Transcript Vectors: Motion and Forces in Two Dimensions

Newton's First Law
• Newton's first law of motion: An object at rest stays at
rest and an object in motion stays in motion with the
same speed and in the same direction unless acted
upon by an unbalanced force.
The Meaning of Force
• A force is a push or pull upon an object resulting from the
object's interaction with another object.
• Force is a quantity that is measured using the standard
metric unit known as the Newton.
• All forces (interactions) between objects can
be placed into two broad categories
– Contact forces - that result when the two
interacting objects are perceived to be
physically touching each other.
– Field forces - that result even when the two
interacting objects are not in physical contact
with each other, yet are able to exert a push
or pull despite their physical separation.
Contact Forces
Action-at-a-Distance
Forces (Field Force)
Applied Force
Gravitational Force
Tension Force
Electrical Force
Normal Force
Magnetic Force
Air Resistance Force
Frictional Force
Spring Force
Gravity Force (Weight) Fgrav
• The force of gravity is the force with which the earth,
moon, or other massively large object attracts another
object towards itself. By definition, this is the weight
of the object. All objects upon earth experience a force
of gravity that is directed "downward" towards the
center of the earth. The force of gravity on earth is
always equal to the weight of the object as found by
the equation:
• Fgrav = m • g
• where g = 9.81 N/kg (on Earth) and m = mass (in kg)
• Note: g is different at different locations
Practice- indicate Fg on each box with an arrow
Fg
Fg
Fg
Fg
Fg
Fg
Comparing Mass and Weight
Weight
• The force of gravity.
• Vector, its direction is
downward.
• W = mg
• The weight of an object
(measured in Newton)
will vary according to
where in the universe
the object is.
Mass
• The mass of an object
refers to the amount of
matter that is contained
by the object;
• Scalar, has no direction
• The mass of an object
(measured in kg) will be
the same no matter where
in the universe that object
is located.
Normal Force (FN )
• The normal force is the support force exerted upon an
object that is in contact with another stable object
(usually a surface). The direction of the normal force is
perpendicular to the surface, from the surface toward
the object and on the object.
Practice- indicate FN on each box with an arrow
FN
FN
FN
Fg
Fg
Fg
FN
FN
FN
Fg
Fg
Fg
Friction Force (Ff)
• The friction force is the force exerted by a
surface as an object moves across it or makes
an effort to move across it. The friction force
often opposes the motion of an object.
• Friction results from the two surfaces being
pressed together closely, causing
intermolecular attractive forces between
molecules of different surfaces. Friction depends
upon the nature of the two surfaces and upon
the degree to which they are pressed
together.
Ff = μFN
Practice- indicate Ff on each box with an arrow
v
FN
Ff
FN
FN
Ff
v
Ff
v
Fg
Fg
Fg
Ff
FN
Ff
Ff
FN
FN
v
v
Fg
v
Fg
Fg
Air Resistance Force (Fair )
• The air resistance is a special type of frictional force that
acts upon objects as they travel through the air. The
force of air resistance is often observed to oppose the
motion of an object. This force will frequently be
neglected due to its negligible magnitude.
•
Tension Force (FT
) force is the force that is transmitted through
The tension
a string, rope, cable or wire when it is pulled tight by
forces acting from opposite ends. The tension force is
directed along the length of the wire and pulls equally on
the objects on the opposite ends of the wire.
Spring Force (Fspring )
• The spring force is the force exerted by a compressed or
stretched spring upon any object that is attached to it. An
object that compresses or stretches a spring is always
acted upon by a force that restores the object to its rest
or equilibrium position – directed toward equilibrium
position.
Balanced and Unbalanced Forces
If two individual forces are of equal
magnitude and opposite direction,
then the forces are said to be
balanced.
When only balanced forces act on
an object, the object is said to be at
equilibrium.
Unbalanced forces
State of Motion
• The state of motion of an object is defined
by its velocity - the speed with a direction.
• Inertia: tendency of an object to resist
changes in its velocity.
• Inertia: tendency of an object to resist
accelerations.
Newton’s First Law
Also known as the “Law of Inertia”
Inertia
Tendency of an object to maintain its STATE OF
MOTION
Forces Don't Keep Objects Moving
Everyday Applications of Newton's First Law
• Blood rushes from your head to your feet while quickly
stopping when riding on a descending elevator.
• The head of a hammer can be tightened onto the
wooden handle by banging the bottom of the handle
against a hard surface.
• A brick is painlessly broken over the hand of a physics
teacher by slamming it with a hammer. (CAUTION: do
not attempt this at home!)
• To dislodge ketchup from the bottom of a ketchup bottle,
it is often turned upside down and thrusted downward at
high speeds and then abruptly halted.
• Headrests are placed in cars to prevent whiplash injuries
during rear-end collisions.
• While riding a skateboard (or wagon or bicycle), you fly
forward off the board when hitting a curb or rock or other
object that abruptly halts the motion of the skateboard.
Inertia is proportional to MASS
Do these guys have a lot of inertia?
MORE MASS
means
MORE INERTIA
LOTS OF INERTIA
hard to…
GET MOVING or
STOP
Drawing Free-Body Diagrams
• Free-body diagrams are used to show the
relative magnitude and direction of all forces
acting upon an object in a given situation.
• The size of the arrow in a free-body diagram
reflects the magnitude of the force. The arrow
shows the direction that the force is acting.
• Each force arrow in the diagram is labeled to
indicate the exact type of force.
• It is generally customary to draw the force arrow
from the center of the box outward in the
direction that the force is acting.
A block of wood is sitting motionless on a table.
What forces are acting on it?
Normal
Weight
FN
Fg
Normal Force is a
REACTION
force that any
object exerts
when pushed on
Weight is the force of
gravity
pulling an object
toward the
CENTER OF THE
EARTH
practice
• A book is at rest on a tabletop. Diagram the
forces acting on the book.
FN
Fg
Determining the Net Force
•The net force is the vector sum of all the forces that act upon an object.
A
400 N
up
30 N
B
C
200 N
down
20 N left
R2 = (30N)2 + (40N)2
θ = tan-1(30/40) = 53.1o
40 N
Net force is 50 N at 53.1o West of North
Net Force
• If there is NO NET FORCE on an object, then it is
at EQUILIBRIUM and either:
MOTIONLESS OR MOVING WITH CONSTANT VELOCITY
• So a “net” or “unbalanced” force will
– CHANGE AN OBJECT’S VELOCITY
• Changing velocity means ACCELERATION
A net force (an unbalanced force) causes an acceleration
Description of Motion
Net Force: Yes or No?
yes
yes
no
no
yes
yes
Force  Acceleration
• How much acceleration?
• Depends on:
– AMOUNT OF FORCE
• MORE FORCE = MORE ACCELERATION
• Acceleration is directly related to force
– MASS OF OBJECT
• MORE MASS = LESS ACCELERATION
• Acceleration is inversely related to mass
Newton’s Second Law
“The acceleration of an object is directly proportional to
the net external force acting on the object and inversely
proportional to the mass of the object.”
Fnet
a
m
Unit of force is the NEWTON (N)
Relationships: a ~ F; a ~ 1/m
Fnet
a
m
a
a
F
m
Fnet
a
m
• If mass is held constant,
• doubling of the net force results in …
• a doubling of the acceleration,
• halving of the net force results in …
• a halving of the acceleration.
• If force is held constant,
• doubling of the mass results in …
• a halving of the acceleration
• halving of the mass results in …
• a doubling of the acceleration.
Example
• A 2 kilogram box is pushed with a net,
unbalanced force of 10 newtons.
• What is the acceleration experienced by
the box?
a = Fnet / m
a = (10 N) / (2 kg)
a = 5 m/s2
The Big Misconception
• The most common misconception is
one that dates back for ages; it is the
idea that sustaining motion requires a
continued force.
• Newton's laws declare loudly that a
net force (an unbalanced force)
causes an acceleration;
Are You Infected with the Misconception?
• Two students discussing an object that is being acted
upon by two individual forces as shown. During the
discussion, Anna Litical suggests to Noah Formula that
the object under discussion could be moving.
• Noah Formula objects, arguing that the object could not
have any horizontal motion if there are only vertical
forces acting upon it.
• Who do you agree with?
Friction
A force that
causes
surfaces friction
to stick together
Ways
to minimize
and opposes motion.
SMOOTH
LUBRICATION
SURFACES
At the MICROSCOPIC level, most surfaces are very
BUMPY and IRREGULAR
Coefficient of Friction (μ)
• How much materials STICK TOGETHER
– DIMENSIONLESS (no units)
– The greater the coefficient, the greater the
tendency to STICK TOGETHER
– The coefficient is lowered if surfaces are SLIDING
past each other
Friction Force
• Static Friction
– STATIONARY OBJECTS – cancels out
applied force - KEEPS OBJECTS IN PLACE
– CAN CHANGE – increases as the applied
force increases until it reaches the maximum
quantity for that specific surface.
– ROLLING OBJECTS
• Kinetic Friction
– SLIDING OBJECTS
– OPPOSES MOTION
Calculating Friction Force
• Amount of friction depends on:
– Coefficient of friction
• Static – the object is motionless, rolling, or pushing off
from a surface
• Kinetic – the object is sliding across a surface
– Normal Force
• Greater normal force  HIGHER friction force
Ff  FN
Kinetic versus Static Friction
• kinetic friction results
when an object moves
across a surface.
Ffrict = μ • Fnorm
• The symbol μ represents
the coefficient of kinetic
friction between the two
surfaces. The coefficient
value is dependent
primarily upon the nature
of the surfaces that are in
contact with each other. It
does not depends on area
of contact, the angle of the
area, or the temperature,
etc.
• Static friction results when
the surfaces of two objects
are at rest relative to one
another and a force exists on
one of the objects to set it
into motion relative to the
other object.
• The static friction force
balances the force that you
exert on the box such that the
stationary box remains at
rest.
Ffrict-static ≤ μfrict-static• Fnorm
Finding the unknowns
• Fnet is the vector sum of all the individual
forces. The three major equations that will
be useful are
– Fnet = m•a,
– Fg = m•g,
– Ff = μ•FN
Example #1
• A man pushes a 50 kilogram crate across a
frictionless surface with a constant force of
100 Newtons.
WhatDraw
isWhat
the
What
What
anormal
free-body
isisisthe
the
thenet
force
crate’s
weight
force
diagram
that
acceleration?
of
on
pushes
the
the
of crate?
the
crate?
on
crate.
the crate?
FN
Fg = mg
Fg = (50 kg)(9.81 m/s2)
Fg = 490.5 N
FA
FN = Fg
FN = 490.5 N
Fg
Fnet will only be
the 100N horizontal
force
a = Fnet / m
a = (100 N) / (50 kg)
a = 2 m/s2
Example #2
• A horse pulls a 500 kilogram sled with a
constant force of 3,000 Newtons. The force of
friction between the sled and the ground is
500 Newtons.
WhatDraw
isWhat
What
the
What
anormal
free-body
isisisthe
the
thenet
sled’s
force
weight
force
diagram
that
acceleration?
of
onpushes
the
the
of sled?
the
sled?
on
sled.
the sled?
Fg = mg
Fg = (500 kg)(9.81 m/s2)
Fg = 4905 N
Ff
FN = Fg
FN = 4905 N
Fg
FN
Fnet = ΣFx
Fnet = 3000 N – 500 N
Fnet = 2500 N
FA
a = Fnet / m
a = (2500 N) / (500 kg)
a = 5 m/s2
Example #3
the object is moving horizontally. Use the diagram to
determine the normal force, the net force, the mass,
and the acceleration of the object.
80 N
8 kg
5 m/s2 right
40 N right
Example #4
• Edwardo applies a 4.25-N rightward force to a 0.765-kg
book to accelerate it across a tabletop. The coefficient of
friction between the book and the tabletop is 0.410.
Determine the acceleration of the book.
Example #5
•
Lee Mealone is sledding with his friends when he
becomes disgruntled by one of his friend's comments.
He exerts a rightward force of 9.13 N on his 4.68-kg sled
to accelerate it across the snow. If the acceleration of the
sled is 0.815 m/s/s, then what is the coefficient of friction
between the sled and the snow?
Free Fall and Air Resistance
Free Fall
Falling with air resistance
• Objects that are said to be
undergoing free fall, are
• not encountering air
resistance;
• falling under the sole
influence of gravity. All
objects will fall with the
same rate of acceleration,
regardless of their mass.
This is due to that the
acceleration is The ratio of
force to mass (Fnet/m)
• As an object falls through air, it
usually encounters some degree of
air resistance - the result of collisions
of the object's leading surface with air
molecules.
• The two most common factors that
have a direct affect upon the amount
of air resistance are
– the speed of the object: Increased
speeds result in an increased
amount of air resistance.
– the cross-sectional area of the
object: Increased cross-sectional
areas result in an increased
amount of air resistance.
Falling with air resistance – terminal velocity
• As an object falls, it picks up speed. The increase in
speed leads to an increase in the amount of air
resistance. Eventually, the force of air resistance
becomes large enough to balances the force of gravity.
At this instant in time, the net force is 0 Newton; the
object will stop accelerating. The object is said to have
reached a terminal velocity.
Newton's Third Law
• For every action, there is an equal and
opposite reaction.
• Forces always come in pairs - equal and
opposite action-reaction force pairs.
• Examples:
– The propulsion of a fish through the water.
– The flying motion of birds.
– The motion of a car on the way to school.
Third Law Examples
• A firefighter directs a stream of water
from a hose to the east. In what
direction is the force on the hose?
There will be a force on the hose to the WEST
• A man getting out of a rowboat
jumps north onto the dock. What
happens to the boat?
The boat will move to the SOUTH
Identifying Action and Reaction
Force Pairs
• Identifying and describing action-reaction force pairs is a
simple matter of identifying the two interacting objects and
making two statements describing who is pushing on
whom and in what direction.
Action/reaction forces vs.
equilibrium forces
• Action and
reactions force act
on different
objects
Force on the car
• Equilibrium
forces act on
same object
FN
Fg
Force on the ground
Check Your Understanding
1. While driving down the road, a firefly strikes the
windshield of a bus and makes a quite obvious
mess in front of the face of the driver. This is a
clear case of Newton's third law of motion. The
firefly hit the bus and the bus hits the firefly.
Which of the two forces is greater: the force on
the firefly or the force on the bus?
2. For years, space travel was believed to be
impossible because there was nothing that
rockets could push off of in space in order to
provide the propulsion necessary to accelerate.
This inability of a rocket to provide propulsion is
because ...
a. ... space is void of air so the rockets have
nothing to push off of.
b. ... gravity is absent in space.
c. ... space is void of air and so there is no air
resistance in space.
d. ... nonsense! Rockets do accelerate in space
and have been able to do so for a long time.
3. Many people are familiar with the fact that a rifle recoils
when fired. This recoil is the result of action-reaction
force pairs. A gunpowder explosion creates hot gases
that expand outward allowing the rifle to push forward
on the bullet. Consistent with Newton's third law of
motion, the bullet pushes backwards upon the rifle. The
acceleration of the recoiling rifle is ...
a. greater than the acceleration of the bullet.
b. smaller than the acceleration of the bullet.
c. the same size as the acceleration of the bullet.
Objectives: Forces in Two
Dimensions
1. Net Force Problems Revisited
2. Equilibrium and Static
3. Inclined Planes
Net Force Problems Revisited
• When forces acting at angles to
the horizontal, Newton’s 2nd law
still applies:
∑F = ma
• Force is a vector quantity.
Adding forces in 2 dimensions
follows the rules for adding
vectors.
• The two ways for adding vectors are:
1. Graphically - Head and tail method
2. Mathematically: Add forces by components and
Pythagorean Theorem to determine magnitude and
tangent function to determine direction
Determine the Fnet graphically
Determine the Fnet mathematically
1. Resolve the vectors at an angle into x
and y components.
2. Add all the x components together
3. Add all the y components together
4. Use Pythagorean Theorem to find the
resultant (hypotenuse)
5. Resultant2 = x2 + y2
6. Use trigonometric function to determine
the direction: tanθ = opp / adj
Determine the Fnet mathematically
21 N
Ax = 20cos(225o) = -14 N
C
Ay = 20sin(225o) = -14 N
B
21 N
-14 N
D
E
A -14 N
Cx = 30cos(45o) = 21 N
Cy = 30sin(45o) = 21 N
Rx = Ax + Bx + Cx + Dx + Ex
Rx = -14N + 21N + 25N = 32N
R2 = Rx2+ Ry2
Ry = Ay + By + Cy + Dy + Ey
R = 39.4 N
Ry = -14N + 20N + 21N -50 N = -23N
θ = tan-1(-23/32) = -36o
Example - Pulling on an Angle
A block is pushed along a
frictionless, horizontal surface with a
force of 100 newtons at an angle of
30° above horizontal.
FN
FAY
FA
30˚
This applied force (FA)
FAxcan
= 100cos(30
) = 87 N
be broken ointo
COMPONENTS
FAy = 100sin(30o) = 50 N
X verticalYforce must
The total
be 0, so
FAY
RyFAX
= FN + FAY
–Fg = 0
FN = Fg F–g FAY
FN
FAX
Fg
Total =R
FAX
= Rx Total
= Fax= 0
Acceleration depends only on
FAX
Example
• A man pulls a 40 kilogram crate across a
smooth, frictionless floor with a force of 20 N
that is 45˚ above horizontal.
What is the net force on the sled?
How could the
Fnetacceleration
= FA cos θ be increased?
Fnet = (20
N)(cos
45°) F greater and
Pushing at a smaller
angle
will make
net
Fnetincrease
= 14.14acceleration.
N
therefore
What is the crate’s acceleration?
a = Fnet / m
a = (14.14 N) / (40 kg)
a = 0.35 m/s2
Pushing on an Angle
A block is pushed along a
frictionless, horizontal surface with a
force of 100 newtons at an angle of
30° below horizontal.
This applied force (FA)
canXbe broken
Y into
COMPONENTS
FAX
F
FN
FAX
FAY
The total verticalg force must
N
be 0, Fso
= FgTotal
+ FAY
Total F
=N
FAX
=0
-30˚
FA
Fg
FAY
Acceleration depends only on
FAX
Example
• A girl pushes a 30 kilogram lawnmower
with a force of 15 Newtons at an angle of
60˚ below horizontal.
Assuming there is no friction, what is the
acceleration of the lawnmower?
Fnet = FA cos θ
Fnet = (15 N)(cos 60°)
Fnet = 7.5 N
a = Fnet / m
a = (7.5 N) / (30 kg)
a = 0.25 m/s2
What could she do to reduce her acceleration?
Push at an greater angle
Example – find acceleration
•
•
•
•
The vertical forces are balanced (Fgrav, Fy, and Fnorm add up to 0 N),
The horizontal forces add up to 29.3 N, right
The net force is 29.3 N, right
a = Fnet / m = 29.3 N / 10 kg = 2.93 m/s2, right
Determine the net force and
acceleration
• Fnet = 69.9 N, right
• m = (Fgrav / g) = 20 kg
• a = (69.9 N) / (20 kg) =3.50 m/s/s, right
Equilibrium and Static
• When all the forces that act upon an object are
balanced, then the object is said to be in a state of
equilibrium.
• An object at equilibrium is either ...
– at rest and staying at rest, or
– in motion and continuing in motion with the same
speed and direction.
• "static equilibrium." refers to an object at rest
Example
• A frame is shown with the given tension. Determine the
weight of the frame.
Rx = Ax + Bx + Cx = 0
Ax = 50cos(150o) = -43 N
A
B
30o
Bx = 50cos(30o) = 43 N
Cx = Rx - Ax - Bx = 0
Ry = Ay + By + Cy = 0
C=?
C2 = Cx2+ Cy2
R = 50. N
Ay = 50sin(150o) = 25 N
By = 50sin(30o) = 25 N
Cy = Ry - Ay - By = -50 N
example
• A sign is shown with the given mass of 5 kg.
Determine the tension of each cable.
Tsin140o
Tsin40o
A=T
Tcos140o
B=T
40o
40o
Fg = Tsin40o + Tsin140o
(5 kg)(9.81 m/s2) = 1.286T
T = 38 N
C = Fg
Tcos40o
An important principle
• As the angle with the horizontal increases, the amount of
tensional force required to hold the sign at equilibrium
decreases.
Fg = 10 N
• A tool used to move objects from one height to
another.
• Allows for the movement of an object without
lifting it directly against gravity.
Down the Slope
• The object accelerate
downward due to the
component gravity that is
parallel to the plane.
Fg on Inclined Plane
Calculations
• Consider forces:
– Perpendicular
• F┴ = Fg cos θ
• Cancel out Normal (FN )
– Parallel
• F// = Fg sin θ
• Could be in the same or
opposite of Friction (Ff )
Tilt you head method
Essential Knowledge
• What happens to the component of weight that is
perpendicular to the plane as the angle is increased?
Decreases – Fg perpendicular
• What happens to the component of weight that
points ALONG the plane as the angle is increased?
Increases – Fg parallel
• What happens to the normal force as the angle is
increased?
Decreases – depends on Fg perpendicular
• What happens to the friction force as the angle is
increased?
Decreases – depends on normal force
• The net force is the vector sum of all the
forces.
– All the perpendicular components (including
the normal force) add to 0 N.
– All the parallel components (including the
friction force) add together to yield the net
force. Which should directed along the
incline.
In the absence of friction
Fnet = F//
mgsinθ = ma
a = gsinθ
Object is at equilibrium – at rest or
moving with constant velocity
Ff
Horizontal:
Fnet = 0
F// = Ff
Vertical:
mgsinθ = μFN
F┴ = FN
mgsinθ = μ∙mgcosθ
mgcosθ = FN
tanθ = μ
Example
Fg = 50N
30°
• What is the magnitude of the normal force?
FN = Fg perpendicular = Fg cos θ = 43.3 N
• If the box is sliding with a constant velocity,
what is the magnitude of the friction force?
Ff = Fg parallel = Fg sin θ = 25 N
example
• The free-body diagram shows the forces acting upon a 100kg crate that is sliding down an inclined plane. The plane is
inclined at an angle of 30 degrees. The coefficient of friction
between the crate and the incline is 0.3. Determine the net
force and acceleration of the crate.
F┴ = Fgrav∙cos30o = 850 N
F// = Fgrav∙sin30o = 500 N
In perpendicular direction:
Fnorm = F┴ = 850 N
In parallel direction:
Fnet = F// - Ff
Fnet = 500 N - µFnorm
Fnet = 235 N
a = Fnet / m = 2.35 m/s2
practice
Double Trouble (a.k.a., Two Body
Problems)
• Two body-problems can typically be approached using
one of two basic approaches.
– One approach is the system analysis, the two
objects are considered to be a single object moving
(or accelerating) together as a whole.
– Another approach is the individual object analysis,
either one of the two objects is isolated and
considered as a separate, independent object.
Example - system analysis
• A 5.0-kg and a 10.0-kg box are touching each other. A
45.0-N horizontal force is applied to the 5.0-kg box in
order to accelerate both boxes across the floor. Ignore
friction forces and determine the acceleration of the
boxes and the force acting between the boxes.
m = 15 kg
Fnet = 45 N
a = Fnet / m = 3 m/s2
Example - individual analysis
In vertical direction:
FN = Fg = (5 kg) (9.81 m/s2) = 49 N
In vertical direction:
FN = Fg = (10 kg) (9.81 m/s2) = 98 N
In horizontal direction:
Fnet = Fapp - Fcontact
(5 kg)a = 45 N - Fcontact
In horizontal direction:
Fnet = Fcontact
(10 kg)∙a = Fcontact
5a = 45 – 10a
a = 3 m/s2
Example: system analysis
• A 5.0-kg and a 10.0-kg box are touching each other. A 45.0N horizontal force is applied to the 5.0-kg box in order to
accelerate both boxes across the floor. The coefficient of
kinetic friction is 0.200. Determine the acceleration and the
contact force.
In vertical direction:
FN = Fg = (15 kg) (9.81 m/s2) = 147 N
In horizontal direction:
Fnet = Fapp - Ffrict = 45 N - μ•Fnorm
Fnet = 15.6 N
a = Fnet / m = (15.6 N/15.0 kg) = 1.04 m/s2
However, in order to find the contact force between the
objects, we must make individual analysis.
Example: individual analysis
In vertical direction:
FN = Fg = (10 kg) (9.81 m/s2) = 98 N
In horizontal direction:
Fnet = Fcontact - Ff
(10 kg)∙(1.04 m/s2) = Fcontact - μ•Fnorm
10.4 = Fcontact – (0.2)(9.8)
Fcontact = 8.44 N