direction as the motion. We often use positive and negative numbers to show the direction of force and acceleration. A common choice is to make velocity, force, and acceleration positive when they point to the right. Velocity, force, and acceleration are negative when they point to the left. You can choose which direction is to be positive, but once you choose, be consistent in assigning values to forces and accelerations. When solving problems, the acceleration always has the same sign as the net force. If the net force is negative, the acceleration is also negative. When both velocity and acceleration have the same sign, the speed increases with time. When velocity and acceleration have opposite signs, speed decreases with time. Careful use of positive and negative values helps keep track of the direction of forces and accelerations. Acceleration from multiple forces Three people are pulling on a wagon applying forces of 100 N, 150 N, and 200 N as shown. Determine the acceleration and the direction the wagon moves. The wagon has a mass of 25 kilograms. 1. You are asked for the acceleration (a) and direction. 2. You are given the forces (F) and mass (m). 3. The second law relates acceleration to force and mass (a = F m). 4. Assign positive and negative directions. Calculate the net force then use the second law to determine the acceleration from the net force and the mass. F = 100N 150 N + 200N = 50N a = ( 50 N) (25 kg) = 2 m/s 2 The wagon accelerates 2 m/s 2 to the left. 106 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

9 NEWTON S LAWS: FORCE AND MOTION Chapter 5 Finding force from acceleration Amount of force needed Forces that must have been Force on a tennis ball striking a racquet Newton s second law allows us to determine how much force is needed to cause a given acceleration. Engineers apply the second law to match the force developed by different engines to the acceleration required for different vehicles. For example, an airplane taking off from a runway needs to reach a certain minimum speed to be able to fly. If you know the mass of the plane, Newton s second law can be used to calculate how much force the engine must supply to accelerate the plane to take-off speed. The second law also allows us to determine how much force must have been present to cause an observed acceleration. Wherever there is acceleration there must also be force. Any change in the motion of an object results from acceleration. Therefore, any change in motion must be caused by force. When a tennis ball hits a racquet, it experiences high acceleration because its speed goes rapidly to zero then reverses direction. The high acceleration is evidence of tremendous forces between the racquet and the ball, causing the ball to flatten and the racquet strings to stretch. Newton s second law can be used to determine the forces acting on the ball from observations of its acceleration. A tennis ball contacts the racquet for much less than one second. High-speed photographs show that the speed of the ball changes from 30 m/s to +30 m/s in seconds. If the mass of the ball is 0.2 kg, how much force is applied by the racquet? 1. You are asked for force (F). 2. You are given the mass (m), the change in speed (v 2 v 1 ), and the time interval (t). 3. Newton s second law (a = F m) relates force to acceleration. Acceleration is the change in speed divided by the time interval over which the speed changed or a = (v 2 v 1 ) t. 4. Use the change in speed to calculate the acceleration. Use the acceleration and mass to calculate the force. a = (60 m/s) (0.006 s) = 10,000 m/s 2 F = (0.2 kg) (10,000 m/s 2 ) = 2,000 N. This force is equal to three times the weight of the tennis player and 1,000 times the weight of the tennis ball! Force to accelerate a plane taking off An airplane needs to accelerate at 5m/s 2 to reach take-off speed before reaching the end of the runway. The mass of the airplane is 5,000 kg. How much force is needed from the engine? 1. You asked for the force (F). 2. You are given the mass (m) and acceleration (a). 3. The second law applies. a = F m 4. Plug in the numbers. Remember that 1 N = 1 kg m/s 2. F = (5,000 kg) (5 m/s 2 ) = 25,000 N 5.2 THE SECOND LAW: FORCE, MASS, AND ACCELERATION 107

10 Chapter 5 NEWTON S LAWS: FORCE AND MOTION Finding forces when acceleration is zero Zero acceleration means zero net force Equilibrium Statics problem When acceleration is zero, the second law allows us to calculate unknown forces in order to balance other forces we know. Think about a gymnast hanging motionless from two rings (Figure 5.5). The force of gravity pulls down on the gymnast. The acceleration must be zero if he is not moving. The net force must also be zero because of the second law. The only way the net force can be zero is if the ropes pull upward with a force exactly equal and opposite the force of gravity pulling downward. If the weight of the gymnast is 700 newtons, then each rope exerts an upward force of +350 newtons. The condition of zero acceleration is called equilibrium. In equilibrium, all forces cancel out, leaving zero net force. Objects that are standing still are in equilibrium because their acceleration is zero. Objects that are moving at a constant speed and direction are also in equilibrium. A statics problem usually means there is no motion. Most statics problems involve using the requirement of zero net force, or equilibrium, to determine unknown forces. Engineers who design bridges and buildings solve statics problems to calculate how much force must be carried by cables and beams. The cables and beams can then be designed so that they safely carry the forces that are required. The net force is also zero for motion at constant speed. Constant speed problems are treated like statics problems as far as forces are concerned. A woman is walking two dogs on a leash. If each dog pulls with a force of 80 newtons, how much force does the woman have to exert to keep the dogs from moving? Figure 5.5: This gymnast is not moving so the net force must be zero. If the weight of the gymnast is 700 N, then each rope must pull upward with a force of 350 N in order to make the net force zero. A static force problem 1. You are asked for force (F). 2. You are given two 80 N forces and the fact that the dogs are not moving (a = 0). 3. Newton s second law says the net force must be zero if the acceleration is zero. 4. The woman must exert a force equal and opposite to the sum of the forces from the two dogs. Two times 80 N is 160 N, so the woman must hold the leash with an equal and opposite force of 160 N. 108 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

11 NEWTON S LAWS: FORCE AND MOTION Chapter The Third Law: Action and Reaction This section is about the often-repeated phrase For every action there is an equal and opposite reaction. This statement is known as Newton s third law of motion. Newton s first and second laws of motion discuss single objects and the forces that act on them. Newton s third law discusses pairs of objects and the interactions between them. This is because forces in nature always occur in pairs, like the top and bottom of a sheet of paper. You cannot have one without the other. Forces always occur in action-reaction pairs Moving in space is a problem Forces always come in pairs Forces on objects at rest Action-reaction pairs The astronauts working on the space station have a serious problem when they need to move around in space: There is nothing to push on. How do you move around if you have nothing to push against? The solution is to throw something opposite the direction you want to move. This works because all forces always come in pairs. If this seems like a strange idea, think through the following example. Suppose an astronaut throws a wrench. A force must be applied to the wrench to accelerate it into motion. The inertia of the wrench resists its acceleration. Because of its inertia, the wrench pushes back against the gloved hand of the astronaut. The wrench pushing on the astronaut provides a force that moves the astronaut in the opposite direction (Figure 5.6). Forces also come in pairs when objects are not moving. For example, consider this book. It is probably lying open on a table. The weight of the book exerts a force on the table, the same as it would exert on your hands if the book was resting on your hands. The table pushes back upward on the book with a force equal and opposite the book s weight. A chain of force pairs keeps going because the table pushes down on the floor and the floor pushes back up on the table (Figure 5.7). The floor pushes down on the walls and Earth pushes back up on the walls to hold up the floor. The two forces in a pair are called action and reaction. Anytime you have one, you also have the other. If you know the strength of one you also know the strength of the other since both forces are always equal. The two forces in an action-reaction pair always point in exactly opposite directions. They do not cancel each other because they act on different objects. Figure 5.6: An astronaut can move in space by throwing an object in the direction opposite where the astronaut wants to go. Figure 5.7: Forces always come in action-reaction pairs. The two forces in a pair are equal in strength and opposite in direction. 5.3 THE THIRD LAW: ACTION AND REACTION 109

12 Chapter 5 NEWTON S LAWS: FORCE AND MOTION Newton s third law of motion The first and second laws The third law operates on pairs of objects Action-reaction forces act on different objects Stopping action and reaction confusion Action and reaction The first and second laws apply to single objects. The first law states that an object will remain at rest or in motion at constant speed and direction until acted upon by an external force. The second law states that net force causes acceleration and mass resists acceleration. In contrast to the first two laws, the third law of motion applies to pairs of objects because forces always come in pairs. Newton s third law states that for every action force there has to be a reaction force that is equal in strength and opposite in direction. For example, to move on a skateboard you push your foot against the ground (Figure 5.8). The reaction force is the ground pushing back against your foot. The reaction force is what pushes you forward, because it is the force that acts on you. Your force against the ground pushes against Earth; however, the planet is so large that there is no perceptible motion resulting from your force. Action and reaction forces act on different objects, not on the same object. For example, the action-reaction pair that is required to move a skateboard in the traditional way includes your foot and Earth. Your foot pushing against the ground is the action force. The ground pushing back on your foot is the reaction force. The reaction force makes you move because it acts on you (Figure 5.8). Why doesn t your foot make the ground move? Simply because the force is too small to accelerate Earth s huge mass. Even though the reaction force that acts on you is the same size, you are much less massive than Earth. The same size reaction force is big enough to accelerate you. It is easy to get confused about action and reaction forces. People often ask, Why don t they cancel each other out? The reason is that the action and reaction forces act on different objects. The action force of your foot acts on Earth and Earth s reaction force acts on you. The forces cannot cancel because they act on different objects. It does not matter which is the action force and which is the reaction. Whichever force you call the action makes its counterpart the reaction. The important thing is to recognize which force acts on which object (Figure 5.9). To apply the second law properly, you need to identify the forces acting on the object for which you are trying to find the acceleration. Figure 5.8: All forces come in pairs. When you push on the ground (action), the reaction of the ground pushes back on your foot. Figure 5.9: It does not matter which force you call the action or the reaction. The action and reaction forces are interchangeable. 110 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

13 Solving problems with action-reaction forces Thinking about which force is acting on which object In many physics problems, you are asked to determine the acceleration of a moving object from the forces acting on it. In the last section, you learned that the net force is the total of all forces acting on an object. Very often one of the forces will be a reaction force to a force created by the object. For example, consider a small cart attached to a spring (Figure 5.10). When you push the spring against a wall, a force is created. When you let the cart go, the force from the spring accelerates the cart away from the wall. But the force from the spring is pushing on the wall, so what force accelerates the cart? The answer is the reaction force of the wall pushing back on the spring. A force created by an object cannot accelerate the object itself, but the reaction force can. NEWTON S LAWS: FORCE AND MOTION Chapter 5 Determining the reaction forces from people pushing a cart Three people are each applying 250 newtons of force to try to move a heavy cart. The people are standing on a rug. Someone nearby notices that the rug is slipping. How much force must be applied to the rug to keep it from slipping? Sketch the action and reaction forces acting between the people and the cart and between the people and the rug. Figure 5.10: Analyzing the action and reaction forces for a cart launched off a wall by a spring. 1. You are asked for how much force (F) it takes to keep the rug from slipping. 2. You are given that three forces of 250 N each are being applied. 3. The third law says that each of the forces applied creates a reaction force. 4. Each person applies a force to the cart and the cart applies an equal and opposite force to the person. The force on the rug is the sum of the reaction forces acting on each person. The total force that must be applied to the rug is 750 N in order to equal the reaction forces from all three people. 5.3 THE THIRD LAW: ACTION AND REACTION 111

14 Chapter 5 NEWTON S LAWS: FORCE AND MOTION Locomotion Locomotion In the water In the air The natural jet engine in a squid The act of moving or the ability to move from one place to another is called locomotion. Any animal or machine that moves depends on Newton s third law to get around. When we walk, we push off the ground and move forward because of the ground pushing back on us in the opposite direction. When something swims, it pushes on water and the water pushes back in the opposite direction. As a result, the animal, submarine, or even microscopic organism moves one way, and a corresponding amount of water moves in the opposite direction. The movement of a boat through water results from a similar application of Newton s third law. When a lone paddler in a kayak exerts an action force pushing the water backward, the reaction force acts on the paddle, pushing the paddle and the kayak forward (Figure 5.11). Whether insect, bat, bird, or machine, any object that flies under its own power moves by pushing the air. Living creatures flap their wings to push air, and the air pushes back, propelling them in the opposite direction. Jets, planes, and helicopters push air, too. In the specific example of a helicopter, the blades of the propeller are angled such that when they spin, they push the air molecules down (Figure 5.12). According to Newton s third law, the air molecules push back up on the spinning blades and lift the helicopter. Squid use jet propulsion to move quickly. A squid fills a large chamber in its body with water. The chamber has a valve the squid can open and close. To move quickly, the squid squeezes the water inside its body with powerful muscles, then opens the chamber valve and shoots out a jet of water. The squid moves with a force equal and opposite in direction to the water jet that leaves its body. Figure 5.11: Action and reaction forces for a kayak moving through the water. Figure 5.12: Action and reaction forces on a helicopter. 112 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

15 Biomechanics Biomechanics is the science of how physics is applied to muscles and motion. Many athletes use principles of biomechanics to improve their performance. People who design sports equipment use biomechanics to achieve the best performance by matching the equipment design to the athlete s body. Physicians, carpenters, people who build furniture, and many others also use biomechanics in their work. Any machine that relies on forces from the human body also relies on biomechanics. The force platform A force platform is a very sophisticated scale that can record how forces change over time. Instead of containing springs, as your bathroom scale might, a force platform contains strain gauges (Figure 5.13). When a person steps or jumps on the platform, each strain gauge produces a reaction force and also a signal proportional to the strength of the reaction force. The force readings given off by the platform are referred to as ground reaction forces or GRFs. Measuring force in three directions Who uses force platforms? The center of pressure There are usually 12 strain gauges, three in each corner of a force platform oriented along the x-, y-, and z-axes. When force is applied to the platform, electrical signals from all 12 strain gauges are sent to a computer. The computer converts the signals to 12 separate force readings. From these readings, data is generated regarding the magnitude, direction, and sequence of GRFs being produced. Based on the relative magnitude of forces on each gauge, the center of pressure, or location of the force, can also be calculated. Force platforms are used in many different fields including medicine and athletics. Physicians, technicians, and therapists use force platforms in clinical settings to help in the diagnosis and rehabilitation of walking disorders. Biomechanists, including athletic trainers, use force platforms for research and to help athletes improve their technique. Equipment designers and manufacturers use information from force platforms in the design of sports equipment such as running shoes. The center of pressure is the place on your foot at which the average force is exerted against the ground. The center of pressure moves as your foot changes its contact point with the ground during walking or running. Force platform analysis is often used to evaluate the differences caused by various types of shoes, different track surfaces, walking versus running, and changes in gait patterns before and after surgery (Figure 5.14). Figure 5.13: A force platform has 12 strain gauges arranged to measure forces in the x, y, and z directions at each of the corners. Figure 5.14: The center of pressure for two runners with different running styles. 5.3 THE THIRD LAW: ACTION AND REACTION 113

16 Force from a vertical jump Jumping is a common sports skill Measuring the forces from a vertical jump The force versus time curve Other characteristics of jumping motion Other biomechanical techniques The vertical jump is a common sport skill. Vertical jumps are seen in many different sports including basketball, volleyball, soccer, football, baseball, and tennis. A force platform makes an excellent tool to analyze the forces between the jumper s foot and the floor. To start the experiment, the athlete stands motionless in the middle of the platform (Figure 5.15). The standing still data is used to measure the weight of the athlete. That weight is converted to mass, using the second law (m = F g). The mass data is stored for later use. When given the command by the researcher, the athlete bends and jumps as high as possible. The force platform measures the force from each strain gauge at a rate of 250 to 1,000 measurements per second. The biomechanist uses the data to generate a force versus time graph. A typical force versus time curve for a vertical jump is shown at the bottom of Figure The total force recorded is the combination of the athlete s weight and the force produced during the jump. In this case, the athlete weighs 550 newtons. The peak GRF recorded is approximately 1,340 newtons. The time from the start of the jump until the athlete leaves the platform is just about one second. Once the athlete takes off and no longer touches the force table, the force readings drop to zero until the athlete lands back on the platform. The total time that the force is zero corresponds to the time in the air, a piece of information that allows other calculations to be made later. The force table data can be used to calculate many characteristics of the jumping motion. The total energy used can be calculated, as well as the maximum height reached. The force generated by the athlete s legs can also be determined along with maximum acceleration, and the balance of force between right and left legs. The technique of electromyography monitors the nerve signals to muscles and can determine the relative strength and sequence of contractions in the muscles being used in jumping. When combined with video equipment, the force, position, and time data can give a complete analysis of the motion that an athlete can watch to improve or evaluate her technique. Figure 5.15: A force platform can be used to measure the vertical force exerted during a vertical jump. 114 UNIT 2 MOTION AND FORCE IN ONE DIMENSION

17 NEWTON S LAWS: FORCE AND MOTION Chapter 5 Chapter 5 Assessment Vocabulary Select the correct term to complete the sentences. force inertia law of inertia Newton s first law net force dynamics equilibrium statics Newton s second law locomotion newton (N) action reaction Newton s third law 1. The measure of the ability of an object to resist a change in its state of motion is called. 2. Any action which has the ability to change an object s motion must be a(n). 3. Newton s first law is sometimes referred to as the. 4. The law of motion which states that all objects tend to resist changes in motion is known as. 5. A problem in which there is no motion is described as. 6. The acceleration of an object is proportional to the net force applied to the object according to. 7. The total force applied to an object as the result of many applied forces is called the. 8. A(n) problem is one involving motion. 9. The force needed to accelerate a mass of one kilogram at a rate of 1 m/s 2 is one. 10. When the net force acting on an object is zero, the condition of zero acceleration is called. 11. Forces always occur in pairs. One force is called the force, the other is called the force. 12. For every action force there is an equal and opposite reaction force, is a statement of. 13. The act of moving can be called. Concept review 1. A glass of milk sits motionless on the kitchen table. a. Describe the forces acting on the glass of milk. Include their direction in the description. b. What word describes the state of motion of the glass of milk? 2. Name two units commonly used to measure force. How are they related? 3. Are the following statements true or false? Explain your answers using an example. a. Applying a force to an object will make it move. b. To keep an object moving, a force must be applied. c. A force must be applied to change the direction of a moving object. 4. To tighten the head of a hammer on its handle, it is banged against a surface as shown to the right. Explain how Newton s first law is involved. 5. How can rolling a bowling ball help you to determine the amount of matter in the ball? 6. List at least three parts of an automobile that are designed to overcome the effects of Newton s first law. Briefly explain the function of each. 7. State Newton s second law in words. Write an equation expressing the law. 8. Explain how the unit of force used by scientists, the newton, is defined. 9. In a space shuttle orbiting Earth, where objects are said to be weightless, an equal arm balance could not be used to measure the mass of an object. How could you measure the mass of an object in this situation? 10. Explain the difference between mass and weight. State common units for each. 11. What is the difference between the terms force and net force? 12. In physics problems, velocities, accelerations, and forces often appear with positive (+) or negative ( ) signs. What do those signs indicate? 13. How does the sign of the force applied to an object compare with the sign of the acceleration? 14. What do motionless objects have in common with objects that are moving in a straight line with constant speed? CHAPTER 5 ASSESSMENT 115

18 Chapter 5 NEWTON S LAWS: FORCE AND MOTION 15. What is the difference between dynamics problems and statics problems? Give an example of each. 16. You and your little 6-year-old cousin are wearing ice skates. You push off each other and move in opposite directions. How does the force you feel during the push compare to the force your cousin feels? How do your accelerations compare? Explain. 17. You jump up. Earth does not move a measurable amount. Explain this scenario using all three of Newton s laws of motion. Problems 1. The box pictured is being pulled to the right at constant speed along a level surface. a. Draw a diagram with arrows to represent the size and direction of all the forces acting on the box. b. Draw a diagram with one arrow to represent the size and direction of the net force acting on the box. 2. Calculate a. the weight in pounds of a 16-newton object. b. the weight in newtons of a 7-pound object. c. the weight in newtons of a 3-kilogram object on Earth. d. the mass in kilograms of an object that weighs 12 newtons on Earth. 3. How does the inertia of a 200-kg object compare to the inertia of a 400-kg object? 4. A constant force is applied to a cart, causing it to accelerate. If the mass of the cart is tripled, what change occurs in the acceleration of the cart? 5. If the net force acting on an object is tripled, what happens to its acceleration? 6. On Venus, the acceleration due to gravity is 8.86 m/s 2. What is the mass of a man weighing 800 N on the surface of that planet? 7. A 60-kilogram boy on a 12-kilogram bicycle rolls downhill. What net force is acting on the boy and his bicycle if he accelerates at a rate of 3.25 m/s 2? 8. A young girl whose mass is 30 kilograms is standing motionless on a 2-kg skateboard holding a 7-kg bowling ball. She throws the ball with an average force of 75 N. a. What is the magnitude of her acceleration? b. What is the magnitude of the acceleration of the bowling ball? 9. On Mercury, a person with a mass of 75 kg weighs 280 N. What is the acceleration due to gravity on Mercury? 10. A baseball player strikes the ball with his 1-kg bat. The bat applies an average force of 500 N on the 0.15-kg baseball for 0.20 seconds. a. What is the force applied by the baseball on the bat? b. What is the acceleration of the baseball? c. What is the speed of the baseball at the end of the 0.20 seconds? 11. The graph represents the motion of a 1,500-kg car over a 20-second interval. a. During which interval(s) is the net force on the car zero? b. What force is being applied to the car during interval C D? 12. Two forces are applied to a 2-kilogram block on a frictionless horizontal surface as shown in the diagram. Calculate the acceleration of the block. Applying your knowledge 1. When a sled pulled by a horse accelerates, Newton s third law says that the sled and horse exert equal and opposite forces on each other. If the horse and sled apply equal but opposite forces on one another, explain how the sled can be accelerated under these circumstances. 2. A bowling ball is positioned near the front of a stationary wagon. If the wagon is suddenly pulled forward, the bowling ball appears to move backwards in the wagon. Use each of Newton s three laws to explain what is actually happening to the wagon-ball system. 3. A 0.1-kilogram ball held at waist height is dropped and bounces back up toward the student s hand. Include all of the words from the chapter vocabulary list in describing the motion of the ball. 116 UNIT 2 MOTION AND FORCE IN ONE DIMENSION