Download Chapter 4 - Newton`s Laws of motion

Chapter 4 - Newton’s Laws of motion
4.1 force and interactions
• A force is a push or a pull. It is an
interaction between two bodies or between
a body and its environment.
• Force as a vector quantity
For simplicity sake, all forces (interactions)
between objects can be placed into two broad
categories
Contact forces
Long range force
Contact force
• Contact forces are those types of forces which
result when the two interacting objects are
perceived to be physically contacting each
other.
• Examples of contact forces include
• frictional forces
tensional forces
normal forces,
Action-at-a-distance forces
• The types of forces which 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.
• Examples of action-at-a-distance forces include
– gravitational forces. The sun and planets exert a
gravitational pull on each other despite their large
spatial separation. Even when your feet leave the earth
and you are no longer in physical contact with the earth,
there is a gravitational pull between you and the Earth.
– Electric forces are action-at-a-distance forces. For
example, the protons in the nucleus of an atom and the
electrons outside the nucleus experience an electrical
pull towards each other despite their small spatial
separation.
Contact Forces
Action-at-a-Distance Forces
Frictional Force
Gravitational Force
Tension Force
Electrical Force
Normal Force
Magnetic Force
Air Resistance Force
Applied Force
Spring Force
The unit of force
• Force is a quantity which is measured using the
standard metric unit known as the Newton.
• A Newton is abbreviated by a "N." To say "10.0
N" means 10.0 Newton of force.
• One Newton is the amount of force required to
give a 1-kg mass an acceleration of 1 m/s/s.
Thus, the following unit equivalency can be
stated:
A force is a vector quantity
• A vector quantity is a quantity which has both
magnitude and direction. To fully describe the
force acting upon an object, you must describe
both the magnitude (size or numerical value)
and the direction. Thus, 10 Newton is not a full
description of the force acting upon an object. In
contrast, 10 Newton, downwards is a complete
description of the force acting upon an object;
both the magnitude (10 Newton) and the
direction (downwards) are given.
Superposition of forces
• When there are more than one force acting on a
body, the effect on the body’s motion is the same
as if a single force R were acting equal to the
vector sum of the original forces:
• R = F1 + F2.
• It’s more convenient to describe a force F in
terms of its x and y components Fx and Fy
• Our coordinate axes does not have to be vertical
and horizontal.
Note: we draw a wiggly line through the force vector F to
show that we have replaced it by its x and y components.
• To fine the vector sum of all the forces (net
force) acting on a body:
Example 4.1 superposition of forces
• Three professional wrestlers are fighting over the same
champion's belt. As viewed from above, they apply the
three horizontal forces the belt that are shown in the
figure. The magnitudes of the three forces are F1 = 250
N, F2 = 50 N, F3 = 120 N. Find the x and y components
of the net force on the belt, and find the magnitude and
direction of the net force.
Test your understanding 4.1
•
1.
2.
3.
4.
5.
6.
The figure shows a force F acting on a crate. With the x- and yaxes shown in the figure, which statement about the
components of the gravitational force that the each exerts on
the crate (the crate’s weight) is correct?
The x- and y-components are both positive
The x-components is zero and the y-component is positive
The x-component is negative and the y-component is positive
The x- and y-components are both negative
The x-component is zero and the y-component is negative
The x-component is positive and the y-component is negative.
4.2 Newton’s 1st Law
• Newton’s first law of motion: a body acted on by
no net force moves with constant velocity (which
may be zero) and zero acceleration.
• The tendency of a body to keep moving once it s set in
motion or the tendency of a body at rest to remain at rest is
due to inertia.
• It’s important to note that Newton’s first law is under the
condition that the net force on the object is zero.
Newton’s 1st Law – a body in
equilibrium
Example 4.2 – zero net force
means constant velocity
• In the classic 1950 science fiction film X-M, a spaceship
is moving in the vacuum of outer space, far from an
planet, when its engine dies. As a result, the spaceship
slows down and stops, what does Newton's first law say
about this event?
According the Newton’s 1st Law, an object in
motion will remain in motion. The spaceship will
not slow down, it will travel at constant velocity
forever.
Example 4.3: constant velocity
means zero net force
You are driving a Porsche Carrera GT on a straight testing
track at a constant speed of 150 km/h. you pass a 1971
Volkswagen beetle doing a constant 75 km/h. For which car
is the net force greater?
Since both cars are in equilibrium because their
velocities are constant, therefore the net force on
each car is zero.
• Suppose you are in a bus that is traveling on a
straight road and speeding up. If you could
stand in the aisle on roller skates, you would
start moving backward relative to the bus as the
bus gains speed. If instead the bus was slowing
to a stop, you would start moving forward down
the aisle. In either case, it looks as though
Newton’s first law is not obeyed; there is no net
force acting on you, yet your velocity change.
What wrong?
Inertial frame of reference
• The bus in accelerating with respect to the earth
and in not a suitable frame of reference for
Newton’s 1st law. A frame of reference in which
Newton's 1st law is valid is called an inertial
frame of reference. The earth is at least
approximately an inertial frame of reference, but
the bus is not.
• Because Newton’s first law is used to determine
what we mean by an inertial frame of reference,
it is sometimes called the law of inertia.
Initially, you and the
vehicle stay at rest. Your
body tends to stay at rest
as the vehicle
accelerating around you.
Initially, you and the
vehicle move at constant
velocity. Your body tends
to move at constant
velocity as the vehicle
slows down you.
• As the vehicle round a
corner at constant speed,
your body tend to continue
moving in a straight line.
• In each case, an observer in the vehicle’s
frame of reference might be tempted to
conclude that there is a net force acting on
the passenger, since the passenger’s
velocity relative the vehicle changes in
each case. This conclusion is simply
wrong because the vehicle is not an
inertial frame and Newton’s law isn’t valid.
• We know the earth’s surface is one inertial frame
of reference. But there are many inertial frames.
Any frame of reference that moves at constant
velocity to the earth is also an inertial frame of
reference. Viewed from this light, the state of
rest and the state of constant velocity are not
very different.
Test your understanding 4.2
•
1.
2.
3.
4.
In which of the following situations is there zero
net force on the body?
An airplane flying due north at a steady 120
m/s and at a constant altitude.
A car driving straight up a hill with a 3o slope at
a constant 90 km/h
A hawk circling at a constant 20 km/h at a
constant height of 15 m above an open field
A box with slick, frictionless surfaces in the
back of a truck as the truck accelerates forward
on a level road at 5 m/s2.
4.3 Newton’s 2nd law
Newton’s 1st law tells us the when a body is acted on by zero
net force, it moves with constant (including zero) velocity and
zero acceleration.
But what happens when the net force is not zero?
Newton’s 2nd law - A net force acting on a body causes the
body to accelerate in the same direction as the net force.
If the magnitude of the net force is constant, then the
magnitude of the acceleration is also constant. In fact, the
magnitude of the acceleration is directly proportional to the
magnitude of the net force acting on the body.
These conclusions about net force and acceleration also
apply to a body moving along a curved path.
Mass and Force
• The ratio of the magnitude |∑F| of the net
force to the magnitude of a = |a| of the
acceleration is constant, regardless of the
magnitude of the net force.
• We call this ratio the inertial mass, or
simply the mass, of the body and denote it
by m.
• Mass is a quantitative measure of inertia,
the greater its mass, the more a body
“resists” being accelerated.
• The SI unit of mass is the kilogram.
Since
One Newton is the amount of net force that gives an
acceleration of 1 m/s/s to a body with a mass of 1
kilogram.
• Suppose we apply a constant net force ∑F to a body
having a known mass m1 and we find an acceleration of
magnitude a1. We then apply the same force to another
body having an unknown mass m2, and we find an
acceleration of magnitude a2.
For the same net force, the ratio of the masses
of two bodies is the inverse of the ratio of their
acceleration.
• When two bodies with masses m1 and m2 are fastened
together, we find that the mass of the composite body is
always m1 + m2.
• Mass is the quantity of matter in a body.
Stating Newton's 2nd law
• If a net external force acts on a body the body
accelerates. The direction of acceleration is the same as
the directions of the net force. The mass of the body
times the acceleration of the body equals the net force
vector.
Using Newton's second law
• Since the equation is a vector equation, we will use it in
component form, with a separate equation for each
component of force and the corresponding acceleration:
Note:
1. ∑F means the sum of all external forces.
2. the equation is only valid if mass is constant.
3. the equation is only valid in the inertial frame of reference
caution
• ma is not a force.
• The vector ma is equal to the vector sum
of all the forces acting on the body (∑F)
Example 4.4 determining
acceleration from force
• A worker applies a constant horizontal force with
magnitude 20 N to a box with mass 40 kg resting on a
level floor with negligible friction. What is the
accelerations of the box?
Example 4.5 determining force
from acceleration
• A waitress shoves a ketchup bottle with mass 0.45 kg to
the right along a smooth, level lunch counter. The bottle
leaves her hand moving at 2.8 m/s, then slows down as
it slides because of the constant horizontal friction force
exerted on it by the counter top. It slides a distance of
1.0 m before coming to rest. What are the magnitude
and direction of the friction force acting on the bottle?
• Find acceleration:
Some notes on units
1 dyne = 10-5 N
1 pound = 4.48 N
Test your understanding
•
a.
b.
c.
d.
Rank the following situations in order of the
magnitude of the object’s acceleration, from
lowest to highest. Are there any cases that
have the same magnitude of acceleration?
A 2.0 kg object acted on by a 2.0 N net force;
A 2.0 kg object acted on by an 8.0 N net force.
An 8.0 kg object acted on by a 2.0 N net force.
An 8.0 kg object acted on by a 8.0 N net force.
4.4 mass and weight
• Mass characterizes the inertial properties
of a body. The greater the mass, the
greater the force needed to cause a given
acceleration; ∑F = ma
• Weight is a force exerted on a body the
pull of the earth. Bodies having large mass
also have large weight.
• The force that makes the body accelerate
downward at 9.8 m/s2 is its weight.
• A body with mass m has weight of
magnitude of w
w = mg
– The magnitude w of a body’s weight is directly
proportional to its mass m.
– The weight of a body is a force, a vector
quantity,
w = mg
Caution:
A body’s weight acts
at all times
Example 4.6 Net force and
acceleration in free fall
• A one-euro coin was dropped from rest from the Leaning
Tower of Pisa. If the coin falls freely, so that the effects of
the air are negligible, how does the net force on the coin
vary as it falls?
Variation of g with location
• The value of g varies from point to point on the earth’s
surface, from about 9.78 to 9.82 m/s2, because the earth
is not perfectly spherical and because of effects due to
its rotation and orbital motion.
• At a point where g = 9.80 m/s2, the weight of a standard
kilogram is w = 9.80 N. At a different point, where g =
9.78 m/s2, the weight is w = 9.78 N but the mass is still 1
kg. the weight of a body varies from one location to
another; the mass does not.
If we take a standard kilogram to the surface of the moon,
its weight is 1.62 N, but is mass is still 1 kg.
An 80.0 kg astronaut weighs 784 N on Earth, but weighs
only 130 N on the moon.
Measuring mass and weight
• The easiest way to measure
the mass of a body is to
measure its weight, often by
comparing with a standard.
• Two bodies that have the
same weight at a particular
location also have the same
mass.
• The equal-arm balance can
determine with great precision
when the weights of two
bodies are equal and hence
their masses are equal.
• In outer space, we can
compute the mass as the ratio
of force to acceleration.
Example 4.7 mass and weight
Test your understanding of section
4.4
• Suppose an astronaut landed on a planet
where g = 19.6 m/s2. Compare to Earth,
would it be easier, harder, or just as each
for her to walk around?
• Would it be easier, harder, or just as easy
for her to catch a ball that is moving
horizontally at 12 m/s (assume that the
astronaut’s spacesuit is a light-weight
model that doesn’t impede her movements
in any way.)
4.5 Newton’s Third Law
• A force acting on a body is always the result of its
interaction with another body, so forces always comes in
pairs.
• Newton’s 3rd law of motion: if body A exerts a force on
body B (an “action”), then body B exerts a force on body
A 9a “reaction”). These to forces have the same
magnitude but are opposite in direction. Theses two
forces act on different bodies.
• In the statement of Newton’s third law, “action” and
“reaction” are the two opposite forces; we sometimes
refer to them as an action-reaction pair. This is not
meant to imply any cause-and effect relationship; we can
consider either force as the “action” and the other as the
“reaction.”
CAUTION: the two forces in an action-reaction pair
act on different bodies. Unlike in Newton's 1st or 2nd
Law, which involve the forces act on one body.
The action and reaction forces can be contact forces or longrange forces.
When you drop a ball, both the ball and the earth accelerate
toward each other. The net force on each body ahs the same
magnitude, but the earth’s acceleration is microscopically
small because its mass is so great. Nevertheless it does
move!
Example 4.8 which force is greater?
• After your sports car breaks down, you start to
push it to the nearest repair shop. While the car
is starting to move, how does the force you exert
on the car compare to the force the car exerts on
you? How do these forces compare when your
are pushing the car along at a constant speed?
In both cases, the force you exert on the car is equal in
magnitude and opposite in direction to the force the car
exerts on you.
Example 4.9 applying Newton's 3rd law
– object at rest
An apple sits on a table in equilibrium. What forces
act on it? What is the reaction force to each of the
forces acting on the apple? What are the actionreaction pairs?
Example 4.10 applying Newton's 3rd law –
object in motion
• A stonemason drags a marble block across a floor by
pulling on a rope attached to the block. The block may or
may not be in equilibrium. How are the various force
related? What are the action-reaction pairs?
Action-Reaction Pair
Action: Man on Rope
Reaction: Rope on Man
Action: Rope on Box
Reaction: Box on Rope
Not Action-Reaction Pair
Man on Rope
Man on Rope
Box on Rope
Rope on Box
Example 4.11
We saw in example 4.10 that the stonemason pulls as
hard on the rope-block combination as that combination
pulls back on him. Why, then, does the block move while
the stonemason remains stationary?
Check your understanding 4.5
•
You are driving your car on a country road when a
mosquito splatters itself on the windshield. Which has
the greater magnitude:
1. The force that the car exerted on the mosquito
2. The force that the mosquito exerted on the car?
They are the same in magnitude and opposite in
direction.
Whyis the mosquito splattered while the car is
undamaged?
The car’s mass is much bigger than the mosquito,
therefore, its acceleration a = Fnet / mass is much smaller
than that of the mosquito.
4.6 free-body diagrams
1.
Newton’s first and seconds laws apply to a specific body.
• Newton’s 1st law, ∑F = 0, (equilibrium)
• Newton’s 2nd law, ∑F = ma, (non equilibrium)
2. Only forces acting on the body matter.
3. Free-body diagram are essential to help
identify the relevant forces. “free” of its
surroundings, with vectors drawn to show
the magnitudes and directions of all the
forces applied to the body by the various
other bodies that interact with it.
4. Label all the forces acting on the body.
Test your understanding 4.6
• The buoyancy force shown in the figure is one half
of an action-reaction pair. What force is the other
half of this pair?
1. The weight of the swimmer;
2. The forward thrust force
3. The backward drag force
4. The downward force that the swimmer exerts on the
water
5. The backward force that the swimmer exerts on the
water by kicking.