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Dynamics
Chapter 4
Expectations
After Chapter 4, students will:



understand the concepts of force and inertia.
use Newton’s laws of motion to analyze situations
involving force, inertia, and acceleration.
understand the meaning of dynamic equilibrium and
identify objects or systems that are, or are not, in
equilibrium.
Expectations
After Chapter 4, students will:

recognize the properties of, and perform calculations
involving, particular kinds of forces:




gravitational
normal
frictional
tension
Force and Inertia
Newton’s second law of motion: If an object
experiences a net (unbalanced) force, it accelerates in
the direction of the force. The magnitude of its
acceleration is directly proportional to the magnitude
of the force, and inversely proportional to its inertia
(mass).
Force and Inertia
Newton’s second law of motion: If an object
experiences a net (unbalanced) force, it accelerates in
the direction of the force. The magnitude of its
acceleration is directly proportional to the magnitude
of the force, and inversely proportional to its inertia
(mass).
In mathematical terms:

 F
a
, or
m


 F  ma
Force and Inertia
Sir Isaac Newton 1642 – 1727
Natural philosopher
(scientist / mathematician);
the original mathematical
physicist, and inventor of
the calculus. He was also
greatly interested in theology.
Force and Inertia
We need more definition of the terms “force” and
“mass.”
Force: a push or a pull, which tends to produce
acceleration.
Mass: the property of an object which resists
acceleration by a force (inertia), and which is
determined by the amount of matter that constitutes
the object.
Force and Inertia
Mass: a scalar quantity
SI unit: the kilogram (kg) … a basic unit
Force: a vector quantity. We get the dimensions and
units of force from Newton’s second law:
dimensions: mass  length
time 2
SI units:
kg  m
 newtons (N)
2
s


 F  ma
Forces and Free-Body Diagrams
We said that an object accelerates as a result of the “net”
or “unbalanced” force acting on it.
More than one force can act on a body at the same time.
Forces are vectors. We can add them together as vectors
to determine the net or total force acting on an object.
The net force is simply the vector sum of all the
forces acting on an object.
Forces and Free-Body Diagrams
A convenient method for keeping track of the forces
acting on an object is to draw a “free-body diagram”
for it.
A free-body diagram:
 is drawn “of” a single object, and shows that object.
 shows every force that acts on an object.
 does not show any force that acts on anything besides
the object.
Forces and Free-Body Diagrams
Example: a wooden crate is being dragged across a level
floor by a rope that makes an angle of 30° with the
horizontal.
N (normal force, 126 N)
T (tension force, 98 N)
30°
F (frictional force, 25 N)
W (weight force, 175 N)
Forces and Free-Body Diagrams
Resolve forces into x and y components (as needed):
N (normal force, 126 N)
TY = 98 N · sin 30° = 49 N
TX = 98 N · cos 30° = 84.9 N
F (frictional force, 25 N)
W (weight force, 175 N)
Forces and Free-Body Diagrams
Add the x-direction forces, and also the y-direction
ones. The resultant or net force is entirely horizontal,
toward the left. The crate accelerates in that
direction. If its mass is 17.9 kg:
59.9 N
2
a
 3.35 m/s
17.9 kg
( TY + N - W = 49 N + 126 N – 175 N = 0 )
TX – F = 84.9 N – 25 N = 59.9 N
Equilibrium: Newton’s First Law
Why did we discuss Newton’s second law before
discussing his first law?
Because the first law is really a special case of the
second.
In words: an object remains at rest, or moves in a
straight line with constant velocity, unless it is acted
on by a net force.
Equilibrium: Newton’s First Law
Expressed mathematically, the second law:
 F  ma
and the first law:
a  0 if
F  0
An object is in equilibrium if the sum of the forces (“net
force”) acting on it is zero. Its velocity is constant. It
does not accelerate.
Action - Reaction: Newton’s Third
Law
If object A exerts a force on object B, Newton’s third
law says that object B exerts a force, having the same
magnitude and the opposite direction, on object A:


FAB   FBA
FAB and FBA are an action-reaction pair.
Kinds of Forces
Three fundamental categories of forces:
 Gravitational
 Weak (electromagnetic)
 Strong nuclear
They are listed here in ascending order of strength.
Gravitational Force
As with many other things, we have Isaac Newton to
thank for the first mathematical description of the
operation of the gravitational force.
Newton’s law of universal gravitation: any two objects
exert equal attractive forces on each other. The
magnitude of the forces is proportional to the product
of the objects’ masses, and inversely proportional to
the square of the distance separating them.
Gravitational Force
Newton’s law of universal gravitation (mathematically):
m1m2
FG  2
r
We can make this proportionality into an equation by
introducing a constant of proportionality, G:
m1m2
FG  G 2
r
G is called the universal gravitational constant.
Gravitational Force
m1m2
FG  G 2
r
Do not confuse G, the gravitational constant, with g, the
acceleration due to gravity!
If we use SI units, the value of G is 6.67×10-11 N·m2/kg2.
If one of the masses is the Earth’s mass, and r is the
Earth’s radius, the gravitational force is called the
weight of an object.
Gravitational Force
We can use this expression for the gravitational force to
calculate the value of g, the acceleration due to gravity
at the Earth’s surface.
m
m m
W  G E 2  mg
rE


GmE
6.67  10-11 Nm2 /kg 2 5.98  1024 kg
g 2 
2
6
rE
6.38  10 m
g  9.80 m/s 2



W

Ordinarily, we express an object’s weight as: W  mg
Normal Force
The atoms of which objects are made are surrounded by
electron clouds. Due to the electromagnetic (“weak”)
force, these electron clouds resist being brought close
together. Their resistance to being actually merged is
extremely powerful.
When objects are in macroscopic contact, each exerts a
force on the other to prevent their occupying
(macroscopically) the same space at the same time.
These forces are called normal forces.
Normal Force
If objects A and B are in contact:
 They exert equal and opposite normal forces on each


other. N AB   N BA , in accordance with Newton’s third
law.
 The directions of the forces are perpendicular to the
contact surface. (“Normal” is another way of saying
“perpendicular.”)
 The magnitude of the forces is as large as necessary to
prevent the objects from merging.
Normal Force
Example: a box rests (equilibrium) on a table.
Since the box is in equilibrium
(does not accelerate), the sum of
the forces acting on it must be zero.
 F  N  mg  0
N  mg
The magnitude of the normal force
on the box is equal to its weight.
N
m
mg
Apparent Weight (Nonequilibrium)
Frame of reference: another name for a coordinate system
A coordinate system that does not accelerate is also called
an inertial reference frame.
What happens in a frame of reference that does accelerate
(a non-inertial frame)?
Apparent Weight (Nonequilibrium)
Consider an elevator car that can accelerate up or down …
or can move with constant velocity or be at rest.
When it accelerates, it is a non-inertial frame of reference.
When it does not accelerate, it is an inertial frame.
We place a scale inside the elevator, and an object on the
scale. A scale reports the normal force that it exerts on
an object (and that the object exerts on it).
Apparent Weight (Nonequilibrium)
When the elevator moves with
constant velocity and is an inertial
a=0
frame …
… the normal force exerted
(and reported) by the scale
is equal to the weight of the
object:
N
N  mg
mg
Apparent Weight (Nonequilibrium)
When the elevator accelerates upward …
… the normal force exerted
and reported by the scale
(the object’s “apparent weight”)
increases:
a
N
N  mg  ma
N  m( g  a )
mg
Apparent Weight (Nonequilibrium)
When the elevator accelerates downward …
… the normal force exerted
and reported by the scale
(the object’s “apparent weight”)
decreases:
N
N  mg  ma
N  m( g  a )
mg
a
Frictional Forces
Another force that objects in contact exert on each other is
the frictional force.
Two kinds of frictional force:
 Static: when the objects are stationary with respect to
each other.
 Kinetic: when the objects move relative to one another.
Frictional Forces
Where do frictional forces come from?
Even objects whose surfaces are macroscopically smooth
have microscopic textures that tend to interlock.
Frictional Forces
Frictional forces are exerted parallel to the contact area
between objects.
Their direction is such as to oppose motion between the
objects.
The static frictional force is as large in magnitude as
necessary to prevent motion … but not larger than a
calculable maximum magnitude.
Frictional Forces
Once motion between the contacting objects takes place,
the frictional force become a kinetic one. The kinetic
frictional force between two objects is smaller than the
maximum static frictional force.
The magnitude of the frictional force depends on the
normal force, and on a coefficient of friction which is
determined by the materials in the objects, as well as
the condition / texture of their contacting surfaces:
F  N
Frictional Forces
Rearranging as a defining equation:
F  N

F

N
We see that the coefficient of friction is a dimensionless
ratio of the magnitudes of two forces.
There are two coefficients of friction for any combination
of surfaces, yielding two frictional forces:
FS  S N (maximum static)
and
FK  K N (kinetic)
Tension Forces
“Tension” means both the forces applied to the ends of a
rope, string, belt, ribbon, wire, chain, cable, thread, etc.
… and the force that the rope-like object (R-LO) exerts
to resist being pulled apart.
The reluctance of R-LOs to be pulled apart means that
when you pull on one end, the other end pulls on
whatever it may be attached to. We can think of R-LOs
as being transmitters of pulling forces.
Tension Forces
If the R-LO has mass in a non-equilibrium situation, the
tension force is diminished in transmission:
a
T
M
m
T = (M + m)a
T’
M
a
T’ = Ma = T - ma
T’
T
m
a
T - T’ = ma
Important to Remember
If an object or system does not accelerate (a = 0), it is in
equilibrium, and
F 0

If the object or system does accelerate (a ≠ 0), it is not in
equilibrium, and
 F  ma
In either case, Newton’s second law applies. In
equilibrium, the right-hand side becomes zero, and the
second law becomes the first law.
Important to Remember
Velocity and acceleration are vector quantities. So,
Newton’s laws apply separately in the X and Y
directions:
F
x
 max
F
y
 may
Often (as in the case of a projectile), an object is in
equilibrium in one direction and not in equilibrium in
the other. Both the dynamic and kinematic situations
must be considered separately in each axis.