Chapter 4 The Laws of Motion

4.1 Classes of Forces Contact forces 接觸力 involve physical contact between two objects Field forces 場力 act through empty space No physical contact is required Fig 4.1

More About Forces A spring can be used to calibrate the magnitude of a force

Forces are vectors, so you must use the rules for vector addition to find the net force acting on an object

Fig 4.3

4.2 Newton’s First Law If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration This is also called the law of inertia慣性定律 It defines a special set of reference frames called inertial frames慣性座標系, We call this an inertial frame of reference

Isaac Newton (1642-1727)

Inertial Frames Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame We can consider the Earth to be such an inertial frame although it has a small centripetal acceleration associated with its motion

Newton’s First Law – Alternative Statement In the absence of external forces, when viewed from an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with a constant velocity Newton’s First Law describes what happens in the absence of a force Also tells us that when no force acts on an object, the acceleration of the object is zero

4.3 Inertia and Mass The tendency of an object to resist any attempt to change its velocity is called inertia Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity

More About Mass Mass is an inherent property of an object independent of the object’s surroundings independent of the method used to measure it a scalar quantity The SI unit of mass is kg

Mass vs. Weight 質量 vs 重量 Mass and weight are two different quantities Weight is equal to the magnitude of the gravitational force exerted on the object Weight will vary with location The mass of an object is the same everywhere

4.4 Newton’s Second Law The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass Force is the cause of change in motion, as measured by the acceleration Algebraically,

More About Newton’s Second Law is the net force This is the vector sum of all the forces acting on the object Newton’s Second Law can be expressed in terms of components:

Units of Force

Fig 4.4

4.5 Gravitational Force The gravitational force, , is the force that the earth exerts on an object This force is directed toward the center of the earth Its magnitude is called the weight of the object Weight = Fg = mg

More About Weight Because it is dependent on g, the weight varies with location g, and therefore the weight, is less at higher altitudes Weight is not an inherent property of the object

Gravitational Mass vs. Inertial Mass In Newton’s Laws, the mass is the inertial mass and measures the resistance to a change in the object’s motion In the gravitational force, the mass is determining the gravitational attraction between the object and the Earth Experiments show that gravitational mass and inertial mass have the same value

4.6 Newton’s Third Law If two objects interact, the force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 on object 1 Note on notation: is the force exerted by A on B

Newton’s Third Law, Alternative Statements Forces always occur in pairs A single isolated force cannot exist The action force is equal in magnitude to the reaction force and opposite in direction One of the forces is the action force, the other is the reaction force It doesn’t matter which is considered the action and which the reaction The action and reaction forces must act on different objects and be of the same type

Action-Reaction Examples, 1 The force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to exerted by object 2 on object 1 Fig 4.5

Fig 4.5

Action-Reaction Examples, 2 The normal force (table on monitor) is the reaction of the force the monitor exerts on the table Normal means perpendicular, in this case The action (Earth on monitor) force is equal in magnitude and opposite in direction to the reaction force (the monitor exerts on the Earth) Fig 4.6(a)

Free Body Diagram In a free body diagram, you want the forces acting on a particular object The normal force and the force of gravity are the forces that act on the monitor Fig 4.6(b)

CONCEPTUAL EXAMPLE Third law clarification CONCEPTUAL EXAMPLE Third law clarification. Michelangelo’s assistant has been assigned the task of moving a block of marble using a sled. He says to his boss, “When I exert a forward force on the sled, the sled exerts an equal and opposite force backward. So how can I ever start it moving? No matter how hard I pull, the backward reaction force always equals my forward force, so the net force must be zero. I’ll never be able to move this load.” Is this a case of a little knowledge being dangerous? Explain.

Fig 4.7

4.7 Applications of Newton’s Law Assumptions Objects can be modeled as particles Masses of strings or ropes are negligible When a rope attached to an object is pulling it, the magnitude of that force, , is the tension in the rope, along the rope away from the object

Objects in Equilibrium If the acceleration of an object that can be modeled as a particle is zero, the object is said to be in equilibrium Mathematically, the net force acting on the object is zero

Fig 4.9

Problem-Solving Hints Newton’s Laws Conceptualize the problem – draw a diagram Categorize the problem Equilibrium (SF = 0) or Newton’s Second Law (SF = m a) Analyze Draw free-body diagrams for each object Include only forces acting on the object

Problem-Solving Hints Newton’s Laws, cont Analyze, cont. Establish coordinate system Be sure units are consistent Apply the appropriate equation(s) in component form Solve for the unknown(s) Finalize Check your results for consistency with your free- body diagram Check extreme values

Newton’s Second Law, Example 1a Forces acting on the crate: A tension, the magnitude of force The gravitational force, The normal force, , exerted by the floor Fig 4.8

Newton’s Second Law, Example 1b Apply Newton’s Second Law in component form: Solve for the unknown(s) If is constant, then a is constant and the kinematic equations can be used to more fully describe the motion of the crate

A traffic light weighing 122 N hangs from a cable tied to two other cables fastened to a support, as in Figure 4.10a. The upper cables make angles of 37.0° and 53.0° with the horizontal. These upper cables are not as strong as the vertical cable and will break if the tension in them exceeds 100 N. Does the traffic light remain in this situation, or will one of the cables break?

Analyze Need two free-body diagrams Apply equilibrium equation to the light and find Apply equilibrium equations to the knot and find and Fig 4.10(b)(c)

Objects Experiencing a Net Force If an object that can be modeled as a particle experiences an acceleration, there must be a nonzero net force acting on it. Draw a free-body diagram Apply Newton’s Second Law in component form

A child on a sled is released on a frictionless hill of angle , as in Figure 4.11a.

Forces acting on the object: Determine the acceleration of the sled after it is released. A Forces acting on the object: The normal force acts perpendicular to the plane The gravitational force acts straight down Choose the coordinate system x along the incline and y perpendicular to the incline Replace the force of gravity with its components

From (2) we conclude that the n = mg cos  .

Suppose the sled is released from rest at the top of the hill and the distance from the front of the sled to the bottom of the hill is d. How long does it take the front of the sled to reach the bottom, and what is its speed just as it arrives at that point? B

Multiple Objects When two or more objects are connected or in contact, Newton’s laws may be applied to the system as a whole and/or to each individual object Whichever you use to solve the problem, the other approach can be used as a check

When two objects with unequal masses are hung vertically over a light, frictionless pulley as in Active Figure 4.12a, the arrangement is called an Atwood machine. The device is sometimes used in the laboratory to measure the free-fall acceleration. Calculate the magnitude of the acceleration of the two objects and the tension in the string.

Active Figure 4.12 NEXT

Forces acting on the objects: Tension (same for both objects, one string) Gravitational force Each object has the same acceleration since they are connected Draw the free-body diagrams Apply Newton’s Laws Solve for the unknown(s) Fig 4.12

To finalize the problem, let us consider some special cases To finalize the problem, let us consider some special cases. (a) m1 = m2, (b) m2 >> m1

Two blocks of masses m1 and m2, with m1 > m2, are placed in contact with each other on a frictionless, horizontal surface, as in Active Figure 4.13a. A constant horizontal force is applied to m1 as shown.

Active Figure 4.13 If you can't see the image above, please install Shockwave Flash Player. If this active figure can’t auto-play, please click right button, then click play. NEXT

Find the magnitude of the acceleration of the system of two blocks. First treat the system as a whole:

Apply Newton’s Laws to the individual blocks Solve for unknown(s) Determine the magnitude of the contact force between the two blocks. B Apply Newton’s Laws to the individual blocks Solve for unknown(s) Check: |P21| = |P12|

Imagine that the force in Active Figure 4 Imagine that the force in Active Figure 4.13 is applied toward the left on the right-hand block of mass m2. Is the magnitude of the force the same as it was when the force was applied toward the right on m1? C

A person weighs a fish on a spring scale attached to the ceiling of an elevator, as shown in Figure 4.14. Show that if the elevator accelerates, the spring scale reads an apparent weight different from the fish’s true weight.

Solution An observer on the accelerating elevator is not in an inertial frame. We need to analyze this situation in an inertial frame, so let us imagine observing it from the stationary ground. Newton’s second law applied to the fish in the vertical direction gives us

For example, if the weight of the fish is 40 For example, if the weight of the fish is 40.0 N and is upward with ay = 2.00 m/s2, the scale reading is

4.8 Forces on Automobiles The force that accelerates an automobile is the friction force from the ground The engine applies a force to the wheels The bottom of the tires apply forces backward on the road surface and the reaction (road on tires) causes the car to move forward

Automobile Performance