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Chapter 5
More Applications of
Newton’s Laws
Forces of Friction

When an object is in motion on a
surface or through a viscous medium,
there will be a resistance to the motion
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This is due to the interactions between the
object and its environment
This resistance is called the force of
friction
Forces of Friction, cont.
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The force of static friction, ƒs, is generally
greater than the force of kinetic friction, ƒk
The coefficient of friction (µ) depends on the
surfaces in contact
Friction is proportional to the normal force
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ƒs  µs n and ƒk= µk n
These equations relate the magnitudes of the
forces, they are not vector equations
Forces of Friction, final
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The direction of the frictional force is
opposite the direction of motion and
parallel to the surfaces in contact
The coefficients of friction are nearly
independent of the area of contact
Static Friction
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Static friction acts to
keep the object from
moving
If increases, so does
If decreases, so does
ƒs  µs n where the
equality holds when the
surfaces are on the
verge of slipping
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Called impending motion
Kinetic Friction
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The force of kinetic
friction acts when
the object is in
motion
Although µk can vary
with speed, we shall
neglect any such
variations
ƒk = µk n
Some Coefficients of Friction
Friction in Newton’s Laws
Problems
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Friction is a force, so it simply is
included in the SF in Newton’s Laws
The rules of friction allow you to
determine the direction and magnitude
of the force of friction
Friction Example, 1
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The block is sliding down
the plane, so friction acts
up the plane
This setup can be used to
experimentally determine
the coefficient of friction
µ = tan q
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For µs, use the angle where
the block just slips
For µk, use the angle where
the block slides down at a
constant speed
Friction Example 2
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Image the ball moving
downward and the cube
sliding to the right
Both are accelerating
from rest
There is a friction force
between the cube and
the surface
Friction Example 2, cont
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Two objects, so two
free body diagrams
are needed
Apply Newton’s
Laws to both objects
The tension is the
same for both
objects
Uniform Circular Motion
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A force, , is directed
toward the center of the
circle
This force is associated
with an acceleration, ac
Applying Newton’s
Second Law along the
radial direction gives
Uniform Circular Motion, cont
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A force causing a
centripetal acceleration
acts toward the center of
the circle
It causes a change in the
direction of the velocity
vector
If the force vanishes, the
object would move in a
straight-line path tangent to
the circle
Centripetal Force
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The force causing the centripetal
acceleration is sometimes called the
centripetal force
This is not a new force, it is a new role
for a force
It is a force acting in the role of a force
that causes a circular motion
Conical Pendulum
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The object is in
equilibrium in the
vertical direction and
undergoes uniform
circular motion in the
horizontal direction
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v is independent of m
Horizontal (Flat) Curve
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The force of static
friction supplies the
centripetal force
The maximum speed at
which the car can
negotiate the curve is
Note, this does not
depend on the mass of
the car
Banked Curve
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These are designed
with friction equaling
zero
There is a component
of the normal force that
supplies the centripetal
force
Loop-the-Loop
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This is an example
of a vertical circle
At the bottom of the
loop (b), the upward
force experienced
by the object is
greater than its
weight
Loop-the-Loop, Part 2
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At the top of the
circle (c), the force
exerted on the
object is less than
its weight
Non-Uniform Circular Motion
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The acceleration and
force have tangential
components
produces the
centripetal acceleration
produces the
tangential acceleration
Vertical Circle with NonUniform Speed
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The gravitational
force exerts a
tangential force on
the object
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Look at the
components of Fg
The tension at any
point can be found
Top and Bottom of Circle
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The tension at the
bottom is a
maximum
The tension at the
top is a minimum
If Ttop = 0, then
Motion with Resistive Forces
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Motion can be through a medium
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Either a liquid or a gas
The medium exerts a resistive force, , on an
object moving through the medium
The magnitude of depends on the medium
The direction of is opposite the direction of
motion of the object relative to the medium
nearly always increases with increasing
speed
Motion with Resistive Forces,
cont
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The magnitude of can depend on the
speed in complex ways
We will discuss only two
is proportional to v
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Good approximation for slow motions or small
objects
is proportional to v2
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Good approximation for large objects
R Proportional To v
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The resistive force can be expressed as
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b depends on the property of the
medium, and on the shape and
dimensions of the object
The negative sign indicates is in the
opposite direction to
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R Proportional To v, Example
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Analyzing the
motion results in
R Proportional To v, Example,
cont
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Initially, v = 0 and dv/dt = g
As t increases, R increases and a
decreases
The acceleration approaches 0 when R
 mg
At this point, v approaches the terminal
speed of the object
Terminal Speed
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To find the terminal speed,
let a = 0

Solving the differential
equation gives
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t is the time constant and t
= m/b
R Proportional To
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2
v
For objects moving at high speeds through
air, the resistive force is approximately
proportional to the square of the speed
R = 1/2 DrAv2
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D is a dimensionless empirical quantity that is
called the drag coefficient
r is the density of air
A is the cross-sectional area of the object
v is the speed of the object
R Proportional To
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Analysis of an object
falling through air
accounting for air
resistance
2
v,
example
R Proportional To
Speed
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The terminal speed
will occur when the
acceleration goes to
zero
Solving the equation
gives
2
v,
Terminal
Some Terminal Speeds
Fundamental Forces
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Gravitational force
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Electromagnetic forces
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Between two charges
Nuclear force
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Between two objects
Between subatomic particles
Weak forces
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Arise in certain radioactive decay processes
Gravitational Force
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Mutual force of attraction between any
two objects in the Universe
Inherently the weakest of the
fundamental forces
Described by Newton’s Law of
Universal Gravitation
Electromagnetic Force
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Binds atoms and electrons in ordinary
matter
Most of the forces we have discussed
are ultimately electromagnetic in nature
Magnitude is given by Coulomb’s Law
Nuclear Force
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The force that binds the nucleons to form the
nucleus of an atom
Attractive force
Extremely short range force
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Negligible for r > ~10-14 m
For a typical nuclear separation, the nuclear
force is about two orders of magnitude
stronger than the electrostatic force
Weak Force
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Tends to produce instability in certain
nuclei
Short-range force
About 1034 times stronger than
gravitational force
About 103 times stronger than the
electromagnetic force
Unifying the Fundamental
Forces
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Physicists have been searching for a
simplification scheme that reduces the
number of forces
1987 – Electromagnetic and weak forces
were shown to be manifestations of one
force, the electroweak force
The nuclear force is now interpreted as a
secondary effect of the strong force acting
between quarks
Drag Coefficients of
Automobiles
Reducing Drag of Automobiles
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Small frontal area
Smooth curves from the front
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The streamline shape contributes to a low
drag coefficient
Minimize as many irregularities in the
surfaces as possible
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Including the undercarriage