Lecture Notes Chapter 4.pdf
- 1. Lecture Notes Chapter 4: Forces & Newton’s Laws Introduction to Dynamics: Newton’s Laws of Motion The study of motion is kinematics, but kinematics only describes the way objects move – their velocity and their acceleration. Dynamics considers the forces that affect the motion of moving objects and systems. Galileo was instrumental in establishing observation as the absolute determinant of truth, rather than “logical” argument. 4.1 Development of Force Concept Dynamics is the study of the forces that cause objects and systems to move. Force is defined as a push or pull, which has both magnitude and direction, therefor it is a vector. Since force is a vector, it adds just like other vectors. (b) Free-body diagram is a technique used to illustrate all the external forces acting on a body. The body is represented by a single isolated point (or free body), and only those forces acting on the body from the outside (external forces) are shown – these forces are the only ones shown, because only external forces acting on the body affect its motion. Take-Home Experiment: Force Standards, p. 131 To investigate force standards and cause and effect, get two identical rubber bands.
- 2. Hang one rubber band vertically on a hook. Find a small household item that could be attached to the rubber band using a paper clip, and use this item as a weight to investigate the stretch of the rubber band. Measure the amount of stretch produced in the rubber band with one, two, and four of these (identical) items suspended from the rubber band. What is the relationship between the number of items and the amount of stretch? How large a stretch would you expect for the same number of items suspended from two rubber bands? What happens to the amount of stretch of the rubber band (with the weights attached) if the weights are also pushed to the side with a pencil? 4.2 Newton’s First Law of Motion: Inertia Define mass and inertia Understand Newton’s first law of motion. Newton’s First Law of Motion A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net external force. We can think of this law as preserving the status quo of motion. This law states that these must be a cause (which is a net external force) for there to be any change in velocity (either a change in magnitude or direction). An object sliding across a table or floor slows down due to the net force of friction acting on the object. If friction disappeared, would the object still slow down? The idea of cause and effect is crucial in accurately describing what happens in various situations. The object would not slow down at all if friction were completely eliminated. If we know enough about the friction, we can accurately predict how quickly the object will slow down. Friction is an external force. Mass The property of a body to remain at rest or to remain in motion with constant velocity is called inertia. Newton’s first law is often called the law of inertia. The inertial of an object is measured by its mass. Mass is a measure of the amount of matter in something. Unlike weight, mass does NOT vary with location.
- 3. CHECK YOUR UNDERSTANDING Which has more mass: a kilogram of cotton balls or a kilogram of gold? Solution: They are equal. A kilogram of one substance is equal in mass to a kilogram of another substance. The quantities that might differ between them are volume and density. 4.3 Newton’s Second Law of Motion: Concept of a System Define net force, external force, and system. Understand Newton’s second law of motion. Apply Newton’s second law to determine the weight of an object. A change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration. Newton’s first law says that a net external force causes a change in motion; thus, a net external force causes acceleration. What do we mean by an external force? An external force acts from outside the system of interest. Only external forces affect the motion of a system, according to Newton’s first laws Different forces exerted on the same mass produce different accelerations. (The internal forces actually cancel.) You must define the boundaries of the system before you can determine which forces are external. (a) The system of interest is the wagon plus the child in it. The two forces exerted by the other children are external forces. The force the child in the wagon exerts to hang onto the wagon is an internal force between elements of the system of interest. The weight W of the system and the support of the ground N are assumed to cancel. The vector f represents the friction action on the wagon, and it acts to the left, opposing the motion of the wagon. (b) All of external forces acting on the system add together to produce a net force, Fnet.
- 4. The free-body diagram shows all of the forces acting on the system of interest. The dot represents the center of mass of the system. Each force vector extends from this dot. Because there are two forces acting to the right, we draw the vectors collinearly. (c) A larger net external force produces a larger acceleration (a’ > a) when an adult pushes the child. Acceleration is directly proportional to and in the same direction as the net (total) external force acting on a system. The same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direction comparison of the two situations. The acceleration of an object depends only on the net external force and the mass of the object. Combining the two proportionalities just given yields Newton’s second law of motion.
- 5. Newton’s Second Law of Motion The acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass. In equation form, Newton’s second law of motion is This is often written in the more familiar form When only the magnitude of force and acceleration are considered, this equation is simply Although these last two equations are really the same, the first gives more insight into what Newton’s second law means. The law is a cause and effect relationship among three quantities that is not simply based on their definitions. The validity of the second law is completely based on experimental verification. Units of Force 1N = 1 kg ∙ m/s2 = 0.225 lb. Weight and the Gravitation Force When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight w. Weight can be denoted as a vector w because it has a direction; down is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is denoted as w. Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration g. Using Galileo’s result and Newton’s second law, we can derive an equation for weight.
- 6. Consider an object with mass m falling downward toward Earth. It experience only the downward force of gravity, which has magnitude w. Newton’s second law states that the magnitude of the net external force on an object is Fnet = ma. Since the object experiences only the downward force of gravity, Fnet = w. The acceleration of an object due to gravity is g, or a = g. Substituting these into Newton’s second law gives When the net external force on an object is its weight, we say that it is in free-fall. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air action on the object. The acceleration due to gravity g varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. The broadest definition of weight is that the weight of an object is the gravitational force on it from the nearest large body, such as Earth, the Moon, the Sun, and so on. It is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much “stuff”) and does not vary in classical physics. Weight is the gravitational force and DOES vary depending on gravity. EXAMPLE 4.1 What Acceleration Can a Person Produce when Pushing a Lawn Mower? Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N (about 11 lb) parallel to the ground. The mass of the mower is 24 kg. What is its acceleration? Solution: The magnitude of the acceleration a is . Entering known values gives
- 7. Substituting the units for N yields Discussion: The direction of the acceleration is the same direction as that of the net force, which is parallel to the ground. The force exerted by the person pushing the mower must be greater than the friction opposing the motion (since we know the mower moves forward) and the vertical forces must cancel if there is to be no acceleration in the vertical direction (the mower is moving only horizontally). EXAMPLE 4.2 What Rocket Thrust Accelerates This Sled? (p. 136) Prior to manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on one or two rails and propelled by several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust T, for the four-rocket propulsion system in the figure below. The sled’s initial acceleration is 49 m/s2 , the mass of the system is 2100 kg, and the force of friction opposing the motion is known to be 650 N. A sled experiences a rocket thrust that accelerates it to the right. Each rocket creates an identical thrust T. As in other situations where there is only horizontal acceleration, the vertical forces cancel – the ground exerts an upward force N on the system that is equal in magnitude and opposite in direction to its weight, W. The system here is the sled, its rockets, and rider, so none of the forces between these objects are considered. The arrow representing friction (f) is drawn larger than scale.
- 8. Strategy: Although there are forces acting vertically and horizontally, we assume the vertical forces cancel since there is not vertical acceleration. Solution: Since we have defined the direction of the force and acceleration as acting “to the right,” we need to consider only the magnitudes of these quantities in the calculations. , where (while the engine thrusts add, friction opposes the thrust). Substituting this into Newton’s second law gives Then Substituting known values yields Newton’s second law of motion is more than a definition; it is a relationship among acceleration, force, and mass. It can help us make predictions. Each of those physical quantities can be defined independently, so the second law tells us something basic and universal about nature. 4.4 Newton’s Third Law of Motion: Symmetry in Forces Understand Newton’s third law of motion. Apply Newton’s third law to define systems and solve problems of motion. This law represents a certain symmetry in nature: Forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself. We sometimes refer to this law loosely as “action-reaction,” where the force exerted is the action and the force experienced as a consequence is the reaction. Newton’s third law has practical uses in analyzing the origin of forces and understanding which forces are external to a system. Consider a swimmer pushing off from the side of a pool as illustrated in the figure below. She pushes against the pool wall with her feet and accelerates in the direction opposite to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems.
- 9. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then is an external force on this system and affects its motion. The swimmer moves in the direction of . In contrast, the force acts on the wall and not on our system of interest. Thus does not directly affect the motion of the system and does not cancel . Note that the swimmer pushes in the direction opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction. When the swimmer exerts a force on the wall, she accelerates in the direction opposite to that of her push. The net external force on her is in the direction opposite to . This opposition occurs because, in accordance with Newton’s third law of motion, the wall exerts a force on her, equal in magnitude but in the direction opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that does not act on this system (the swimmer) and thus, does not cancel . Thus the free-body diagram shows only . w, the gravitational force, and BF, the buoyant force of the water supporting the swimmer’s weight. The vertical forces w and BF cancel since there is not vertical motion.
- 10. Other Examples of Newton’s third law As a professor paces in front of a whiteboard, she exerts a force backward on the floor. The floor exerts a reaction force forward on the professor that causes her to accelerate forward. A car accelerates because the ground pushes forward on the drive wheels in reaction to the drive wheels pushing backward on the ground. o You can see evidence of the wheels pushing backward when tires spin on a gravel road and throw rocks backward. Rockets move forward by expelling gas backward at high velocity. o The rocket exerts a large backward force on the gas in the rocket combustion chamber, and the gas therefore exerts a large reaction force forward on the rocket. o This reaction force is called thrust. o It is a common misconception that rockets propel themselves by pushing on the ground or on the air behind them – they actually work better in vacuum, where they can more easily expel the exhaust gases. Helicopters similarly create lift by pushing air down, thereby experiencing an upward reaction force. Birds and airplanes also fly by exerting force on air in a direction opposite to that of whatever force they need. o The wings of a bird force air downward and backward in order to get lift and move forward. An octopus propels itself in the water by ejecting water through a funnel from its body, similar to a jet ski. Professional cage fighters experience reaction forces when they punch, sometimes breaking their hand by hitting an opponent’s body. EXAMPLE 4.3 Getting Up to Speed: Choosing the Correct System A physics professor pushes a cart of demonstration equipment to a lecture hall, as seen in the figure below. Her mass is 65.0 kg, the cart’s is 12.0 kg, and the equipment’s is 7.0 kg. Calculate the acceleration produced when the
- 11. professor exerts a backward force of 150 N on the floor. All forces opposing the motion, such as friction on the cart’s wheels and air resistance, total 24.0 N. Solution: Since they accelerate as a unit, we define the system to be the professor, cart, and equipment (System 1 in the above figure). The professor pushes backward with a force Ffoot of 150 N. According to Newton’s third law, the floor exerts a forward reaction Ffloor of 150 N on System 1. Because all motion is horizontal, we can assume there is no net force in the vertical direction. The problem is therefore one-dimensional along the horizontal direction. o f opposes the motion and is thus in the opposite direction of Ffloor. We do not include the forces Fprof or Fcart because these are internal forces, and we do not include Ffoot because it acts on the floor, not on the system. Newton’s second law is given by The net external force on System 1 is deduced from the figure and the discussion above to be The mass of the System 1 is So the acceleration is EXAMPLE 4.4 Force on the Cart – Choosing a New System Calculate the force the professor exerts on the cart in the same figure. Solution: If we define the system of interest to be the cart plus equipment (System 2 in the figure above), then the net external force on System 2 is the force the professor exerts on the cart minus friction. The force she exerts on the car, Fprof, is an external force acting on System 2. Fprof was internal to System 1, but it is external to System 2 and will enter Newton’s second law for System 2.
- 12. Newton’s second law can be used to find Fprof, starting with The magnitude of the net external force on System 2 is Solve for Fprof, The value of f is given as 24.0 N, and we can calculate Fnet since both the acceleration and mass of System 2 are known. So This force is significantly less than the 150-N force the professor exerted backward on the floor. Not all of that 150-N force is transmitted to the cart; some of it accelerates the professor. 4.5 Normal, Tension, and Other Examples of Forces Define normal and tension forces. Apply Newton’s laws of motion to solve problems involving a variety of forces. Use trigonometric identities to resolve weight into components. NORMAL FORCE Weight (force of gravity) is a pervasive force that acts at all times and must be counteracted to keep an object from falling. But how do inanimate objects like a table support the weight of a mass placed on them? Unless the object is deformed beyond its limit, it will exert a restoring force much like a deformed spring (or trampoline or diving board). The greater the deformation, the greater the restoring force. So when the load is placed on the table, the table sags until the restoring force
- 13. becomes as large as the weight of the load. At this point the net external force on the load is zero. We must conclude that whatever supports a load, be it animate or not, must supply an upward force equal to the weight of the load. If the force supporting a load is perpendicular to the surface of contact between the load and its support, this force is defined to be a normal force (FN). The word normal means perpendicular to a surface. The normal force can be less than the object’s weight if the object is on an incline. EXAMPLE 4.5 Weight on an Incline, a Two-Dimensional Problem Consider the skier on a slope shown in the figure. Her mass including equipment is 60.0 kg. (a) What is her acceleration if friction is negligible? (b) What is her acceleration if friction is known to be 45.0 N? Solution: Since motion and friction are parallel to the slope, it is most convenient to project all forces onto a coordinate system where one axis is parallel to the slope and the other is perpendicular. FN (N) is perpendicular to the slope and f is parallel to the slope, but W has components along both axes, FN (N) is equal in magnitude to , so there is no motion perpendicular to the slope, but f is less than , so that there is a downslope acceleration (along the parallel axis). The magnitude of the component of the weight parallel to the slope is , and the magnitude of the component of the weight perpendicular to the slope is
- 14. (a) Neglecting friction. Since the acceleration is parallel to the slope, we need only consider forces parallel to the slope. (Forces perpendicular to the slope add to zero, since there is no acceleration in that direction.) The forces parallel to the slope are the amount of the skier’s weight parallel to the slope and friction f. Using Newton’s second law, with subscripts to denote the quantities parallel to the slope , where Assuming no friction for this part, so the acceleration is (b) Including friction. We now have a given value for friction and its direction is parallel to the slope and it opposes motion between surfaces in contact. So the net force is now And substituting this into Newton’s second law, gives Substituting known values gives Which yields which is the acceleration parallel to the incline when there is 45.0 N of opposing friction. DISCUSSION Since friction always opposes motion between surfaces, the acceleration is smaller when there is friction than when there is none. It is a general result that if friction on an incline is negligible, then the acceleration down the incline is
- 15. This is related to the fact that all objects fall with the same acceleration in the absence of air resistance. Similarly, all objects, regardless of mass, slide down a frictionless incline with the same acceleration (if the angle is the same). RESOLVING WEIGHT INTO COMPONENTS An object rests on an incline that makes an angle 𝜃 with the horizontal. The perpendicular force of weight, is typically equal in magnitude and opposite in direction to the normal force, FN (N). The force acting parallel to the plane, , causes the object to accelerate down the incline. The force of friction, f, opposes the motion of the object, so it acts upward along the plane. TENSION A tension is a force along the length of a medium, especially a force carried by a flexible medium, such as a rope or cable. The word “tension” comes from a Latin word meaning “to stretch”. Not coincidentally, the flexible cords that carry muscle forces to the parts of the body are called tendons. Any flexible connector, such as a string, rope, chain, wire, or cable, can exert pulls only parallel to its length; thus, a force carried by a flexible connector is a tension with direction parallel to the connect. Tension is a pull in a connector – “You can’t push a rope.” The tension force pulls outward along the two ends of a rope.
- 16. When a perfectly flexible connector (one requiring no force to bend it) such as this rope transmits a force T, that force must be parallel to the length of the rope, as shown. The pull such a flexible connect exerts is a tension. The rope pulls with equal force but in opposite directions on the hand and the supported mass (neglecting the weight of the rope). This is an example of Newton’s third law. The rope is the medium that carries the equal and opposite forces between the two objects. The tension anywhere in the rope between the hand and the mass is equal – once you have determined the tension in one location, you have determined the tension at all locations along the rope. Tension in the rope must equal the weight of the supported mass. If the 5.00-kg mass in the figure is stationary, then its acceleration is zero, and thus Fnet = 0. The only external forces acting on the mass are its weight w and the tension T supplied by the rope. where T and w are the magnitude of the tension and weight. For a 5.00 kg-mass, neglecting the mass of the rope, EXAMPLE 4.6 What is the Tension in a Tightrope? Calculate the tension in the wire supporting the 70.0-kg tightrope walker shown in the figure in the left: The weight of a tightrope walker causes a wire to sag by 5.0 degrees. The system of interest here is the point in the wire at which the tightrope walker is standing. The wire is bent under the person’s weight.
- 17. Thus, the tension on either side of the person has an upward component that can support the weight. The system is the tightrope walker, and the only external forces acting on him are his weight w and the two tensions TL (left tension) and TR (right tension). It is reasonable to neglect the weight of the wire itself. The net external force is zero since the system is stationary. The magnitudes of the tensions TL and TR must be equal to cancel each other out because there is no horizontal acceleration in the rope. Solution: When the vectors are projected onto vertical and horizontal axes, their components along those axes must add to zero, since the tightrope walker is stationary. The small angle results in T being much greater than w. The horizontal components of the forces (denoted with subscript x) The net horizontal force since the person is stationary. Thus Using trigonometry, Equating TLx and TRx:
- 18. Thus, . Considering vertical components (denoted by a subscript y), we can solve for T. Again, since the person is stationary, according to Newton’s second law, the net Fy = 0. Thus, Substituting the values for TLy and TRy, into the net force equation in the vertical direction: So DISCUSSION The vertical tension in the wire acts as a normal force that supports the weight of the tightrope walker. The tension is almost six times the 686-N weight of the tightrope walker. Since the wire is nearly horizontal, the vertical component of its tension is only a small fraction of the tension in the wire. The large horizontal components are in opposite directions and cancel, and so most of the tension in the wire is not used to support the weight of the tightrope walker.
- 19. If we wish to create a very large tension, all we have to do is exert a force perpendicular to a flexible connector. The tension in the rope related to the weight of the tightrope walker in the following way: When a perpendicular force is exerted at the middle of a flexible connector: Tension becomes very large as 𝜃 approaches zero. Even the relatively small weight of any flexible connector will cause it to sag, since an infinite tension would results if it were horizontal (i.e., 𝜃= 0 and sin 𝜃 = 0). 4.6 Problem-Solving Strategies 4.7 Further Applications of Newton’s Laws of Motion 4.8 Extended Topic: The Four Basic Forces –An Introduction Understand the four basic forces that underline the processes in nature.