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Chapter 4

The Laws of Motion

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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 requiredFig 4.1

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More About Forces

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

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

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4.2 Newton’s First Law

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

If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration

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Isaac Newton (1642-1727)

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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

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Newton’s First Law – Alternative Statement

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

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

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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

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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

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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

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4.4 Newton’s Second Law

Force is the cause of change in motion, as measured by the acceleration

Algebraically,

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass

m

Fa

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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:

F

x xF ma

y yF ma

z zF ma

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Units of Force

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Fig 4.4

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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

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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

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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

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4.6 Newton’s Third Law

Note on notation: is the force exerted by A on B

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

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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

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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

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Fig 4.5

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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)

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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)

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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.

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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

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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

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Fig 4.9

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Problem-Solving Hints Newton’s Laws Conceptualize the problem – draw a

diagram Categorize the problem

Equilibrium (F = 0) or Newton’s Second Law (F = m a)

Analyze Draw free-body diagrams for each object Include only forces acting on the object

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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

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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

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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

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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?

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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)

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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

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

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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

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From (2) we conclude that the

n = mg cos .

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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

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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

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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.

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Active Figure4.12

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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

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

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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.F

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Active Figure4.13

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Find the magnitude of the acceleration of the system of two blocks.A

First treat the system as a whole:

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Determine the magnitude of the contact force betweenthe two blocks.B

Apply Newton’s Laws to the individual blocks Solve for unknown(s) Check: |P21| = |P12|

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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 F

12P

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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.

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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

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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

a

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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

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Automobile Performance

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