Chapter 4

The Laws of Motion

Classical Mechanics

Describes the relationship between the motion of objects in our everyday world and the forces acting on them

Conditions when Classical Mechanics does not apply

very tiny objects (< atomic sizes)

objects moving near the speed of light

Forces

Usually think of a force as a push or pull

Vector quantity

May be a contact force or a field force Contact forces result from physical contact

between two objects

Field forces act between disconnected objects Also called “action at a distance”

Contact and Field Forces

Fundamental Forces

Types Strong nuclear force

Electromagnetic force

Weak nuclear force

Gravity

Characteristics All field forces

Listed in order of decreasing strength

Only gravity and electromagnetic in mechanics

External and Internal Forces

External force Any force that results from the

interaction between the object and its environment

Internal forces Forces that originate within the object

itself

They cannot change the object’s velocity

Net Force

Net force is

the combination of all forces that change an object’s state of motion.

example: If you pull on a box with 10 N and a friend

pulls oppositely with 5 N, the net force is 5N in the direction you are pulling.

Net Force

Vector quantity

a quantity whose description requires both magnitude (how much) and direction (which way)

can be represented by arrows drawn to scale, called vectors

length of arrow represents magnitude and arrowhead shows direction

examples: force, velocity, acceleration

The Equilibrium Rule

The equilibrium rule

the vector sum of forces acting on a non-accelerating object equals zero

in equation form: F = 0

The Equilibrium Rule

example: a string holding up a bag of flour

two forces act on the bag of flour:

–tension force acts upward

–weight acts downward

equal in magnitude and opposite in

direction

when added, cancel to zero

bag of flour remains at rest

Inertia

Is the tendency of an object to continue in its original motion

Galileo (1564– 1642)

Inertia

Galileo’s Concept of Inertia

Italian scientist Galileo demolished Aristotle’s assertions in early 1500s.

Galileo’s discovery • objects of different weight fall to the ground

at the same time in the absence of air resistance

• a moving object needs no force to keep it moving in the absence of friction

Galileo’s Concept of Inertia

Force

is a push or a pull

Inertia

is a property of matter to resist changes in motion

depends on the amount of matter in an object (its mass)

The use of inclined planes for Galileo’s experiments helped him to discover the property called inertia. Intertia is a property of matter.

Galileo’s Inclinded Plane

Mass—A Measure of Inertia

Mass

a measure of the inertia of a material object

independent of gravity

greater inertia greater mass

unit of measurement is the kilogram (kg)

Weight

the force on an object due to gravity

scientific unit of force is the Newton (N)

unit is also the pound (lb)

Mass—A Measure of Inertia

Mass and weight in everyday conversation are interchangeable.

Mass, however, is different and more fundamental than weight.

Mass versus weight • on Moon and Earth

– weight of an object on Moon is

less than on Earth (1/6th)

– mass of an object is the same

in both locations

Mass—A Measure of Inertia

One Kilogram Weighs 9.8 Newtons

Relationship between kilograms and pounds

1 kg = 2.2 lb = 9.8 N at Earth’s surface

1 lb = 4.45 N

Mass

A measure of the resistance of an object to changes in its motion due to a force

Scalar quantity

SI units are kg

Sir Isaac Newton

1642 – 1727

Formulated basic concepts and laws of mechanics

Universal Gravitation

Calculus

Light and optics

Isaac my man

Newton’s First Law

An object moves with a velocity that is constant in magnitude and direction, unless acted on by a nonzero net force

The net force is defined as the vector sum of all the external forces exerted on the object

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.

F and a are both vectors

Can also be applied three-dimensionally

Units of Force

SI unit of force is a Newton (N)

US Customary unit of force is a pound (lb)

1 N = 0.225 lb

2s

mkg1N1

Problem 4.1

A 6.0-kg object undergoes an acceleration of 2.0 m/s2. (a) What is the magnitude of the resultant force acting on it? (b) If this same force is applied to a 4.0-kg object, what acceleration is produced?

Problem 4.12

The force exerted by the wind on the sails of a sailboat is 390 N north. The water exerts a force of 180 N east. If the boat (including its crew) has a mass of 270 kg, what are the magnitude and direction of its acceleration?

Problem 4.19

A 2 000-kg car is slowed down uniformly from 20.0 m/s to 5.00 m/s in 4.00 s. (a) What average force acted on the car during that time, and (b) how far did the car travel during that time?

Gravitational Force

Mutual force of attraction between any two objects

Expressed by Newton’s Law of Universal Gravitation:

2

21g

r

mmGF

Weight

The magnitude of the gravitational force acting on an object of mass m near the Earth’s surface is called the weight w of the object w = m g is a special case of Newton’s

Second Law g is the acceleration due to gravity

g can also be found from the Law of Universal Gravitation

More about weight

Weight is not an inherent property of an object

mass is an inherent property

Weight depends upon location

Newton’s Third Law

If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force exerted by object 2 on object 1.

Equivalent to saying a single isolated force cannot exist

12 21 F F

Newton’s Third Law cont.

F12 may be called the action force and F21 the reaction force

Actually, either force can be the action or the reaction force

The action and reaction forces act on different objects

Some Action-Reaction Pairs

is the normal force, the force the table exerts on the TV

is always perpendicular to the surface

is the reaction – the TV on the table

'andn n

' n n

n

n

'n

More Action-Reaction pairs

is the force the Earth exerts on the object

is the force the object exerts on the earth

'

g gandF F

gF

'

gF

'

g g F F

Forces Acting on an Object

Newton’s Law uses the forces acting on an object

are acting on the object

are acting on other objects

gandn F

'' gandn F

Applications of Newton’s Laws

Assumptions

Objects behave as particles

can ignore rotational motion (for now)

Masses of strings or ropes are negligible

Interested only in the forces acting on the object

can neglect reaction forces

Free Body Diagram

Must identify all the forces acting on the object of interest

Choose an appropriate coordinate system

If the free body diagram is incorrect, the solution will likely be incorrect

Free Body Diagram, Example

The force is the tension acting on the box The tension is the same

at all points along the rope

are the forces exerted by the earth and the ground

gandn F

Free Body Diagram, final

Only forces acting directly on the object are included in the free body diagram

Reaction forces act on other objects and so are not included

The reaction forces do not directly influence the object’s motion

Solving Newton’s Second Law Problems

Read the problem at least once

Draw a picture of the system

Identify the object of primary interest

Indicate forces with arrows

Label each force

Use labels that bring to mind the physical quantity involved

Solving Newton’s Second Law Problems

Draw a free body diagram If additional objects are involved, draw

separate free body diagrams for each object

Choose a convenient coordinate system for each object

Apply Newton’s Second Law The x- and y-components should be taken

from the vector equation and written separately

Solve for the unknown(s)

Equilibrium

An object either at rest or moving with a constant velocity is said to be in equilibrium

The net force acting on the object is zero (since the acceleration is zero)

0F

Equilibrium cont.

Easier to work with the equation in terms of its components:

This could be extended to three dimensions

0 0x yF and F

Equilibrium Example – Free Body Diagrams

Problem 4.45

(a) What is the resultant force exerted by the two cables supporting the traffic light in Figure P4.45? (b) What is the weight of the light?

Problem 4.26 (a) An elevator of mass m moving upward has two forces

acting on it: the upward force of tension in the cable and the downward force due to gravity. When the elevator is accelerating upward, which is greater, T or w? (b) When the elevator is moving at a constant velocity upward, which is greater, T or w? (c) When the elevator is moving upward, but the acceleration is downward, which is greater, T or w? (d) Let the elevator have a mass of 1 500 kg and an upward acceleration of 2.5 m/s2. Find T. Is your answer consistent with the answer to part (a)? (e) The elevator of part (d) now moves with a constant upward velocity of 10 m/s. Find T. Is your answer consistent with your answer to part (b)? (f) Having initially moved upward with a constant velocity, the elevator begins to accelerate downward at 1.50 m/s2. Find T. Is your answer consistent with your answer to part (c)?

Inclined Planes

Choose the coordinate system with x along the incline and y perpendicular to the incline

Replace the force of gravity with its components

Multiple Objects – Example

When you have more than one object, the problem-solving strategy is applied to each object

Draw free body diagrams for each object

Apply Newton’s Laws to each object

Solve the equations

Multiple Objects – Example, cont.

Problem 4.20

Two packing crates of masses 10.0 kg and 5.00 kg are connected by a light string that passes over a frictionless pulley as in Figure P4.20. The 5.00-kg crate lies on a smooth incline of angle 40.0°. Find the acceleration of the 5.00-kg crate and the tension in the string.

Connected Objects

Apply Newton’s Laws separately to each object

The magnitude of the acceleration of both objects will be the same

The tension is the same in each diagram

Solve the simultaneous equations

More About Connected Objects

Treating the system as one object allows an alternative method or a check

Use only external forces

Not the tension – it’s internal

The mass is the mass of the system

Doesn’t tell you anything about any internal forces

Problem 4.52

Three objects are connected by light strings as shown in Figure P4.52. The string connecting the 4.00-kg object and the 5.00-kg object passes over a light frictionless pulley. Determine (a) the acceleration of each object and (b) the tension in the two strings.

Forces of Friction

When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion

This is due to the interactions between the object and its environment

This is resistance is called friction

More About Friction

Friction is proportional to the normal force

The force of static friction is generally greater than the force of kinetic friction

The coefficient of friction (µ) depends on the surfaces in contact

The direction of the frictional force is opposite the direction of motion

The coefficients of friction are nearly independent of the area of contact

Static friction acts to keep the object from moving

If F increases, so does ƒs

If F decreases, so does ƒs

ƒs µ n

Static Friction, ƒs

Kinetic Friction, ƒk

The force of kinetic friction acts when the object is in motion

ƒk = µ n Variations of the

coefficient with speed will be ignored

Block on a Ramp, Example

Axes are rotated as usual on an incline

The direction of impending motion would be down the plane

Friction acts up the plane Opposes the motion

Apply Newton’s Laws and solve equations

Problem 4.41

Find the acceleration reached by each of the two objects shown in Figure P4.41 if the coefficient of kinetic friction between the 7.00-kg object and the plane is 0.250.

Problem 4.48

(a) What is the minimum force of friction required to hold the system of Figure P4.48 in equilibrium? (b) What coefficient of static friction between the 100-N block and the table ensures equilibrium? (c) If the coefficient of kinetic friction between the 100-N block and the table is 0.250, what hanging weight should replace the 50.0-N weight to allow the system to move at a constant speed once it is set in motion?

Friction of Air

When acceleration of fall is less than g—non-free fall • occurs when air resistance is non-negligible

• depends on two things: speed and frontal surface area

Friction of Air

example: A skydiver jumps from plane.

Weight is the only force until air resistance acts.

As falling speed increases, air resistance on diver builds up, net force is reduced, and acceleration becomes less.

When air resistance equals the diver’s weight, net force is zero and acceleration terminates.

Diver reaches terminal velocity, then continues the fall at constant speed.

Friction of Air

Terminal speed

occurs when acceleration terminates (when air resistance equals weight and net force is zero)

Terminal velocity

same as terminal speed, with direction implied or specified

When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is

A. less than g.

B. more than g.

C. g.

D. terminated.

Newton’s Second Law of Motion CHECK YOUR NEIGHBOR

When a 20-N falling object encounters 5 N of air resistance, its acceleration of fall is

A. less than g.

B. more than g.

C. g.

D. terminated.

Comment:

Acceleration of a nonfree-fall is always less than g. Acceleration will actually be (20 N – 5 N)/2 kg = 7.5 m/s2.

Newton’s Second Law of Motion CHECK YOUR ANSWER

If a 50-N person is to fall at terminal speed, the air resistance needed is

A. less than 50 N.

B. 50 N.

C. more than 50 N.

D. none of the above

Newton’s Second Law of Motion CHECK YOUR NEIGHBOR

If a 50-N person is to fall at terminal speed, the air resistance needed is

A. less than 50 N.

B. 50 N.

C. more than 50 N.

D. none of the above

Explanation:

Then, F = 0 and acceleration = 0.

Newton’s Second Law of Motion CHECK YOUR ANSWER

As the skydiver falls faster and faster through the air, air resistance

A. increases.

B. decreases.

C. remains the same.

D. not enough information

Newton’s Second Law of Motion CHECK YOUR NEIGHBOR

As the skydiver falls faster and faster through the air, air resistance

A. increases.

B. decreases.

C. remains the same.

D. not enough information

Newton’s Second Law of Motion CHECK YOUR ANSWER

As the skydiver continues to fall faster and faster through the air, net force

A. increases.

B. decreases.

C. remains the same.

D. not enough information

Newton’s Second Law of Motion CHECK YOUR NEIGHBOR

As the skydiver continues to fall faster and faster through the air, net force

A. increases.

B. decreases.

C. remains the same.

D. not enough information

Newton’s Second Law of Motion CHECK YOUR ANSWER

As the skydiver continues to fall faster and faster through the air, her acceleration

A. increases.

B. decreases.

C. remains the same.

D. not enough information

Newton’s Second Law of Motion CHECK YOUR NEIGHBOR

As the skydiver continues to fall faster and faster through the air, her acceleration

A. increases.

B. decreases.

C. remains the same.

D. not enough information

Newton’s Second Law of Motion CHECK YOUR ANSWER

Newton’s Second Law of Motion A situation to ponder…

Consider a heavy and light person jumping together with the same-size parachutes from the same altitude.

Who will reach the ground first?

A. the light person

B. the heavy person

C. both will reach at the same time

D. not enough information

A situation to ponder… CHECK YOUR NEIGHBOR

Who will reach the ground first?

A. the light person

B. the heavy person

C. both will reach at the same time

D. not enough information

Explanation:

The heavier person has a greater terminal velocity. Do you know why?

A situation to ponder… CHECK YOUR ANSWER

Newton’s Second Law of Motion

Coin and feather fall

feather reaches terminal velocity very quickly and falls slowly at constant speed, reaching the bottom after the coin does

coin falls very quickly and air resistance doesn’t build up to its weight over short-falling distances, which is why the coin hits the bottom much sooner than the falling feather

How to be a flying squirrel: Jeb Corliss "Grinding the Crack"

Air Resistance

Felix Baumgartner: Record high sky dive

Air Resistance

24 miles high 725 mph

Backup

Problem 4.42

A 2.00-kg block is held in equilibrium on an incline of angle θ = 60.0° by a horizontal force applied in the direction shown in Figure P4.42. If the coefficient of static friction between block and incline is μs = 0.300, determine (a) the minimum value of and (b) the normal force exerted by the incline on the block.

Forces on an Elevator

Tension T

Force of gravity

Force of elevator acceleration

mg mg

T ma T

ma