Transcript Physics 151: Principles of Physics: Mechanics & Heat (Honors)

III. Newton’s Laws of Motion
A. Kinematics and Mechanics
B. Newton’s Laws of Motion
C. Common Forces in Nature
D. Problem Solving Using Newton’s Laws
E. Frictional Forces
F. Circular Motion
A. Kinematics and Mechanics
Kinematics is concerned with the relationship between
position, velocity, and acceleration of an object without
reference to the origin of the motion.
Mechanics is the study of the relationship between the
forces experienced by an object and the motion
resulting from these forces.
B. Newton’s Laws of Motion
From the time of the Greeks until Galileo, the “natural
state” of an object was thought to be rest, so it was
assumed that there was a direct connection between the
force on an object and the object’s velocity:
F v
A nonzero force was thought to imply a nonzero
velocity, and vice-versa.
Although this idea seems to be just common sense,
Galileo and Newton showed that it is wrong!
Galileo Galilei
(1564-1642)
His genius lie in carrying out “thought experiments”, by which he was able to
simplify physical situations and infer the important causal relationships. He realized
that a moving object, which seems to slow down as a matter of necessity, would not
slow down in the absence of frictional forces.
Isaac Newton
(1642-1727)
A towering genius in both mathematics and physics, he was the co-inventor of
differential calculus as well as the discoverer of the inverse square law of universal
gravitation.
First Law: An object moves with constant velocity (or
remains at rest) unless acted upon by a net external
force.
Also called the law of inertia, this principle assumes
that the proper connection is between the force on an
object and its change in velocity, or its acceleration:
F a
Thus, if an object moves with a constant velocity, the
net external force acting on it is ZERO!
Second Law: The acceleration of an object is in the
direction of the net force it experiences and is inversely
proportional to its mass, or inertia.

a

 Fi
i
m


or just  Fi  ma
i
Third Law: All forces occur in equal and opposite pairs.
If object A exerts a force on object B, then object B
exerts an equal and opposite force on object A:

FBA

FAB
A


FBA  FAB
B
This principle is important in problems involving two or more
objects in contact with each other.
An important qualification:
Newton’s Laws are only valid in inertial (nonaccelerating) frames of reference.
Thus, the accelerations and forces that we use to write
down these laws must be measured by an observer in
an inertial reference frame.
C. Common Forces in Nature
Weight is the force with which a massive object (like
the earth) attracts other objects. If an object with mass
m is near the surface of the earth, it experiences a force
directed toward the center of the earth with magnitude
W  mg
m
W
C. Common Forces in Nature, contd.
Two objects that are touching exert contact forces on
each other. The direction of such forces is always
normal (perpendicular) to the surface of contact, so
they are sometimes called normal forces.
Fn = force of table on box
Fn' = force of box on table
C. Common Forces in Nature, contd.
When a 1-D spring (or similar elastic cord) is elongated
or compressed from its equilibrium position by a
distance x, it exerts a restoring force (or spring force)
Fs which is given approximately by
x
Fs  kx
x
Fs
where k is the spring constant. Note that the force is
always opposite to the displacement.
D. Problem Solving Using Newton’s Laws
We clearly indicate the forces acting on an object by representing
the object as a point particle and drawing arrows from the point in
the direction of the force. On such a free-body diagram, be sure
you only draw arrows for forces! Velocity and acceleration are not
forces!
Fn
m
frictionless surface
mg
Problem Solving Steps (p.97):
1. Draw a neat diagram that includes important features.
2. Draw separate free-body diagrams for each object of interest.
3. Choose a convenient coordinate system for each object.
4. Write down Newton’s second law in component form and
use Newton’s third law if you have more than one object.
5. Solve the resulting equations for desired unknown(s).
6. Check your answers for units, plausibility, and familiar
limiting cases.
Example: Two accelerating blocks
Two blocks, with masses m1 and m2, are in contact on a frictionless horizontal
surface. A force F is applied to m1, causing both blocks to accelerate along the
surface. Find the force of contact between the two blocks.
Fn1
F
m1
m2
Fc
m1g
Fn2
F
Fc
m2g
Note carefully the notation for the normal forces and the weights of the blocks.
Also note that force F acts only on m1 and NOT on m2!
Another example: two blocks connected by a string
Two blocks, with masses m1 and m2, are connected by a string of negligible
mass on a frictionless horizontal surface. A force F is applied to m1 at an angle
 above the horizontal, causing both blocks to accelerate along the surface. Find
the acceleration of the two blocks and the tension in the string, in terms of the
masses, the angle  and the force F.
T
m2
F
m1

An inclined plane problem
Two blocks with mass m and 2m are connected by a string of negligible mass
over a pulley as shown. The plane is a frictionless surface inclined at an angle 
from the horizontal. After the block 2m is released, find the acceleration of the
system and the tension in the string, in terms of m and .
2m

E. Frictional Forces
The two basic cases we need to understand are static friction and
sliding friction. Consider a block of mass m on a surface and
subject to a horizontal force F, but still at rest:
Fn
F
m
fs
F
mg
The symbol fs represents the force of static friction . We know it
must be present because the net horizontal force must equal zero if
the block is not accelerating.
If the force F is increased, the block will remain at rest until F
is large enough to cause it to accelerate from rest. At this
point the static friction force is a maximum (fs)max.
Graphically we can represent this:
f
(fs)max
fk
fk= force of kinetic friction
F
static
friction
kinetic
friction
What does the frictional force depend on?
fk
F
m
fk
m
2fk
2m
Experiment shows the same frictional
force in both cases.
F
Thus, fk does not depend on contact area.
F
Experiment also shows that as the block
mass increases, the frictional force
increases proportionally.
Thus, fk depends linearly on normal force
between block and surface.
Approximate quantitative relationships
f k  k Fn
( f s ) max  s Fn
k = coefficient of kinetic friction
(approximately a constant, independent of
velocity)
s = coefficient of static friction
(s > k)
This equation only holds when the object
is on the verge of sliding
F. Circular Motion
Consider a particle of mass m moving with constant speed in a
circular path of radius r. By using a simple vector diagram we can
calculate its average acceleration over a time interval t:

vi

vf
s
r

vf

v


 vi



Since these triangles are similar, and the magnitudes v f  vi  v
s v
 
r
v
To find the acceleration:
vs
 v
r
v v s

t r t
 v  v
 s 
lim    lim  
t 0 t
  r t 0 t 
v2
ac 
r
centripetal acceleration
(center-seeking)
Newton’s Second Law in Circular Motion: an example
A car of mass m drives over a hill of height h whose shape near
the top approximates a circular arc. How fast can the car be
moving and stay in contact with the road at all times?
vmax = ?
h
Circular Motion with Frictional Forces: an example
An automobile has mass m = 1000 kg, and the coefficient of friction s
between its tires and a dry road surface is approximately 1.0. Find the
maximum speed at which it can safely (i.e., without sliding) navigate a
circular curve with radius of curvature 25 m.
r
v
r
m
top view of path
center
of circle
side view
v into the
screen
Remember:
Problems worthy
of attack
Prove their worth
by hitting back
--Piet Hein