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Newton’s Laws
1st Law: A body acted on by no net force moves with
constant velocity (which may be zero)
2st Law: The acceleration of an object is directly
proportional to the net force acting on it and is inversely
proportional to its mass. The direction of the acceleration
is in the direction of the net force acting on the object.


 Fi  ma
3rd Law: For every action there is an equal, but
opposite reaction
The First Law states that an object will
remain at rest or in uniform motion in a
straight line unless acted upon by an external
force. It may be seen as a statement about
inertia, that objects will remain in their state
of motion unless a force acts to change the
motion.
Aristotle: a natural state of an object is
at rest; a force is necessary to keep an
object in motion. It follows from
common sense.
384-322 B.C.
Galileo: was able to identify a hidden
force of friction behind commonsense experiments, abstracted from
it a fundamental law of inertia and
the principle of relativity
1564-1642
First Law is identical to Galilean Principle of relativity
Galileo Galilei
Galilean principle of relativity
(Galileo’s ship!)
Laws of physics look the same for all
observers who move with a constant
velocity with respect to each other, i.e.
in all inertial frames of reference.
1564-1642
Indeed, if no force in needed to keep the body in motion with constant
velocity, all such states of motion are equivalent. For different inertial
observers the object will appear moving with different but constant velocity
(which may be zero).
The First Law contains implications about the fundamental symmetry
of the universe in that a state of motion in a straight line must be just
as "natural" as being at rest. If an object is at rest in one frame of
reference, it will appear to be moving in a straight line to an observer
in a reference frame which is moving by the object. There is no way to
say which reference frame is "special", so all constant velocity
reference frames must be equivalent.
The first law is valid only with respect to an inertial
observer, i.e. in inertial frames of reference. It is
violated in accelerated reference frames.
2nd Law
From experiments we know:
1. A force is needed to change the state of motion
2. Force is a vector; obeys superposition principle: the net
force is a vector sum of all forces acting on an object
3. The direction of acceleration vector is the same as the
direction of the force vector
4. The magnitude of the force and acceleration are related
by a constant which intuitively is a “quantity of matter”.
This is the inertial mass.
Newton’s 2nd Law of Motion
2. The acceleration a
of a body is
inversely
proportional to its
mass m, directly
proportional to the
net force F, and in
the same direction
as the net force.
a = F/m  F = m a
1 N = 1 kg m/s2
Newton’s 3rd Law of Motion
3. To every action,
there is an equal
and opposite
reaction.
The same force that
is accelerating the
boy forward, is
accelerating the
skateboard
backward.
Clockwork universe
Types of forces
•
•
•
•
Gravity and weight
Normal force
Friction
Tension and spring force
Contact versus long-range forces
All are manifestation of four Fundamental “forces”
•Gravity
•Electromagnetic
•Strong
•Weak
Gravity is a strange force. It has a unique property:
All bodies in the same point in space experience the same acceleration!
Galileo, about 1600
mi  mg !!!
m
mM
F G 2
R
F G
mM
2

mg
;
g

9
.
8
m/s
RE2
mi a  G
a
R
M
mg M
R2
F
M
G 2
m
R
Weight, “apparent weight”, the force of
gravity, and the normal force
• Riding an elevator
Units of Force
1 Newton  1N  1kg  m / s
2
British system:
units of mass: 1 slug  14.59 kg
units of force: 1 pound  1lb  1 slug  ft / s 2  4.448N
One pound is 0.4536 kg
One pound is the weight of 0.4536 kg on the
Earth
Box on an inclined plane
A box with mass m is placed on top of a frictionless incline
with angle q and height H and is allowed to slide down.
a) What is the normal force?
b) What is the acceleration of the box?
c) What is the velocity of the box when it reaches the
bottom?
q
Friction
Two types of friction:
1. Kinetic: The friction force
that slows things down
2. Static: The force that
makes it hard to even get
things moving
Refrigerator
• If you push a refrigerator when
there is no friction what happens?
• In the real world what happens?
Especially when it’s fully loaded
and on a sticky kitchen floor?
– When does static friction kick in?
– When does kinetic friction kick in?
Friction
There is some maximum value the friction force can
achieve, and once we apply a force greater than this
maximum there is a net force on the object, so it
accelerates.
The maximum of the force of friction varied linearly
with the amount that the block pushes on the table.


Ffriction   N

 - coefficient of friction, N is the vertical force exerted by
the block on the table
The friction force only exists when there is another
force trying to move an object
Kinetic Friction
• For kinetic friction, it turns out that the
larger the Normal Force the larger the
friction. We can write
• FFriction = KineticFNormal
Here  is a constant
• Warning:
– THIS IS NOT A VECTOR EQUATION!
Static Friction
• This is more complicated
• For static friction, the friction force can vary
FFriction  StaticFNormal
Example of the refrigerator:
– If I don’t push, what is the static friction
force?
– What if I push a little?
Is it better to push or pull a
sled?
You can pull or push a sled with the same force
magnitude, FP, and angle Q, as shown in the figures.
Assuming the sled doesn’t leave the ground and has
a constant coefficient of friction, , which is better?
FP
FP
A Recipe for Solving Problems
1. Sketch
Isolate the body, draw a free-body diagram (only
external forces but not forces that one part of the
object exert on another part)
2. Write down 2nd Newton’s law


F  ma
Choose a coordinate system
Write 2nd Newton’s law in component form:





F  Fx i  Fy j  max i  ma y j
Fx  max , Fy  ma y
3. Solve for acceleration
Pulling Against Friction
A box of mass m is on a surface with coefficient of kinetic and
static friction . You pull with constant force FP at angle Q.
The box does not leave the surface.
1. Find the minimum force you need to apply in order to move
the block
2. What is the magnitude of the acceleration?
3. What angle maximizes the acceleration?
Q
Box on an inclined plane with friction
A box with mass m is placed on an incline with
angle q and is allowed to slide down.
a) What is the acceleration of the box?
q
Tension and pulleys
Force of tension
Massless, unstretchable string; massless, frictionless pulley
The advantage of a pulley
What minimum force F is needed
to lift the piano of mass M?
Conical pendulum
A ball of mass m is swung around a circle at the end of
a string of length L. The string will break if the tension
in it exceeds a critical value, Tc.
What is the largest constant angular velocity the ball
can have without breaking the string?
What is the largest period the ball can have without the
string becoming slack?
A mass m1 is going around in a circle on a
string on a frictionless table and the string goes
through a hole where it is attached to a hanging
mass m2. If the mass m1 is going around with
constant   0 , what must the distance from
the mass m1 to the hole be if the mass m2 is to
remain at rest?
m1
m2
Playing with weight:
• A car on an arched bridge
• Your weight in a rotating space station
or on the rotating Earth
A race track designer wants to have the cars
able to maintain a speed vmax without skidding
on a circular track. If the track is flat with a
coefficient of friction  what does the radius
have to be?
A race track designer wants to have the cars
able to maintain a speed vmax without skidding.
At what angle must the track of radius R be
banked assuming no friction? Assuming a
coefficient of friction ?
A satellite of mass m is attracted to the Earth of mass
M with a force of gravity proportional to the inverse
square of the distance to the Earth center, r:

Mm 
F   2 ir
r

is a gravitational constant
Find the velocity of a satellite on circular orbit of radius r.
Find the radius of the orbit for a geostationary satellite