Chapter 5. Force and Motion I 5.1. What is Physics? 5.2. Newtonian Mechanics 5.3. Newton's First Law 5.4. Force 5.5. Mass 5.6. Newton's Second Law 5.7. Some Particular Forces 5.8. Newton's Third Law 5.9. Applying Newton's Laws What Causes Acceleration? Dynamics—the study of causes of motion. The central question in dynamics is: What causes a body to change its velocity or accelerate as it moves? Newtonian Mechanics Newton’s laws fail in the following two circumstances: • 1. When the speed of objects approaches (1% or more) the speed of light in vacuum (c = 8×108 m/s). In this case we must use Einstein’s special theory of relativity (1905). • 2. When the objects under study become very small (e.g., electrons, atoms, etc.). In this case we must use quantum mechanics (1926). Force A force is a push or a pull. Force is a vector. All forces result from interaction. •Contact forces: forces that arise from the physical contact between two objects. •Noncontact forces: forces the two objects exert on one another even though they are not touching. • External forces include only the forces that the environment exerts on the object of interest. • Internal forces are forces that one part of an object exerts on another part of the object. Combining Forces Principle of superposition for forces When two or more forces act on a body, we can find their net force or resultant force by adding the individual forces as vectors taking direction into account. Note: The net force involves the sum of external forces only (internal forces cancel each other). Exercise 1 The figures that follow show overhead views of four situations in which two forces acting on the same cart along a frictionless track. Rank the situations according to the magnitudes of the net force on the cart, greatest first. Newton’s FIRST LAW Newton's First law: Consider a body on which no net force acts. If the body is at rest, it will remain at rest. If the body is moving, it will continue to moving with a constant velocity. – “Net force” is crucial. Often, several forces act simultaneously on a body, and the net force is the vector sum of all of them – An inertial reference frame is the one has zero acceleration. – All newton’s laws are valid only in the inertia reference frames. Mass • The larger the mass, the harder is to cause its motion • Mass and weight are different concepts DEFINITION OF INERTIA AND MASS 1. Inertia is the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line. 2. The mass of an object is a quantitative measure of inertia. SI Unit of Inertia and Mass: kilogram (kg) NEWTON’S SECOND LAW OF MOTION When a net external force F acts on an object of mass m, the acceleration a that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the = same as the direction of the net force. F a SI Unit of Force: kg·m/s2 =newton (N) m Only External forces are considered in the Newton’s second law. Example 1 Pushing a Stalled Car Two people are pushing a stalled car, as Figure 4.5a indicates. The mass of the car is 1850 kg. One person applies a force of 275 N to the car, while the other applies a force of 395 N. Both forces act in the same direction. A third force of 560 N also acts on the car, but in a direction opposite to that in which the people are pushing. This force arises because of friction and the extent to which the pavement opposes the motion of the tires. Find the acceleration of the car. Example 2 Hauling a Trailer A truck is hauling a trailer along a level road, as Figure 4.32a illustrates. The mass of the truck is m1=8500 kg and that of the trailer is m2=27 000 kg. The two move along the x axis with an acceleration of ax=0.78 m/s2. Ignoring the retarding forces of friction and air resistance, determine (a) the tension T in the horizontal drawbar between the trailer and the truck and (b) the force D that propels the truck forward. Questions • The net external force acting on an object is zero. Is it possible for the object to be traveling with a velocity that is not zero? If your answer is yes, state whether any conditions must be placed on the magnitude and direction of the velocity. If your answer is no, provide a reason for your answer. • Is a net force being applied to an object when the object is moving downward (a) with a constant acceleration of 9.80 m/s2 and (b) with a constant velocity of 9.80 m/s? Explain. • Newton’s second law indicates that when a net force acts on an object, it must accelerate. Does this mean that when two or more forces are applied to an object simultaneously, it must accelerate? Explain. Newton's Third Law Newton's Third Law If one object is exerting a force on a second object, then the second object is also exerting a force back on the first object. The two forces have exactly the same magnitude but act in opposite directions. FBA FAB •Forces always exist in pairs. •It is very important that we realize we are talking about two different forces acting on two different objects. Question A father and his seven-year-old daughter are facing each other on ice skates. With their hands, they push off against one another. (a) Compare the magnitudes of the pushing forces that they experience. (b) Which one, if either, experiences the larger acceleration? Account for your answers. EXAMPLE 3: Pushing Two Blocks In Fig. 3-29a, a constant horizontal force of magnitude 20 N is applied to block A of mass 4.0 kg, which pushes against block B of mass 6.0 kg. The blocks slide over a frictionless surface, along an x axis. a) What is the acceleration of the blocks? b) What is the force acting on block B from block A Example 4 The Accelerations Produced by Action and Reaction Forces Suppose that the mass of the spacecraft in Figure 4.7 is mS=11 000 kg and that the mass of the astronaut is mA=92 kg. In addition, assume that the astronaut exerts a force of P=+36 N on the spacecraft. Find the accelerations of the spacecraft and the astronaut. NEWTON’S LAW OF UNIVERSAL GRAVITATION NEWTON’S LAW OF UNIVERSAL GRAVITATION Every particle in the universe exerts an attractive force on every other particle. For two particles that have masses m1 and m2 and are separated by a distance r, the force that each exerts on the other is directed along the line joining the particles and has a magnitude given by: The symbol G denotes the universal gravitational constant, whose value is found experimentally to be DEFINITION OF WEIGHT The weight of an object on or above the earth is the gravitational force that the earth exerts on the object. The weight always acts downward, toward the center of the earth. On or above another astronomical body, the weight is the gravitational force exerted on the object by that body. SI Unit of Weight: newton (N) When the height of object H above the Earth is small M Em M Em M Em W G 2 G G 2 r ( RE H ) RE 2 ME W (G 2 )m mg RE ME g G 2 9.8 m/s 2 RE The Gravitational Acceleration Constant When air resistance can be ignored and H any object under only gravitational force will free fall with a constant acceleration: g 9.8 m/s 2 RE RELATION BETWEEN MASS AND WEIGHT • Mass is an intrinsic property of matter and does not change as an object is moved from one location to another. • Weight, in contrast, is the gravitational force acting on the object and can vary, depending on how far the object is above the earth’s surface or whether it is located near another body such as the moon. Questions When a body is moved from sea level to the top of a mountain, what changes—the body’s mass, its weight, or both? Explain. Questions The force of air resistance acts to oppose the motion of an object moving through the air. A ball is thrown upward and eventually returns to the ground. • (a) As the ball moves upward, is the net force that acts on the ball greater than, less than, or equal to its weight? Justify your answer. • (b) Repeat part (a) for the downward motion of the ball. The Normal Force The normal force FN is one component of the force that a surface exerts on an object with which it is in contact—namely, the component that is perpendicular to the surface. APPARENT WEIGHT The Friction Force When the object moves or attempts to move along a surface, there is a component of the force that is parallel to the surface. This parallel force component is called the frictional force, or simply friction. It is always against the relative motion or the attempts of the motion between object and surface. Tension Tension is the force exerted by a rope or a cable attached to an object 1. Tension in a Nonaccelerating rope: the magnitude of tention is the same everywhere in the rope. 2. An Accelerating rope: the magnitude of tension is not the same everywhere in the rope that has a mass; however, the magnitude of tension is the same everywhere in the rope that is massless. For a massless string 1. It is always directed along the rope. 2. It is always pulling the object. 3. It has the same value along the rope Applying Newton's Laws Newton's Second Law: It can be written as two (or three) component equations: Equilibrium Applications of Newton's Laws of Motion DEFINITION OF EQUILIBRIUM: An object is in equilibrium when it has zero acceleration. Fnet 0 Fx 0 and Fy 0 and Fz 0 EXAMPLE 5: Three Cords In Fig. 6-25a, a block B of mass M = 15 kg hangs by a cord from a knot K of mass mK, which hangs from a ceiling by means of two other cords. The cords have negligible mass, and the magnitude of the gravitational force on the knot is negligible compared to the gravitational force on the block. What are the tensions in the three cords? Free-Body Diagrams (1) Identify the object for which the motion is to be analyzed and represent it as a point. (2) Identify all the forces acting on the object and represent each force vector with an arrow. The tail of each force vector should be on the point. Draw the arrow in the direction of the force. Represent the relative magnitudes of the forces through the relative lengths of the arrows. (3) Label each force vector so that it is clear which force it represents. Example 6 Replacing an Engine An automobile engine has a weight W, whose magnitude is W=3150 N. This engine is being positioned above an engine compartment, as Figure 4.29a illustrates. To position the engine, a worker is using a rope. Find the tension T1 in the supporting cable and the tension T2 in the positioning rope. Example 7 Equilibrium at Constant Velocity A jet plane is flying with a constant speed along a straight line, at an angle of 30.0° above the horizontal, as Figure indicates. The plane has a weight W whose magnitude is W=86 500 N, and its engines provide a forward thrust T of magnitude T=103 000 N. In addition, the lift force L (directed perpendicular to the wings) and the force R of air resistance (directed opposite to the motion) act on the plane. Find L and R. Non-quilibrium Applications of Newton's Laws of Motion DEFINITION OF NONEQUILIBRIUM: An object is in nonequilibrium when it has non-zero acceleration. Fnet ma Fx max and Fy may and Fz maz Example 8 Applying Newton’s Second Law Using Components A man is stranded on a raft (mass of man and raft=1300 kg), as shown in Figure a. By paddling, he causes an average force P of 17 N to be applied to the raft in a direction due east (the +x direction). The wind also exerts a force A on the raft. This force has a magnitude of 15 N and points 67° north of east. Ignoring any resistance from the water, find the x and y components of the raft’s acceleration. Example 9 Towing a Supertanker A supertanker of mass m=1.50×108 kg is being towed by two tugboats, as in Figure. The tensions in the towing cables apply the forces T1 and T2 at equal angles of 30.0° with respect to the tanker’s axis. In addition, the tanker’s engines produce a forward drive force D, whose magnitude is D=75.0×103 N. Moreover, the water applies an opposing force R, whose magnitude is R=40.0×103 N. The tanker moves forward with an acceleration that points along the tanker’s axis and has a magnitude of 2.00×10–3 m/s2. Find the magnitudes of the tensions T1 and T2. Conceptual Questions • According to Newton’s third law, when you push on an object, the object pushes back on you with an oppositely directed force of equal magnitude. If the object is a massive crate resting on the floor, it will probably not move. Some people think that the reason the crate does not move is that the two oppositely directed pushing forces cancel. Explain why this logic is faulty and why the crate does not move. • A stone is thrown from the top of a cliff. As the stone falls, is it in equilibrium? Explain, ignoring air resistance. • Can an object ever be in equilibrium if the object is acted on by only (a) a single nonzero force, (b) two forces that point in mutually perpendicular directions, and (c) two forces that point in directions that are not perpendicular? Account for your answers. • A circus performer hangs stationary from a rope. She then begins to climb upward by pulling herself up, hand over hand. When she starts climbing, is the tension in the rope less than, equal to, or greater than it is when she hangs stationary? Explain. • A weight hangs from a ring at the middle of a rope, as the drawing illustrates. Can the person who is pulling on the right end of the rope ever make the rope perfectly horizontal? Explain your answer in terms of the forces that act on the ring.