Download Y12 Mechanics Notes - Cashmere

Mechanics
L2 NCEA
Achievement Standard 2.4
Text Book reference:
Chapters 2-8
Scalars and Vectors
A scalar quantity is one
that has a size (or
magnitude) only
Eg. Mass, energy, time
A vector quantity is one
that has a size and a
direction
Eg. Force, velocity,
momentum
Motion
Distance
(scalar)
Symbol: d
How far in
total the
object has
moved
Unit: m
Displacement How far the
object ends
(vector)
up from it’s
starting
position
Symbol:d or Unit: m
s
Motion
Speed
(Scalar)
How fast an
object is
travelling
Symbol:s
Unit: ms-1
Velocity
(Vector)
The speed and
direction that
an object is
travelling
Unit: ms-1
Symbol: v
Motion
Velocity is calculated by
Dd
velocity(v) 
Dt
Where d is displacement, t is
time and D means “the change
in”.
Velocity may refer to either
average velocity or
instantaneous velocity.
Constant velocity means that
neither the speed nor the
direction of the objects motion
is changing.
Motion
Acceleration
(Vector)
Symbol: a
The rate at
which the
velocity of an
object is
changing
Unit: ms-2
Acceleration can be calculated
by
Dv
a
Dt
Motion
Acceleration is used to
describe motion where the
object slows down as well
as when it speeds up.
Sometimes the word
deceleration is used.
Acceleration is given a
negative value when the
object is slowing down.
Objects are accelerating
when their direction
changes, even though their
speed may remain
constant.
(Do Page 27Questions 2A)
Graphs of Motion
Distance / diplacement versus
time…
Displacement(m)
C
B
A
D
Time (s)
E
A=Constant velocity (slow)
B=Constant velocity (faster)
C=Stopped
D=Constant velocity (backward)
E=Constant velocity (backward
past starting point)
Graphs of Motion
Speed / velocity versus time…
Velocity (ms-1)
C
B
A
D
Time (s)
E
A=Constant acceleration (low)
B=Constant acceleration (high)
C=Constant velocity
D=Constant deceleration to stop
E=Constant acceleration in
opposite direction
(Do Page 32 Questions2B)
Kinematic Equations
To solve problems involving
objects moving in straight
lines with constant
acceleration.
Terms used:
d=distance/displacement
(m)
vi=initial velocity (ms-1)
vf=final velocity (ms-1)
a=acceleration (ms-2)
t=time (s)
Kinematic Equations
v f  vi  at
 vi  v f
d  
 2

t

2
1
d  vi t  at
2
v  v  2ad
2
f
2
i
(Do Page 33 Questions 2C)
Vectors
A vector is drawn as a
straight, arrowed line.
The arrow points in the
direction of the vector
The length of the line
represents the size of the
quantity
Vectors
Vectors can be multiplied
or divided by a scalar
This will change the length
of the vector
A negative scalar will
reverse the direction
Eg Force F=
So
2F=
1/2F=
-3F=
Vectors
Vectors can be added together.
This is done by drawing them
“head to tail”.
The result is a vector called a
resultant. The resultant has the
same effect as the 2 vectors
combined.
The order in which they are
added does not matter.
Eg d1+d2
d1
d1+ d2
d2
Vectors
Vectors can be subtracted.
This is done by adding a
negative vector
Order does matter.
Eg. v1- v2
v1
v 1 - v2
v2
-v2
Vectors
Vectors can be resolved
into components.
This is done using
SOHCAHTOA and/or
a2+b2=c2
Vertical
F
Component
40°
Horizontal component
(Do Page 43 Questions 3A)
Relative Velocity
The velocity of one object in
relation to another object.
The velocity an object
appears to move at may
change if the object
measuring is also moving.
The velocity of B relative to A
can be calculated by doing
this vector subtraction….
vBrelA  vB  v A
(Do Page 49 Questions 3B)
Forces
A force causes the motion
or shape of an object to
change.
Force is a vector quantity
so must have both a size
and a direction
Force is measured in
Newtons N.
A resultant (or net) force
is produced when 2 or
more forces act on an
object. These forces can be
added to find the resultant.
Forces
Newtons First Law Of Motion:
An object will remain in it’s
current state of motion until a
force acts to change it.
Newton’s Second Law Of
Motion:
The acceleration of an object
is proportional to the net force
applied.
Law 2 can be written like this
for short:
Fnet  ma
Forces
Newton’s Third Law Of
Motion:
For every action there is an
equal and opposite
reaction.
Forces
Friction:
Friction occurs when two
surfaces move past each
other. One of these
surfaces could be air – eg
air resistance is a frictional
force.
Friction is a force that
always opposes the
direction of the motion.
Friction is sometimes
called: drag, water
resistance, air resistance
or the retarding force.
Forces
Tension:
This is the force that occurs
in connecting strings and
ropes
Tension pulls in both
directions along the string
or rope.
Weight:
This is the force of gravity
pulling downwards on an
object.
Weight can be calculated
by:
Fw  mg
g is acceleration due to
gravity and has a value of
10ms-2 on Earth
(Do Page 55 Questions 4A)
Torque
Torque causes things to
spin.
Symbol: t (Greek letter
Tau)
Units: Nm
The size of a torque
depends on the size of the
force and the
perpendicular distance
from the pivot to where
the force is applied.
t  Fd
Equilibrium
An object is at equilibrium if
it is at rest or moving
uniformly (First Law)
Two conditions apply:
All the forces acting on the
object must add to zero,
All the torques acting on
the object must add to
zero.
(Do page 63 Questions 4B)
Momentum
The amount of “ooomph”
an object has.
Momentum depends on the
mass of an object and it’s
velocity.
Symbol: p
Unit: kgms-1
Momentum is a vector.
Momenum can be
calculated using:
p  mv
Momentum
If a force acts on an
object, it’s momentum will
change.
The change in momentum
can be calculated by
subtracting vectors.
Change in momentum
=final momentum – initial
momentum.
Dp  p f  pi
Impulse
When a resultant force acts on
an object, the amount it
changes the object’s
momentum by depends on
how long the force acts for.
The force multiplied by the
time it acts for is called
impulse.
Units: Ns
Impulse equals the change in
momentum.
FDt  Dp
(Do Page 73 Questions 5A)
Conservation of
Momentum
The conservation of
momentum principle
states: Momentum is
conserved in collisions and
explosions as long as
there is no net external
force acting.
This means the momentum
before equals the
momentum after.
m1vi1  m2 vi 2  m1v f 1  m2v f 2
Conservation of
Momentum
The same principle applies
in 2 dimensions.
The vector representing
the sum of the
momentums before must
be the same vector as the
one representing the sums
after.
Do Page 79 Questions 5B
Motion due to Gravity
All objects accelerate
towards the ground at (-)
10ms-2 because of gravity
when dropped.
This acceleration is fairly
constant at the Earth’s
surface, but varies at
great altitudes or on
other planets.
Gravity is always an
attractive force unlike
magnetism or electric
forces.
Do Page 86 Questions 6A
Projectile Motion
Projectile motion is a
parabolic shaped motion
experienced by moving
objects that have only the
force due to gravity acting
on them.
Eg.
Bullets,shotputs,netballs,
water jets
Projectile Motion
When dealing with
projectiles, the horizontal
and vertical components
are treated separately.
The horizontal motion is
constant velocity (as there
are no forces acting in this
direction).
The vertical motion is
constant acceleration of
10ms-2 due to the force of
gravity.
Kinematic Equations are
used to solve problems.
Do Page 89 Questions 6B
Circular Motion
Period of Rotation T - time
it takes to make one
rotation (revolution, cycle)
Measured in seconds s.
Frequency f – number of
rotations completed per
second.
Measured in Hertz Hz or s-1
T and f are inverses of
each other.
1
T
f
Circular Motion
Circumference – distance
travelled in one rotation
(m)
Circumference  2r
The speed of an object
moving in a circle can be
calculated by:
d 2r
Speed (v)  
t
T
Circular Motion
An object moving in a
circle may be travelling at
constant speed, but
because its direction is
always changing, its
velocity is changing….
If velocity is changing, the
object is accelerating….
If an object is accelerating,
there must be a net force
acting on it….
Circular Motion
The force acting on an
object in circular motion is
in towards the centre of
the circle, changing the
objects direction but not its
speed.
This is called centripetal
force.
This force causes a
centripetal acceleration
towards the centre of the
circle.
Circular Motion
Centripetal force
and acceleration
can be
calculated using
the following
formulae:
v=speed(ms-1)
r=radius of
motion
2
v
ac 
r
2
v
Fc  m
r
Do Page 98 Questions 7A
Energy
The three kinds of
mechanical energy are:
kinetic, gravitational and
elastic.
Energy E is measured in
Joules J
Work W is the process of
transforming energy from
one kind to another.
W  Fd
d is the distance moved in
the direction of the force.
Energy
If an object is lifted against
gravity, work is done
transforming chemical
energy(muscles) into
gravitational potential
energy.
The force needed is the
weight force of the object,
the distance moved is the
change in height:
W  Fd
 DE p  (mg)Dh
so
Energy
Any moving object has
kinetic energy.
Doubling the speed
increases the energy by
four. (Squared relationship)
When moving objects stop,
this energy is transformed
into other forms, eg sound,
heat
Ek  mv
1
2
2
Power
Power P is the rate at
which work is done.
Measured in Watts W (or
Js-1)
W
P
t
Conservation of Energy
The conservation of energy
principle states: Energy
cannot be created or
destroyed, only
transformed from one kind
to another.
Energy Efficiency
Often some of the forms it is
transformed into are not
useful. The energy is “lost”
to us
The efficiency of an object is
a measure of the ratio of
input energy to useful output
energy
useful output energy
Efficiency% 
100
total input energy
Springs
Energy can be stored in a
spring as elastic potential
energy.
Hookes Law: F=kx
F=force
k=spring constant (Nm-1) –
a measure of how stiff or
soft a spring is.
x=extension (m) – the
amount a spring is
stretched or compressed
when the force is applied.
Springs
Hooke’s Law as a graph:
Force (N)
Gradient = k
Area under graph =
energy stored in
spring
Extension (m)
Springs
Elastic
potential
energy can
be found by
calculating
the area
under a
Hooke’s
Law graph.
Area  12 b  h

 E  12 Fx
(  F  kx)
and
 E  kx
so
Do Page 107 Questions 8A
1
2
2
A person sits in a car with
a suspension of spring
constant 104Nm-1. If the
suspension is compressed
1.0 cm how much energy
is stored in the springs?
Ep=1/2kx2
=0.5 x 104 x (0.01)2
=0.5 J