Chapter 4

Forces and Mass

Classical Mechanics

does not apply for

• very tiny objects (< atomic sizes)

• objects moving near the speed of light

Newton’s First Law

• If the net force !F exerted on an object is

zero the object continues in its original state of

motion. That is, if !F = 0, an object at rest

remains at rest and an object moving with some

velocity continues with the same velocity.

• Contrast with Aristotle!

Forces

• Usually a push or pull

• Vector

• Either contact or field force

Contact and Field ForcesFundamental (Field) Forces

Types

• Strong nuclear force

• Electromagnetic force

• Weak nuclear force

• Gravity

Strong Nuclear Force

• QCD (Quantum chromodynamics) confines quarks

by exchaning gluons

• Nuclear force: binds protons and neutrons

by exchanging pions

Electromagnetic Forces

• Opposites attract, like-signs repel

• Electric forces bind electrons in atoms

• Magnetic forces arise from moving charges

Weak Nuclear Force

• Involves exchange of heavy W or Z particle

• Responsible for decay of neutrons

Gravity

• Attractive force between any two bodies

• Proportional to both masses

• Inversely proportional to square of distance

F = Gm1m2

r2

Inertia (Newton’s First Law)

• Tendency of an object to continue in its original

motion

Mass

• A measure of the resistance of an object to

changes in its motion due to a force

• Scalar

• SI units are kg

Newton’s Second Law

• Acceleration is proportional to net force and

inversely proportional to mass.

!F! = m

!a

Units of Force

• SI unit is Newton (N)

• US Customary unit is pound (lb)

• 1 N = 0.225 lb

F = ma

1 N = 1kg !m

s2

Weight

Weight is magnitude of gravitational force

w = mg

w = GMearthm

r2

g =GMearth

Rearth2

weight

mass

Weight vs. Mass

• Mass is inherent property

• Weight depends on location

Newton’s Third Law

• Single isolated force cannot exist

• For every action there is an equal and opposite

reaction

Force on “1” due to “2”

!F12

= !!F21

Newton’s Third Law cont.

• F12 is action force F21

is reaction force• You can switch

action <-> reaction

• Action & reaction forces act on different objects

Action-Reaction Pairs

!n = ! "

!n

Fg= !F

g

'

Define the OBJECT (free body)

• Newton’s Law uses the

forces acting ON object

• n and Fg act on object

• n’ and Fg’ act on other

objects

Assumptions for F=ma

• Objects behave as particles

• ignore rotational motion (for now)

• Consider only forces acting ON object

• neglect reaction forces

Definition of Equilibrium

!F! = 0

Example 4.1a

A Ford Pinto is parked in a parking lot

There is no net force on the Pinto

A) True

B) False

Example 4.1b

A Ford Pinto is parked in a parking lot

The contact force acting on the Pinto from the

parking lot surface ______________ .

A) Points upwards

B) Is zero

C) Points downward

Example 4.1c

A Ford Pinto drives down a highway on the moon

at constant velocity (where there is no air

resistance)

The Pinto’s acceleration is __________

A) Less than zero

B) Equal to zero

C) Greater than zero

Example 4.1d

A Ford Pinto drives down a highway on the moon

at constant velocity (where there is no air

resistance)

The force acting on the Pinto from the contact with

the highway is vertical.

A) True

B) False

Mechanical Forces

• Strings, ropes and Pulleys

• Gravity

• Normal forces

• Friction

• Springs (later)

Some Rules for Ropes and Pulleys

• Force from rope points AWAY from object

• Magnitude of the force is tension

• Tension does not change when going over

frictionless pulley

Example 4.2

a) Find acceleration

b) Find T, the tension above

the bowling ball

c) Find T3, the tension in the

rope between the pails

d) Find force ceiling must exert

on pulley

a) a = g/6 = 1.635 m/s2

b) T = 57.2 N

c) T3=24.5 N

d) Fpulley=2T = 114.5 N

Example 4.3a

2) Which statements are correct?

Assume the objects are static.

T1 is _____ T2

cos(10o)=0.985

sin(10o)=0.173

A) Less than

B) Equal to

C) Greater than

Example 4.3b

2) Which statements are correct?

Assume the objects are static.

T2 is ______ T3

cos(10o)=0.985

sin(10o)=0.173

A) Less than

B) Equal to

C) Greater than

Example 4.3c

2) Which statements are correct?

Assume the objects are static.

cos(10o)=0.985

sin(10o)=0.173

A) Less than

B) Equal to

C) Greater than

T1 is _____ Mg

Example 4.3d

2) Which statements are correct?

Assume the objects are static.

T1+T2 is ______ Mg

cos(10o)=0.985

sin(10o)=0.173

A) Less than

B) Equal to

C) Greater than

Example 4.4

Given that Mlight = 25 kg, find all three tensions

T3 = 245.3 N, T1 = 147.4 N, T2 = 195.7 N

Cable Pull Demo

Inclined Planes

• Choose x along the

incline and y

perpendicular to incline

• Replace force of gravity

with its components

Fg,x = mgsin!

Fg,y = mgcos!

Example 4.5

Find the acceleration and the tension

a = 4.43 m/s2, T= 53.7 N

M

Example 4.6 (Skip)

Find M such that the box slides at constant v

M=15.6 kg

Forces of Friction

• RESISTIVE force between object and neighbors

or the medium

• Examples:

• Sliding a box

• Air resistance

• Rolling resistance

Sliding Friction

• Parallel to surface, opposite toother forces

• ~ independent of the area of contact

• Depends on the surfaces in contact

f ! µsN

f = µkN

µs > µk

Coefficients

of Friction

f ! µsN

f = µkN

µs > µk

Static Friction, ƒs

• µs is coefficient of

static friction

• N is the normal force

f

F

fs ! µsN

Kinetic

Friction, ƒk

• µk is coefficient of

kinetic friction

• Friction force opposes F

• n is the normal force F

f

f = µkn

Friction Demo

Example 4.7

The man pushes/pulls with a force of 200 N. The

child and sled combo has a mass of 30 kg and the

coefficient of kinetic friction is 0.15. For each case:

What is the frictional force opposing his efforts?

What is the acceleration of the child?

f=59 N, a=3.80 m/s2 / f=29.1 N, a=4.8 m/s2

Example 4.8

Given m1 = 10 kg and m2 = 5 kg:

a) What value of µs would stop the block from sliding?

b) If the box is sliding and µk = 0.2, what is the

acceleration?

c) What is the tension of the rope?

a) µs = 0.5 b) a=1.96 m/s2 c) 39.25 N

Example 4.9

What is the minimum µs required to

prevent a sled from slipping down a

hill of slope 30 degrees?

µs = 0.577

Other kinds of friction

• Air resistance, F ~ Area " v2

• Rolling resistance, F ~ v

Terminal velocity:

Fresistance = CAv2

= mg at terminal velocity

Coffee Filter Demo Example 4.9

An elevator falls with acceleration a = 8.0 m/s2.

If a 100-kg person stood on a bathroom scale

during the fall, what would the scale read?

18.45 kg

Accelerating Reference Frames

• Equivalent to “Fictitious” gravitational force

g fictitious = !a frame

Fictitious Force: Derivation

Eq. of motion in fixed frame

x = v0t +1

2at2

= v0t +1

2

F

mt2

F-maf looks like force in new frame,

maf acts like fake gravitational force!

x0 (t) =1

2a f t

2

x ! x0 (t) = v0t +1

2

(F ! maf )

mt2

Example 4.10

You are calibrating an accelerometer so that you can

measure the steady horizontal acceleration of a car by

measuring the angle a ball swings backwards.

If M = 2.5 kg and the acceleration, a = 3.0 m/s2:

a) At what angle does the ball swing backwards?

b) What is the tension in the string?

# = 17 deg

T= 25.6 N#

Example 4.11a

A fisherman catches a 20 lb trout (mass=9.072

kg), and takes the trout in an elevator to the

78th floor to impress his girl friend, who is the

CEO of a large accounting firm. The fish is

hanging on a scale, which reads 20 lb.s while the

fisherman is stationary. Later, he returns via the

elevator to the ground floor with the fish still

hanging from the scale.

In the instant just after the elevator begins to

move upward, the reading on the scale will be

______________ 20 lbs. a) Greater thanb) Less thanc) Equal to

Example 4.11b

A fisherman catches a 20 lb trout (mass=9.072 kg), and

takes the trout in an elevator to the 78th floor to impress

his girl friend, who is the CEO of a large accounting firm.

The fish is hanging on a scale, which reads 20 lb.s while

the fisherman is stationary. Later, he returns via the

elevator to the ground floor with the fish still hanging

from the scale.

On the way back down, while descending at

constant velocity, the reading on the scale will

be ________________ 20 lbs. a) Greater than

b) Less than

c) Equal to

Example 4.11c

A fisherman catches a 20 lb trout (mass=9.072 kg),

and takes the trout in an elevator to the 78th floor

to impress his girl friend, who is the CEO of a large

accounting firm. The fish is hanging on a scale, which

reads 20 lb.s while the fisherman is stationary. Later,

he returns via the elevator to the ground floor with

the fish still hanging from the scale.

In the instant just before the elevator comes to

a stop on the 78th floor, the mass of the fish

will be ______________ 9.072 kg. a) Greater than

b) Less than

c) Equal to