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Newton’s Laws
Newton’s first law:
An object will stay at rest or in a state of uniform motion with constant velocity, in a straight line,
unless acted upon by an external force.
In other words, the bodies resist any change of their state of motion. This resistance is called
inertia, and the measure of inertia of the body is its mass, a.k.a. the amount of matter in the body.
The more mass a body has, the harder it is to change its motion. Mass is an intrinsic property of
matter and does not change as an object is moved from one location to the next.
A force is simply a push or a pull. Forces are vectors; they have both size and direction.
There are two major types of forces.
Contact forces: Forces that arise when two objects are in contact (tension, friction, and elastic
forces).
Action-at-distance forces: Forces that do not require the bodies to be in contact (gravitation,
electromagnetic force, or the strong nuclear force).
Forces cause masses to accelerate. Newton’s second law relates force acting upon a body to
G
G
G
G
the mass of the body and the acceleration produced by this force, Fnet = m a or ∑ F = m a . The
sum of the forces acting on a body is equal to the objects mass multiplied by its acceleration. The
sigma (Σ) is the symbol for sum.
The result of adding up all the forces on an object is called the resultant or net force.
Make sure you know that the “Force” in the problems in this course refers to the overall “Net”
force acting on the objects in the problems.
Remember that forces are vectors, so we need to use the rules of vector addition.
G G
F1 + F 2
G
F1
G
F2
However, just as it was the case with two-dimensional motion, we can solve all problems by
breaking down the forces into components.
For example, if there are three forces acting on our object, and we know the components of these
three force vectors [ F1x, F1y , F1z ,] [F2x , F2y , F2z ] [F3x , F3y , F3z ], then the net force in 'x' is F1x +
F2x + F3x , in 'y' the net force is F1y + F2y + F3y, and in 'z' F1z + F2z + F3z .
If all of these sums are not equal to zero, then by Newton’s second law, there will be an
acceleration of the object on which the forces act:
ax = [F1x + F2x + F3x] /m,
ay = [F1y + F2y + F3y]/m,
az = [F1z + F2z + F3z]/m
where ax, ay, az are components of the acceleration vector of the object of mass m.
The origin of forces is always other objects. For example, the force of gravity is exerted by Earth.
The force acting on one body and exerted by another body is one side of interaction between
them. According to Newton’s Third Law of Motion, Whenever one object exerts a force on a
second object, the second exerts an equal and opposite force on the first.
Four forces that we usually work with in Newtonian mechanics problems are
1. Gravitational Forces
2. Normal Forces
3. Tension Forces
4. Friction Forces
1. The gravitational force on an object is attractive force. The gravitational force exerted by Earth
is what we call weight. By Newton’s second law, Weight = mass x acceleration. In this case, ‘a’ is
g = 9.8 m/s2 and, therefore, W = mg. The direction of the weight vector is always vertically
downward.
2. Newton's third law is the reaction force on a body. This force results from the contact between
the body and another surface. The direction of normal is perpendicular to the surface with which
the body is in contact.
Mass and weight are often confused. It is important to realize that mass and weight are
different quantities. Mass is the measure of inertia of the body; it is measured in kilograms.
Weight is the force of gravity on an object. The amount of matter in an object does not change;
hence, the mass of an object is always the same. However, 'g' varies from place to place on the
Earth's surface, depending on the distance from the center of the Earth, so weight does change.
If a body is on a horizontal surface, then a force, exerted by the support, prevents it from falling
down. The net force acting on the body at rest is the sum of the weight caused by the force of
G G
gravity and the normal force, F + W = 0 . Since the forces in this case all act in a single vertical
direction, instead of continuing to write this as vector equation, we will simply write this equation
as F + W = 0 where the normal force will be positive and acting upwards and the weight will be a
negative quantity acting downward.
Normal force
If the person is not accelerating, then
the net force is zero; thus, the normal
force must equal the weight.
W = mg
Note that an object's apparent weight will change if the supporting surface is accelerating, as is
the case with weight measurements made in an elevator.
In this case, the person is
accelerating down,
ΣF = ma ≠ 0 or mg – N = ma;
N = mg - ma
N < mg, so the person feels
lighter.
N
W = mg
G
G
Σ F = ma
a
3. Tension forces are exerted by strings, cables, cords, etc. when these strings and cords are
used to apply forces or make connections between objects in the problem. In an ideal situation,
our strings and cords will be massless and inextensible such that forces at one end are
transmitted undiminished to the other end, keeping the tension uniform throughout. Tension is
always a pulling force; it is always directed along the rope or cable or string. If a pulling force T is
exerted on a rope, this force is transmitted all the way along the rope to the other end.
4. Friction is a contact force. It is always present when two bodies touch. From an atomic
perspective, when two surfaces are in contact with each other, intermolecular forces form contact
bonds which momentarily bond the surfaces together.
Static friction is friction between any two surfaces when they are not moving across each other.
Static friction always resists a force that is trying to slide the surfaces across each other. The
direction of static friction is always opposite to that of the applied force. Another important thing
about static friction is that it does not have certain magnitude. As you push on an object with
increasing force, the force of static friction increases as well. If an object is not moving under an
applied force F , then f s and the magnitude of the component of
F that is parallel to the surface
are equal.
However, static friction cannot continue pushing back without limit. At some point, static friction
“breaks down.” When the component of F that is parallel to the surface exceeds f s , max , the
object starts to move. Up until this point, the force of static friction is equal to whatever the
magnitude of the parallel to the surface component of F is, that is insufficient to move the object.
This point indicates the maximum possible static friction force. This (maximum) force is
proportional to the normal force N between the two surfaces:
f s ≤ μs N
where μ s is the coefficient of proportionality, called the coefficient of static friction. The static
friction force is LESS THAN or equal to μsN.
When the surfaces begin to slide across one another, the force opposing the motion is called
kinetic or sliding friction. The direction of sliding friction is always opposite to that of the
velocity. This force is also proportional to the normal force f k = μ k N , where μ k is the coefficient of
kinetic friction. The coefficient of kinetic friction is usually less than that of static friction. These
coefficients depend only on the nature of the materials in contact and their condition (roughness,
wetness, dust, etc). Neither static nor sliding friction depends on the total contact area between
the surfaces.
Drag Force and Terminal Speed: When objects are moving through a liquid or gas, a drag force
D opposes the objects’ motion.
The magnitude of the drag force mostly depends on the object’s velocity and shape.
In the case of an object falling toward earth, there will be a speed VTerminal = Vt for which the
magnitude of D is equal to the object’s weight in kg.