Lecture 2

Newton’s Laws

Acceleration due to gravity

Forces

We have worked out mathematical

relationships arising from the definitions

of velocity and acceleration.

Free Falling Objects

Probably the most familiar system where

we observe acceleration is that due to

gravity.

0v v at

2 2

0 2v v as

2

0

1

2s v t at

Hammer and a paper tissue

Dropped from same height,

Hammer will hit the ground first

Air resistance will slow the tissue down.

However if we neglect air resistance both

objects will hit the ground at the same time

Free Fall Definition:

Freely falling object is one moving freely

under the influence of gravity alone,

The “free fall condition” considers gravity only:

•neglects other effects such as air resistance

Objects considered to be freely falling

propelled upwards

propelled downwards

released from rest

The acceleration due to gravity near the

earths surface is known as g. This has

been measured to be g = 9.80 ms-2.

A time delay image of

two spheres of very

different mass

falling in a vacuum. .

Free Falling Objects

It can be seen that, in

the absence of air

resistance both

accelerate at the

same rate,

independent of mass.

If we assume objects falling near the

earths surface are affected only by

gravity (air resistance is negligible)

two basic facts govern their motion:

1. Objects accelerate at the same rate,

independent of their

mass,

size

composition.

Free Falling Objects

2 . This gravitational acceleration is

constant and so does not change as

the object falls.

Ignoring air resistance, an object in free fall

experiences an acceleration of magnitude

9.8 ms-2.

In other words the downward directed velocity

increases by 9.8 ms-1 each second.

So if released from rest an object has a velocity

downwards

after 1 second of 9.8m/s.

Acceleration due to gravity

Since by convention displacement upwards is

positive, but gravity acts downwards, then

g = -9.8 ms-2.

after 2 seconds of 19.6m/s

after 3 seconds of 29.4m/s

0v v at

Example:

A ball is dropped from a window 10m above

ground. What will be its velocity just before it

hits the ground?

2

0

2 2

1

2

0 (2) ( 9.8 )( 10 )

14

v v as

v ms m

v ms

Since acceleration due to gravity is constant,

motion under the action of gravity is

uniformly accelerated motion,

so we can use the equations relating position,

displacement, velocity and acceleration

already derived

Acceleration due to gravity

2 2

0 2v v as

114v ms

Up to now we have discussed kinematics i.e.

methods for describing motion (without

reference to the causes).

We will now study motion and the causes of

motion – dynamics.

The basic physical quantities used in dynamics

are

Force, Acceleration &

Newton’s Laws

force, mass and acceleration.

Force: push or a pull

•Strength or magnitude

•Direction

Force is a vector quantity

Characteristics:

A force resulting from direct contact with

another object is called a contact force.

For example when you push or pull an

object you exert a force on it.

Force

There are also non-contact forces.

Gravitational, electrical and magnetic forces

act through empty space.

You don’t have to be standing on the

surface of the earth to experience the

effects of gravity.

Orthodontics:

contact force applied

Force is a vector quantity:

magnitude and direction

Force pushes or pulls

teeth in a particular direction

Application of force

break the periodontal ligaments

Extends width of the socket

http://www.dentistrytoday.com/oral-

medicine/oral-surgery/1536

this web site concerns the mechanics of

extraction including the Physics forceps

Force

Tooth extraction

The force due to gravity exerted on an

object is known as its weight.

Force

The SI unit of force is the Newton, N.

Force can be measured with a spring balance.

When a force pulls on the spring, the spring

extends. A pointer attached to the end of the

spring can indicate the force on a scale.

Answers are contained in

NEWTON’S three LAWS.

Force causes Acceleration

1/ What happens to an object when there

is no net force exerted on it.?

Questions:

2/ What links force and acceleration?

3/ What happens to an object that

exerts a force on another object?

Isaac Newton ( 1643-1727)

Credited with establishing a mathematical

basis for the laws of motion

Earlier Galileo Galilei (1564 –1642) established

theories concerning moving (falling) objects

1. When the vector sum of forces on an

object is zero then the acceleration of that

object is zero.

Force must be applied to an object to change

its velocity.

All dynamics is based on Newton’s Laws.

These are three empirical laws which cannot

be derived from anything more fundamental.

2. When the vector sum of forces is NOT zero

force is related to acceleration.

Force = mass x acceleration.

3.The third law describes the pairs of forces

that interacting objects exert on each other.

If we push an object it pushes back with an

equal force but in the opposite direction.

Newton’s Laws

“Any object will remain at rest or in motion

in a straight line with constant velocity

unless acted upon by an outside force”

} v = 0

v = const. No net

Force

Constant velocity means both constant

magnitude (speed) and constant direction.

NEWTON’S FIRST LAW

There is no distinction between an object at

rest and an object moving with constant

velocity.

This is not as self evident as it may seem.

It actually seems counter intuitive because it

means that:

once an object is set in motion with

uniform velocity, no force is needed to

keep it moving.

This seems contrary to everyday

experience.

Newton’s First Law

“Any object will remain at rest or in motion in

a straight line with constant velocity unless

acted upon by an outside force”

For example,

If you push a book across a table, the book

does not keep moving indefinitely after it has

left your hand. It slows down.

BUT as we will see later, this is due to

frictional forces slowing it down.

Both teams pull on the rope with equal strength

they each exert the same magnitude of force

on it, but in opposite directions.

Imagine a tug of war match with each team

equally matched.

In this case the knot in the middle of the

rope does not move. It does not

accelerate. The rope is in equilibrium.

Equilibrium and Newton’s First Law

We can write - Fleft = Fright

Σ F = 0

This means Fleft + Fright = 0

In equilibrium

Fright Fleft

Greek letter “S” represents the sum

Acceleration produced by forces acting on

an object is

directly proportional to and in the same

direction as the net external force

inversely proportional to the mass of the

object

m = mass of the object

F

a=F/m

Newtons (N) (kg) (ms-2)

extnetF

ma

Newton’s Second Law

extnetF ma

The greater the mass of a body, the less effect

a given force has.

The unit of force, the Newton, is defined as

follows:

Equation F=ma means that mass, in addition

to being a measure of the amount of matter

in an object, is a measure of how difficult it is

to move an object or its inertia.

Newton’s Second Law

A force of 1N acting on a mass of 1kg

produces an acceleration of 1ms-2.

F = ma so (1N) = (1kg)(1ms-2) = 1kgms-2

Inertia is the tendency

of an object at rest to remain at rest

of an object in motion to remain in motion

with its original velocity.

If a number of forces act on an object at

the same time,

Newton’s second law applies to the sum

of the forces and is written.

Thus when working out problems involving

a number of forces, it is best to calculate

the resultant force and then set that

equal to ma.

Newton’s Second Law

F maS

Mass and Weight

Weight however is the force exerted by

gravity on a body.

Thus a heavy truck is

difficult to push because of its mass

difficult to lift due to its weight.

When an object falls under the influence of

gravity it accelerates downwards at the rate:

a = g = 9.80 m/s2

Force produces an acceleration given by

F = ma

Mass is a measure both of

how much matter an object contains

how difficult it is to move.

This equation gives the gravitational force

on an object whether it is in freefall or not.

Any object with mass “near” the surface of

the earth feels a gravitational force (weight),

w = mg.

But if the force on an object due to gravity

is weight, w, and it accelerates at

g = 9.8 ms-2 then, we can write,

Mass and Weight

w mg

1 2

2

m mF G

r

Newtonian Gravity

29.81g ms

G = 6.67 10-11 N m2 kg-2

2

EM mF G ma mg

r

2

EMg G

r

2E

EMg G

R

Mass of a person is 65kg. What is his weight?

Example

On the surface of the moon the force of

gravity is approximately 1/6 of that on earth.

What is the weight of the same person on

the moon.

Mass of a person on Earth is 65kg

Weight of this person on the moon is

w = mgm

weight (w) = 65kg * {(1/6) 9.8m/s2}

w =106.2kg.m/s2 = 106.2N

Example

His weight is given by w =mg

w = 65kg * 9.8m/s2

= 637kg m/s2 = 637N

A tennis ball and a golf ball are simultaneously

dropped from a tall building of height 120m.

Neglecting air resistance, determine

(a) the speed with which each ball hits the

ground.

(b) The time taken for each to reach the

ground (g = 9.8 ms-2)

a = -g and s = -120m

Acceleration and displacement

are in the direction such that

2 2

0 2v v as v0 = 0

Example.

all objects regardless of their mass or size

fall freely with an acceleration g = 9.8ms-2

(b) The time taken for each to reach the ground

v = v0+ at

-48.5m/s = 0 + (-9.8m/s2)t

t = (-48.5m/s )/ (-9.8m/s2) = 5s

v2 = 2*(-g)(-120)

v = ±√2*(-9.8m/s2)(-120m)

= ±48.5m/s

(a) the speed with which each ball hits

the ground. 2 2

0 2v v as

But since the direction is downwards

v = - 48.5m/s

And since speed is the magnitude of velocity

Speed = 48.5m/s

A car has a maximum acceleration of 4 ms-2.

What will its maximum acceleration be while

towing a second car of the same mass.

F=ma

F=MaN

where M is the combined mass of the cars

a = 4ms-2

2N

F Fa

M m

2 2N

ma aa

m

224

22

N

msa ms

Example

EXAMPLE

A ball is thrown upward at 20m/s from a

window 60m above the ground.

(a) How high does it go?

(b) When does it reach its highest point?

(c) When does it hit the ground?

Here we will take the upward direction as the

positive direction. This means any vector

quantities pointing upward (initial velocity) are

positive while vector quantities pointing down

(acceleration due to gravity) are negative.

or t = (v-v0)/a

t = (0-20m/s)/(-9.80m/s2) = 2.04 s

It reaches its highest point after 2.04s

2 2

0( )

2

v vs

a

or

(a) To find the highest point. We note at this

point the velocity is zero. We use

v=0 so

s = [0-(20m/s)2]/[2(-9.80m/s2)] = 20.4 m

Highest point is 20.4m above the window

(c) The ball hits the ground when s = -60m.

t ?

2 2

0 2v v as

0v v at

(b) When does it reach its highest point

(-60m) = (20m/s) t+(1/2)(-9.8m/s2) (t)2

Rearranging gives

(4.9m/s2) (t)2 - (20m/s) (t) – (60m) = 0

This is a quadratic equation whose solution is.

The roots of this equation are then

t = -2s or 6.1s

As the ball could not hit the ground before it is

thrown, The correct answer must be the positive

one t = 6.1s

2 2

2

(20 / ) ( 20 / ) 4(4.9 / )( 60 )

2(4.9 / )

m s m s m s mt

m s

2

(20 / ) (40 / )

(9.8 / )

m s m st

m s

Reducing to

2

0

1

2s v t at