Tension and Pulleys
The Tension in a Rope and Pulley System
This diagram represents a small block on the left in the act of being lifted by
a larger block on the right.
Notice the forces T and -T: even when used in addition to a pulley, the rope
Perhaps the most significant concept derived in dynamics is that of work.
The understanding of work greatly simplifies many physical situations.
Work, in a sense, introduces a dynamic understanding to mechanics.
It allows us to evaluate forces over distan
Newton's third law also gives us a more complete definition of a force.
Instead of merely a push or a pull, we can now understand a force as the
mutual interaction between two bodies.
Whenever two bodies interact in the physical world, a force results.
x(t) = cos wt , where w is a constant.
An object with this position function is undergoing simple
harmonic motion, which means its position is oscillating back and
forth in a special fashion.
Since the range of the cosine function is (- 1, 1) , the object
Newton's First Law
So how exactly does a force relate to motion? Intuitively, we can say that a
force, at least in some way, causes motion.
When I kick a ball, it moves. Newton makes this relation more precise in his
An object moves with consta
Work is commonly misunderstood because of its common definition.
Most people think that it takes a lot of work to hold a 100 pound weight in
The weight is not moving, though, so in the sense of physics no work is
It is important to realize
W = (F cos)x
This new equation has similar form to the old equation, but provides a more
If = 0 , then cos = 1 and we have our first equation.
Also, this equation ensures that it does not take into account any forces
acting on a movi
Motion with Constant Acceleration in Two and Three Dimensions
We have already seen that motion in more than one dimension that
undergoes constant acceleration is given by the vector equation:
a t 2 + v 0 t + x 0,
where a , v 0 and x 0 are constant
Work Done by a Variable Force
Consider a force acting on an object over a certain distance that varies
according to the displacement of the object.
Let call this force F(x) , as it is a function of x .
Though this force is variable, we can break the inter
massless ropes or cables
Almost all situations you will be presented with in classical mechanics deal
with massless ropes or cables.
If a rope is massless, it perfectly transmits the force from one end to the
other: if a man pulls on a massless rope with
x(t) = 1/2at 2 , where a is a constant.
At t = 0 , this object is situated at the origin, but its position
changes quadratically with time (since the exponent of t in the
above equation is 2 ).
For positive a , the graph of this position function looks li
These three forces, normal, tension and friction, are present in a surprising
number of physical situations.
Often the three forces are all present, as we shall see in the problems.
Though Newton's Laws apply in some sense to everything from
One-dimensional Motion with Constant Acceleration
In the previous section on position, velocity, and acceleration we found that
motion with constant acceleration is given by position functions of the form:
at 2 + v 0 t + x 0
where a is the accelera
Now we plug in our expression for force into our work equation:
W net =
F net dx =
Integrating from v o to v f :
W net =
mv f 2 -
mv o 2
This result is precisely the Work-Energy theorem. Since we have proven it
with calculus, this theo
Notice that this means we can write: v(t) = x'(t) .
Similarly, we can also take the derivative of the derivative of a function,
which yields what is called the second derivative of the original function:
Though it is rather surprising that frictional force and normal force are
related in such a simple manner, physical intuition tells us that they should
be directly related.
Consider again a block of wood on a concrete platform.
The normal for
Any force which does not conserve mechanical energy, as opposed to a
Property of conservative forces which states that the work done on any path
between two given points is the same.
Newton's Third Law
All forces result from the interaction of two bodies.
One body exerts a force on another.
Experience tells us that there is in fact a force.
When we push a crate across the floor, our hands and arms certainly feel a
force in the opposit
Newton's Second Law
Newton's Second Law gives us a quantitative relation between force and
F = ma
Stated verbally, Newton's Second Law says that the net force (F) acting upon
an object causes acceleration (a), with the magnitude of the a
Newtons law and dynamics as a whole, provide us with fundamental
axioms for the study of classical mechanics.
Once these foundations are laid, we can derive new concepts from the
axioms, furthering our understanding of mechanics and allowing us to
Newton's Third Law and Units of Force
Stated in words, Newton's third law proclaims: to every action there is an
equal and opposite reaction.
This law is quite simple and generally more intuitive than the other two.
It also gives us a reason for many obse
Mass is defined as the amount of matter in a given body.
This definition seems a little vague, and needs some explanation.
Mass is a scalar quantity, meaning it has no direction, and is a property of
the object itself, not its location.
Mass is measu
With Newton's second law, we can take a given physical situation and find
the acceleration, and thus the motion, of an object in the situation.
In addition, using the method of free-body diagrams, we can evaluate any
number of distinct forces.
Such an abi
With a knowledge of physical forces such as tension, gravity and friction,
centripetal force becomes merely an extension of Newton's Laws.
It is special, however, because it is uniquely defined by the velocity and
radius of the uniform circular motion.
kinetic and static
These two types of friction, like the normal force, arise whenever two
objects are in direct contact.
Often both kinetic and static friction apply to a given situation, as an object
might start at rest (when static friction applies) the
What does Newton mean by a net force?
Consider a rope being used in a tug of war.
There are definitely forces being applied to the rope but, if the two sides pull
with the same force, the rope won't move.
In this example, the two forces on the rope exactl
Kinematics Kinematics is concerned with describing the way in which objects move.
Displacement An object's total change in position.
If a man runs around an oval 500 meter track, stopping at the precise
location he began, though he ran a distance of 500 m
Kinematics, the part of physics we have studied up to this point, deals with
We have looked at position, velocity and acceleration as the three basic
properties of a particle in motion.
In Dynamics, we look at the causes of the motion t
What are the units of work?
The work done by moving a 1 kg body a distance of 1 m is defined as a
A joule, in terms of fundamental units, is easily calculated:
W = Fx =
The joule is a multipurpose unit.
It serves not only as a unit of work, b