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Chapter 4

Forces and Newton’s Laws of Motion

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Forces and Newton’s Laws of Motion

• Forces

• Newton’s Three Laws of Motion

• The Gravitational Force

• Contact Forces (normal, friction, tension)

• Application of Newton’s Second Law

• Apparent Weight

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The net force is the vector sum of all the forces acting on a body.

321net FFFFF

Net Force

The net force is the resultant of this vector addition.

Bold letters represent vectors. The units of Force are Newtons, or the abbreviation N, which represent the SI units: kg-m/s2

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Free Body DiagramsThe free body diagram (FBD) is a simplified representation of an object, and the forces acting on it. It is called free because the diagram will show the object without its surroundings; i.e. the body is “free” of its environment.

We will consider only the forces acting on our object of interest. The object is depicted as not connected to any other object – it is “free”. Label the forces appropriately. Do not include the forces that this body exerts on any other body.

The best way to explain the free body diagram is to describe the steps required to construct one. Follow the procedure given below.

(1) Isolate the body of interest. Draw a dotted circle around the object that separates our object from its surroundings.

(2) Draw all external force vectors acting on that body.

(3) You may indicate the body’s assumed direction of motion. This does not represent a separate force acting on the body.

(4) Choose a convenient coordinate system.

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Free Body Diagram

T F

w1

N1

x

y

The force directions are as indicated in the diagram. The magnitudes should be in proportion if possible.

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Newton’s First Law of Motion: Inertia and Equilibrium

Newton’s 1st Law (The Law of Inertia):

If no force acts on an object, then the speed and direction of itsmotion do not change.

Inertia is a measure of an object’s resistance to changes in its motion.

It is represented by the inertial mass.

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If the object is at rest, it remains at rest (velocity = 0).

If the object is in motion, it continues to move in a straight line with the same velocity.

No force is required to keep a body in straight line motion when effects such as friction are negligible.

An object is in translational equilibrium if the net force on it is zero and vice versa.

Newton’s First Law of Motion

Translational Equilibrium

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Newton’s Second Law of Motion Net Force, Mass, and Acceleration

Newton’s 2nd Law:

The acceleration of a body is directly proportional to the net force acting on the body and inversely proportional to the body’s mass.

Mathematically: aFF

a mm

netnet or

This is the workhorse of mechanics

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An object’s mass is a measure of its inertia. The more mass, the more force is required to obtain a given acceleration.

The net force is just the vector sum of all of the forces acting on the body, often written as F.

Newton’s Second Law of Motion

If a = 0, then F = 0. This body can have:

Velocity = 0 which is called static equilibrium, or

Velocity 0, but constant, which is called dynamic equilibrium.

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Newton’s Third Law of Motion Interaction Pairs

Newton’s 3rd Law:

When 2 bodies interact, the forces on the bodies, due to each other, are always equal in magnitude and opposite in direction.

In other words, forces come in pairs.

Mathematically: .1221 FF

designates the force on object 2 due to object 1.

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Types of Forces

Contact forces: Normal Force & Friction

Tension

Gravitational Force

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Contact Forces

Contact forces: these are forces that arise due

to of an interaction between the atoms in the

surfaces of the bodies in contact.

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Normal force: this force acts in the direction perpendicular to the contact surface.

Normal force of the ramp on the box

N

w

Normal force of the ground on the boxN

w

Normal Forces

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Example: Consider a box on a table.

FBD for box

mgwN

wNFy

that So

0

This just says the magnitude of the normal force equals the magnitude of the weight; they are not Newton’s third law interaction partners.

Apply Newton’s 2nd law

N

w

x

yNormal Forces

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Friction: a contact force parallel to the contact surfaces.

Static friction acts to prevent objects from sliding.

Kinetic friction acts to make sliding objects slow down. Sometimes called Dynamic friction.

Frictional Forces

maxs sf = μ N

d df = μ N

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Frictional Forces

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Tension

An ideal cord has zero mass, does not stretch, and the tension is the same throughout the cord.

This is the force transmitted through a “rope” from one end to the other.

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Example (text problem 4.77): A pulley is hung from the ceiling by a rope. A block of mass M is suspended by another rope that passes over the pulley and is attached to the wall. The rope fastened to the wall makes a right angle with the wall. Neglect the masses of the rope and the pulley.

Find the tension in the rope from which the pulley hangs and the angle .

FBD for the mass M

w

T

x

y

MgwT

wTFy

0Apply Newton’s 2nd Law to the mass M.

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FBD for the pulley:

0sin

0cos

TFF

TFF

y

x

Apply Newton’s 2nd Law:

sincos FFT

This statement is true only when = 45 and

MgTF 2 2

Example continued:

x

y

TF

T

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Gravitational Forces

221

r

MGMF

Gravity is the force between two masses. Gravity is a long-range force. No contact is needed between the bodies. The force of gravity is always attractive!

r is the distance between the two masses M1 and M2 and G = 6.671011 Nm2/kg2.

M2

r

M1F21 F12 .1221 FF

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Let M1 = ME = mass of the Earth. 22M

r

GMF E

Here F = the force the Earth exerts on mass M2. This is the force known as weight, w.

.222gMM

r

GMw

E

E

22

m/s 8.9N/kg 8.9 where E

E

r

GMg

Near the surface of the Earth

km 6400

kg 1098.5

E

24E

r

M

Gravitational Forces

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What is the direction of g?

Note thatm

Fg

is the gravitational force per unit mass. This is called

the gravitational field strength. It is also referred to as

the acceleration due to gravity.

What is the direction of w?

Gravitational Forces

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Example: What is the weight of a 100 kg astronaut on the surface of the Earth (force of the Earth on the astronaut)? How about in low Earth orbit? This is an orbit about 300 km above the surface of the Earth.

On Earth: N 980mgw

In low Earth orbit: N 890)( 2

hR

GMmrmgw

E

Eo

The weight is reduced by about 10%. The astronaut is NOT weightless!

Gravitational Forces

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Applying Newton’s Second Law

aF mThe one equation everyone remembers!

This equation is just the tip of the “iceberg” of the mechanics problem. The student will need to anlyze the forces in the problem and sum the force vector components to build the left hand side of the equation.

Sumof the forces acting on the objects in the system

“m” is the System

Mass

“a” is the System

Response

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Example: A force of 10.0 N is applied to the right on block 1. Assume a frictionless surface. The masses are m1 = 3.00 kg and m2 = 1.00 kg.

Find the tension in the cord connecting the two blocks as shown.

F block 2 block 1

Assume that the rope stays taut so that both blocks have the same acceleration.

Applying Newton’s Second Law

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FBD for block 2:

T F

w1

N1

x

y

x

T

w2

N2

y

FBD for block 1:

011

1

wNF

amTFF

y

x

022

2

wNF

amTF

y

x

Apply Newton’s 2nd Law to each block:

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amTF 1

amT 2

To solve for T, a must be eliminated. Solve for a in (2) and substitute in (1).

(1)

(2)

Example continued:

These two equations contain the unknowns: a and T.

N 5.2

kg 1kg 3

1

N 10

1

1

2

1

2

1

21

211

mm

FT

Tm

mT

m

TmF

m

TmamTF

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Pick Your System Carefully

x

T

w2

N2

y

T F

w1

N1

x

y

Include both objects in the system. Now when you sum the x-components of the forces the tensions cancel. In addition, since there is no friction, y-components do not contribute to the motion.

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Apparent Weight

Stand on a bathroom scale.

FBD for the person: Apply Newton’s 2nd Law:

y

yy

mamgN

mawNF

w

N

x

y

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The normal force is the force the scale exerts on you. By Newton’s 3rd Law this is also the force (magnitude only) you exert on the scale. A scale will read the normal force.

yagmN is what the scale reads.

When ay = 0, N = mg. The scale reads your true weight.

When ay 0, N > mg or N < mg.

Apparent Weight

In free fall ay = -g and N = 0. The person is weightless.

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Example (text problem 4.128):

A woman of mass 51 kg is standing in an elevator. The elevator pushes up on her feet with 408 newtons of force.

What is the acceleration of the elevator?

FBD for woman:

y

yy

mamgN

mawNF

Apply Newton’s 2nd Law: (1)

w

N

x

y

Apparent Weight

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Solving (1) for ay:2m/s 8.1

m

mgNay

The elevator could be (1) traveling upward with decreasing speed, or (2) traveling downward with increasing speed.

The change in velocity is DOWNWARD.

Given: N = 408 newtons, m = 51 kg, g = 9.8 m/s2

Unknown: ay

Example continued:

Apparent Weight

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N

mg

F

Free Body Diagram

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N

mg

aF m

This is not a methodolgy to solve for the acceleration. It is just graphically demonstrating that the net force is ma

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N

mg

mg

N

Same problem but the applied force is angled up

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mg

N

The normal force, N, is smaller in this case because the upward angled applied force reduces the effective weight of the sled.

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Equilibrium Problem

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Equilibrium Problem

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Equilibrium Problem

This is an example of three-vector equilibrium problem. It lends itself to a simple solution because the vector sum of the three vectors closes on itself (equilibrium) and forms a triangle

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Milk Carton

Max static friction force

Non-slip limit on applied force

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Hanging Problems

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Hanging Picture

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Hanging Picture - Free Body Diagram

T1

mg

60o

30o

T2

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Hanging Picture

T 1

m g

6 0 o

3 0 oT 2

01

02

cos(60 )

sin(60 )

T mg

T mg

• Since this turned out to be a right triangle the simple trig functions are that is needed to find a solution.

• If the triangle was not a right triangle then the Law of Sines would have been needed.

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Atwood Machine and Variations

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Atwood’s Machine

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A 2-Pulley Atwood Machine

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Three Body Problem

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Incline Plane Problems

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Single Incline Plane Problem

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Double Incline Plane Problem

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Force SummaryFriction

Opposes motion

Proportional to normal force

Non-conservative

Static & dynamic

Normal Forces

Perpendicular to surface at point of contact.

Magnitude needed to maintain equilibrium

Tension

No mass

No stretching

Pulleys

Massless

No friction (bearing)

Tension in rope continuously changes direction

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Summary

• Newton’s Three Laws of Motion

• Drawing free body diagrams

• Contact forces versus long-range forces

• Different forces (gravity, friction, normal, tension, air resistance)

• Application of Newton’s Second Law