Mechanics

1
Mechanics

• L2 NCEA
• Achievement Standard 2.4
• Text Book reference Chapters 7-13

2
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

3
Motion
Distance (scalar) How far in total the object has moved
Symbol d Unit m
Displacement (vector) How far the object ends up from its starting position
Symbol d or s Unit m
4
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
Symbol v Unit ms-1
5
Motion

• Velocity is calculated by

• 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.

6
Motion
Acceleration (Vector) The rate at which the velocity of an object is changing
Symbol a Unit ms-2

• Acceleration can be calculated by

Acceleration is always in the direction of
7
The concorde flies at an average velocity of1440
km hr-1. How long in seconds does it take to fly
100km?
1440km hr-1 is 400ms-1 and 100km is
100,000mUsing vd/t ? td/v ? 100,000/400 and
t250s

• How far will the concorde fly in a minute?

24km Anyone not sure how to get that?
8
A car travels 500m to the right turns around and
travels another 1000m to the left.The car
travelled with a uniform speed and the time taken
was 150s.Find total distance travelled total
displacement average speed of the car average
velocity of the car
1500m -500m 10ms-1 -3.3ms-1
9
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.

10
A car accelerates from 10ms-1 to 20ms-1 in 4.0s
Calculate its acceleration
A2.5ms-2Does everyone know how that was solved?

• The same car can brake from 20ms-1 to rest in
  5.0s
• Find the acceleration.

a 4.0ms-2 What is wrong with that
answer? a-4.0ms-2
11
Task

• Measure the speed and velocity of your birds.
• I need to see times distances and calculations
• The birds seem to move between 3-12 cms-1

Eric the snail moves at 3.0mms-1 N sees a bird
and takesoff West at 4mms-1. What is his change
in velocity?
12
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

13
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

14
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 d1d2

Sp p137
15
Vectors

• Vectors can be subtracted.
• This is done by adding a negative vector
• Order does matter.
• Eg. v1- v2

16
DV

• DVVf-Vi

try the ball thing
If the velocity changes this means the object
is If the object is accelerating there must
be a ..applied in the direction of the
acceleration.
17
Working out DV when Non Linear

• Draw a vector representing each motion.
• Draw the Vi vector.
• Draw a vector diagram of
• DVVf-Vi or DVVf -Vi
• 4. Using trig and/or pythagarosfind the
  magnitude and
• DIRECTION
• ofDV

Example on the board
18
Graphs of Motion

• Distance / diplacement versus time

• AConstant velocity (slow)
• BConstant velocity (faster)
• CStopped
• DConstant velocity (backward)
• EConstant velocity (backward past starting point)

19
Graphs of Motion

• Speed / velocity versus time

• area under the graph?

• AConstant acceleration (low)
• BConstant acceleration (high)
• CConstant velocity
• DConstant deceleration to stop
• EConstant acceleration in opposite direction

20
Kinematic Equations

• To solve problems involving objects moving in
  straight lines with constant acceleration.
• Terms used
• ddistance/displacement (m)
• viinitial velocity (ms-1)
• vffinal velocity (ms-1)
• aacceleration (ms-2)
• ttime (s)

21
Kinematic Equations
If you know 3 variables you can work out the
other 2
22
Tricks of the Trade

• It is assumed you know gravity in any problem
  which involves rising or falling.
• Look out for Vi0 or Vf0 in other words from
  rest or stops.
• Make sure you get the signs correct.A rising
  object will have acceleration due to gravity
  acting in the opposite direction to motion.

23

• A grasshoppers legs extend by 2.0cm in 0.020s
  when jumping from rest. Assuming the jump is
  vertical
• What is the average acceleration of the
  grasshopper while extending its legs?
• With what velocity does the grasshopper leave the
  ground?
• What is the maximum height the grasshopper can
  jump?

24

• A flea takes 1.0 millisecond to reach take off
  speed of 1.2 ms-1 in a jump.
• What is its average acceleration?
• Assuming vertical take off how high does the flea
  reach?

25

• A jet plane lands on one end of a runway 1.0km
  long. Its maximum stopping acceleration is
  -4.0ms-2 and it takes 20s to come to rest. Does
  the plane stop in time?

26
Vectors

• Vectors can be resolved into components.
• This is done using SOHCAHTOA and/or a2b2c2

Vertical Component
F
40
Horizontal component
27
A ship sails from Lyttelton and sets a straight
course of 130km in a direction N230E from the New
Brighton pier.How far North of the pier is the
ship?
130 cos 230 120km

• How far east of the pier is the ship?

130 sin230 51km
28

• A supermarket trolley is pushed with a force of
  200N acting at an angle of 400 to the ground.
  Find the effective horizontal force pushing the
  trolley along.

200N400
FHFcos 200 cos 400 153N
29

• How fast is the ball rising after being hit?
• How fast is the ball moving horizontally?

Vv V sin 47.10 52.0 sin 47.10
38.1ms-1
VHVcos 47.10 52.0 cos 47.10
35.4ms-1
52.0ms-1
47.10
30
Kiwi Bobsled

• When a green light shows the team accelerates at
  2.0ms-2 for 5.0s and then they all jump in.
  Acceleration still the same.
• How fast is it going after 5.0s?
• What is the distance after 5.0s?
• What is the average speed _at_ 5.0s?
• What distance is covered when v40ms-1

31
R.McLeod the Cyclist

• If he rides at 6.0ms-1 for 6.0s then 12ms-1 for
  12s what is his average speed.

• Clue the answer is not 9.0ms-1

10ms-1
32

• Mr KK runs athletically up the stairs at 5.5ms-1.
  A bunch of chemists are lazily traveling on the
  esculator at 2.3ms-1. What is the relative speed
  of Mr KK w.r.t. The
  chemists? The ground?
  A group of shoppers
  going down a
  similar esculator?

33

• A train goes by at 95ms-1

• A Man is walking forward at 1.2ms-1

• How fast will the man be moving to an observer on
  the ground? In the train?

34

• A train goes by at 95ms-1

• The Man is walking towards the back now at
  1.2ms-1

• How fast will the man be moving to an observer on
  the ground? In the train?

35

• A train goes by at 95ms-1

• A Man is walking forward at 1.2ms-1

• How fast will a bird flying 10ms-1 in the same
  direction see the man moving? How fast will the
  man see the bird flying?

36

• A train goes by at 95ms-1

A bird is flying 10ms-1 in the same direction as
the train. How fast will these people see the
bird flying?
37

• A train goes by at 95ms-1

A bird is flying 10ms-1 in the opposite direction
as the train. How fast will these people see the
bird flying?
38
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.

(Do Page 49 Questions 3B)
39
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, rugby
  balls

40
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 for vertical motion

Do Page 89 Questions 6B
41
The canon ball travels 25ms-1 when fired
horizontally from the top of a 45m cliff.
t0s
t1.0s
t2.0s
For each position find the horizontal, vertical
and resultant velocity.
t3.0s
42
t1.0s
t2.0s
t0s
t3.0s
43
A projectile path is the movement of an object
under the action of gravity only.

• Type 1 Questions

What is a projectile path?
Explain the motion of the golf ball in
the vertical direction. Give a reason for your
explanation
44
The ball is moving with a constant acceleration

• Type 1 Questions

acting vertically downwards (constantly
decelerating upwards).
The golf balls acceleration is due to gravity.
The golf balls weight is the unbalanced force
acting on it.
45
Back to Golf-Still Type 1

• Draw clearly on the diagram a velocity vector to
  represent the size and direction of the initial
  velocity (U) of the golf ball.

46
Golf Analysis
U

• What is the size and direction of the
  ballsspeed ?
• Quote the answer to the correct number
  ofsignificant figures.

47
u2 202 322? u 37.735925 38 m
s?1tan ? 20/32 ? ? 320.
Golf Analysis
What is the time taken for the ball to reach the
topof its flight?
vf vi at ? 0 20 10t ? t 2.0 s
48
Golf Analysis
Calculate the maximum vertical height (H)
reached by the golf ball at the top of its
flight. Acceleration due to gravity is 10 m s-2.
vf 2 vi 2 2ad ? 02 202 2 x(- 10)H ? H
20 m
49
Golf Analysis
Explain the motion of the golf ball inthe
horizontal direction, Give a reason for your
explanation.
Motion is constant velocity (speed and direction)
as there is no unbalanced force acting on the
golf ball in the horizontal direction.
What is the velocity of the ball at the top of
its flight?
32ms-1 horizontal
50
Golf Analysis
Calculate the horizontal distance (R) travelled
by the golf ball.
v ?d / ?t ? 32 R/2t where ?t 2t ? R 2 x
32 x 2 128m
What is the velocity of the ball when it lands?
38ms-1 _at_320 to the ground
What is a force? (type 1)
51
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.

52
Forces

• Newtons First Law Of Motion
• An object will remain in its current state of
  motion until a force acts to change it.
• Newtons 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

53
Forces

• Newtons Third Law Of Motion
• For every action there is an equal and opposite
  reaction.

54
Not What it seems!!!
A 500kg hot air balloon rises at a rate of
0.75ms-2in a cool Christchurch morning air.
What is the total force lifting the balloon?
5375N
5000N to overcome gravity 375N to cause the
acceleration
55
Type 1
Draw a force diagram of an aeroplane
acceleratingin level flight.
Draw a force diagram of Jim standing with
bothfeet on a skate board traveling between B
and D block.
What will be happening to a cyclist
experiencingno net force?
56
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.

Heat?
57
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
• g is acceleration due to gravity and has a value
  of 10ms-2 on Earth

(Do Page 55 Questions 4A)
58
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.

59
Equilibrium

• An object is at equilibrium if it is at rest or
  moving uniformly (First Law)
• Two conditions apply

(Do page 63 Questions 4B)
60
Equilibrium

• All the forces acting on the object must add to
  zero

SF0
(Do page 63 Questions 4B)
61
Equilibrium

• All the torques acting on the object must add to
  zero.

St0
(Do page 63 Questions 4B)
62
Momentum

• The amount of oooomph an object has.
• Momentum depends on the mass of an object and
  its velocity.
• Symbol p
• Unit kgms-1
• Momentum is a vector.
• Momentum can be calculated using

1 of 5 on momentum
63
Momentum

• If a force acts on an object, its momentum will
  change.
• The change in momentum can be calculated by
  subtracting vectors.
• Change in momentum final momentum initial
  momentum.

2 of 5 on momentum
64
Impulse

• When a resultant force acts on an object, the
  amount it changes the objects 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.

(Do Page 73 Questions 5A)
3 of 5 on momentum
65
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.

4 of 5 on momentum
66
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
Whew! last one on momentum
67
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.

68
Circular Motion

• Circumference distance travelled in one
  rotation (m)
• The speed of an object moving in a circle can be
  calculated by

69
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.

70
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.

71
Circular Motion

• Centripetal force and acceleration can be
  calculated using the following formulae
• vspeed(ms-1)
• rradius of motion

Do Page 98 Questions 7A
72
Solve these
The minute hand of the clock is 10cm, second
hand 9.0cm, hour hand 7.0cm. How fast is the
tip of each hand going?
A Swift, the worlds fastest bird, of mass
0.10kgis seen to complete a circular turn of
radius 10mwithout changing speed of 72kmhr-1.
i) Find the centripetal acceleration of the bird
and compare it to gravity. ii) Find the
centripetal force on the bird.
73
Solve These
A record player has speeds of 33rpm, 45rpm
and78rpm. Find the period of revolution for a
record playedat each speed. If a fly lands 30cm
from the centre of rotation. What is the
tangential velocity if the record is going
33rpm?
74
A cricket ball is thrown from the boundary.
Describe the path it takes.

• Type 1 Questions

What are the units for momentum, torque,
tangential speed, work, acceleration?
Draw a force diagram for the ball travelling
through the air.
What is the direction of the net force on the
ball?
Timy and Cameroon sit on a bench outside A2 at
playtime. Draw a diagram showing their weightthe
weight of the bench and the support
forces Through the bench legs
75
Solve This

• A stone of mass 750g is tied to the end of a
  string and spun. The string has a breaking
  strain of 35N and is 1.0m long. It is spun in a
  planehorizontal to the earth at a rate of 60
  timesa minute.
• What is the tangential velocity of the stone?
• What is the centripetal acceleration of the
  stone?
• Show whether the string will break.
• If the stone is now spun in a vertical plane
  atthe same speed show whether the string
  willbreak now.

76
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
  Earths 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
77
Energy

• The three kinds of mechanical energy are
  kinetic, gravitational and elastic.
• Work W is the process of transforming energy from
  one kind to another.

• Energy E is measured in ..
• d is the . moved in the direction of the
  force.

1 of 6 on energy
78
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

2 of 6 on energy
79
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

How many more man-3
power
80

• Type 1 Questions

When Timy and Cameroon sat on a bench outside A2
at playtime you could work out the forces
acting. Discuss the Physics principles you
would have to assume to work out the
forces? Have to mention equilibrium That means
the forces add up to 0, nothing, zip, nada The
torques clockwise equal the anticlockwise torques
81

• Type 1 Questions

Brody running with the union ball at 2.5ms-1
North changes direction and goes 3.2ms-1West to
avoid a tackler.Draw labeled vector diagrams
showing his initialand final velocity.
Draw a vector diagram showing his changein
velocity.
He then crashes into another player and
rebounds.Discuss the Physics principle that will
allow you tocalculate his rebound speed.
82
Power

• Power P is the rate at which work is done.
• Measured in Watts W (or Js-1)

4 of 6 get over it
conservation
83
Conservation of Energy

• The conservation of energy principle states
  Energy cannot be created or destroyed, only
  transformed from one kind to another.

One more after this
Efficiency/lost
84
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
• Elastic collisions-stop talking and listen then
  make a note

End of the energy story for today
85
Solve This
80 of the electricity going into a light bulb
getsturned into heat. How much energy does a
100W bulb use in10 minutes and how much of this
is turned into light?
12kJ light
86
Solve This
A 60kg woman runs up a set of stairs in 15s. She
rises 10m in her climb. Calculate her power. The
important message with this problem is to
realise that the majority of energy is used
against gravity as opposed to the horizontal
motion
87
Solve This
At what speed must a 50 gram squashball travel
if it is to have energy of 0.50J?
4.5ms-1
If the energy is doubled to 1.0J whatmust the
speed be?
6.3ms-1
88
Solve This
A sports car with mass ¾ of a tonne
acceleratesto 108 kmhr-1 in 8.0s. Ignoring
friction what poweris exerted by the motor?
89
Solve This
A 200W motor is used to lift a 15kg bucket of
cement 40m. How long will it take?
90
Springs
91
Springs

• Energy can be stored in a spring as elastic
  potential energy.
• Hookes Law Fkx
• Fforce
• kspring constant (Nm-1) a measure of how stiff
  or soft a spring is.
• xextension (m) the amount a spring is
  stretched or compressed when the force is
  applied.

92
Springs

• Hookes Law as a graph

93
Springs

• Elastic potential energy can be found by
  calculating the area under a Hookes Law graph.

Do Page 107 Questions 8A
94
Solve This
A person sits in a car with a suspension of
spring constant 104 Nm-1. If the suspension is
compressed 1.0 cm how much energy is stored in
the springs?
Ep1/2kx2 0.5 x 104 x (0.01)2 0.5 J
95

• You know everything required to get an excellence
  in Thursdays test now
• But will you.
• Revise the problems in your text and
• The 2.4 exam on the NCEA website for 2004.