Download Polynomial Functions - Energy & Motion

To define work by relating force and
displacement.
 To calculate the net work done when
many forces are applied to an object.
 To identify several forms of mechanical
energy.
 To distinguish between kinetic and
potential energy and calculate them.
 To solve problems using conservation of
energy.

In Physics, work is the meaning of what is accomplished when a force
acts on an object, and the object moves through a distance.
Work is the product of the magnitude of the displacement times the
component of the force parallel to the displacement.
W = 𝑭|| *d
Mariano Rivera exerts a
force on the ball over a
displacement
until
he
releases the ball. The total
work done on the ball will
be equal to the kinetic
energy acquired by the ball.
Force
Displacement
Work is a scalar quantity, it has only magnitude which can be positive or
negative.
For example if you pull a loaded box a distance of 50 m by exerting a
horizontal force of 30 N you do 30 N x 50 m = 1500 N*m of work on the box.
A special name is given to this unit (N*m), the joule.
Find the work a hiker must do on a 15.0 kg backpack to carry it up a hill of
height h=10.0 m as shown in the figure. For simplicity, assume the motion is
smooth and at a constant velocity. Also find the work done by gravity on the
backpack and the net work done on the backpack.
Applying Newton’s laws
ΣF=m𝑎𝑦 in the y axis
we have:
𝐹𝐻−𝑚𝑔=0
Hence,
𝐹𝐻=𝑚𝑔
(15kg)(9.8m/𝑠 2 ) = 147N
Now we find the work done on the backpack by the hiker.
𝑊𝐻 = 𝐹𝐻 𝑑 𝑐𝑜𝑠𝜃
And we note from the figure that d cosθ = h so the work done by the hiker is:
𝑊𝐻 = 147𝑁 10𝑚 = 1470 𝐽
We can conclude that the work done depends only on the change on
elevation and not on the angle of the hill. The hiker will do the same work to
lift the pack vertically the same height.
The work done on the back pack by gravity is:
𝑊𝐺 = 𝐹𝐺 𝑑 cos(180 − 𝜃)
Since Cos(180-θ) =-Cosθ then: 𝑊𝐺 = 𝐹𝐺 𝑑 −𝑐𝑜𝑠𝜃 = −𝑚𝑔ℎ = −1470 𝐽
And finally the net work done on the backpack is 1470 J -1470 J = 0
In small groups discuss for a pair of minutes.
The Moon revolves around the Earth in a nearly circular orbit, kept there by
the gravitational force exerted by the Earth. Does gravity do positive work,
negative work, or no work on the moon?
The gravitational force exerted by Earth on the Moon acts towards the Earth.
The Moon displacement at any moment is tangent to the circle, in the direction
of its velocity, perpendicular to the radius and thus perpendicular to the force of
gravity. (90̊ angle)
Cos90̊=0
So the work done on the Moon by Earth’s gravity is zero.
This is why the Moon, as well
as artificial satellites, can stay
in orbit without the expenditure
of fuel: No net work needs to
be done against the force of
gravity.
If the force acting on the object is not constant we can determine the work
done graphically.
The procedure is like that for determining displacement when the velocity
is known as a function of time. To determine the work done by a variable
force we plot the parallel component of the force to the direction of the
motion as a function of distance. We divide the distance into small
segments and find the area of the rectangles formed. As the value of the
segments approaches zero the total area of the many rectangles
approaches the area under the curve.
Will you release the
ball and let it swing
back to you?
Energy is one of the most important concepts in science.
Energy as “the ability to do work” is a valid definition for mechanical energy.
A moving object can do work on
another object it strikes. A flying
cannonball does work on a brick wall it
knocks down, a hammer does work on
a nail it drives through the wood. In
either case, a moving object exerts a
force on the second one and moves it.
The energy of motion is called kinetic energy from the Greek kinetikos
meaning motion.
Using Newton’s second law equation, and the equations for accelerated
movement we get the following formula for net work done on an object:
1
1
𝑊𝑛𝑒𝑡 = m𝑣 2 2 − m𝑣 21
2
2
1
We define the quantity 2 m𝑣 2 as the translational kinetic energy (KE)
𝟏
𝑲𝑬 = 𝐦𝒗𝟐
𝟐
The net work done on an object is equal to the change in the object’s kinetic
energy.
𝟏
𝟏
So: 𝑾𝒏𝒆𝒕 = 𝟐 𝐦𝒗𝟐 𝟐 − 𝟐 𝐦𝒗𝟐 𝟏 = 𝑲𝑬𝟐 − 𝑲𝑬𝟏 = ∆𝑲𝑬
This means that if the net work on an object is positive, the kinetic energy
increases. If it is negative, its kinetic energy decreases.
Because of its direct connection, Energy is measured in the same units as
Work. Joules for the SI units.
In addition to kinetic energy, it is also possible to have Potential Energy,
which is the energy associated with forces that depend on the position or
configuration of the object relative to its surroundings.
Various types of potential energy exist, each associated with a particular kind
of force.
Note that there is a single universal formula for translational kinetic energy of
an object, but there is no single formula for potential energy because it
depends on the forces involved.
Perhaps the most common example
of potential energy. A heavy brick
held high in the air has potential
energy because of its position
relative to Earth. It has the ability to
do work, for if it’s released it will fall
to the ground due to the
gravitational force.
It is defined as: 𝑷𝑬𝑮𝒓𝒂𝒗 = 𝒎𝒈𝒉
With a polystyrene ball attached to a cord create a pendulum and attach it to
the ceiling.
Stand close to a wall holding the ball as shown on the picture and release it
without pushing it.
Will the ball come back
and hit you?
What kind of energies can
you observe on the
exercise?
If energy cannot be
created, explain why the
ball did not hit you back.
Another sort of potential energy
is associated with elastic
materials. For example a spring
has potential energy when
compressed or stretched for it
can then move an object when
released.
Elastic Potential Energy can be
expressed as:
1
Elastic PE = 2 𝑘𝑥 2
Where k is the spring stiffness
constant and x the distance the
spring
is
stretched
or
compressed.
Watch the following video and try answering the question presented.
www.youtube.com/watch?v=wGIZKETKKdw&feature=related
Then perform the experiment with a real slinky. After discussing your
observations watch the following videos.
www.youtube.com/watch?v=QSoZlomFXlo&feature=related
www.youtube.com/watch?v=oKb2tCtpvNU&feature=related
Forces such as gravity, for which the work done does not depend on the
path taken but only on the initial and final positions, are called
Conservative Forces.
Friction on the other hand, is a Non-conservative force since the work it
does depends on the path.
Several forces act on an object which can undergo a translational motion.
Suppose only some of this forces are conservative. Then the Net Work done
can be defined as:
𝑊𝑁𝑒𝑡 = 𝑊𝐶 + 𝑊𝑁𝐶
Work from conservative forces can be written as potential energy.
And with this the Work-Energy Principle can be expressed as:
𝑊𝑁𝐶 = ∆𝐾𝐸 + ∆𝑃𝐸
Thus the work done by non conservative forces acting on an object is
equal to the total change in kinetic and potential energies.
www.youtube.com/watch?v=Jnj8mc04r9E&feature=related
We can define E, the total mechanical energy of a system as the sum of
potential and kinetic energies at any moment.
E = KE+PE
If only conservative forces are acting, the total mechanical energy of a
system neither increases nor decreases in any process. It stays
constant, it is conserved.
This is called the PRINCIPLE OF CONSERVATION OF MECHANICAL
ENERGY.
If the original height of the rock in the
figure is 3.0 m, calculate the speed
when it has fallen to 1.0 m above the
ground.
1
1
2
𝑚𝑣 1 + 𝑚𝑔𝑦1 = 𝑚𝑣 2 2 + 𝑚𝑔𝑦2
2
2
In this case m cancel out. And 𝑉1 = 0.
So solving for the final 𝑉2 we find that:
𝑉2 = 6.3 m/s
A dart of mass 0.100 kg is pressed against a spring of a toy dart gun. The
spring has a stiffness constant k=250 N/m and is compressed 6.0 cm and
released. If the dart detaches from the spring when the spring reaches its
natural length (x=0), what speed does the dart acquire?
1 2
1
0 + 𝑘𝑥 1 = 𝑚𝑣 2 2 + 0
2
2
𝑉2 = 3.0 𝑚/𝑠
The total energy is neither increased
nor decreased in any process.
Energy can be transformed from one
form to another, and transferred from
one object to another, but the total
amount remains constant.
From this we can conclude that
work is done when energy is
transferred from one object to
another.
Power is defined as the rate at which work is done. Average power equals the
work done divided by the time needed to do it. We can define it also as the
rate at which energy is transformed.
𝑊𝑜𝑟𝑘 𝐸𝑛𝑒𝑟𝑔𝑦 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑑
𝑃 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑤𝑒𝑟 =
=
𝑇𝑖𝑚𝑒
𝑇𝑖𝑚𝑒
In SI power is measured in Joules per second and this special unit is called
watt (W)
Often a larger unit is used for practical purposes like motor engines’ power.
The horsepower (hp) and is equivalent to 746 W.
A 60.0 kg jogger runs up a set of stairs in 4 seconds. The vertical height of
the stairs is 4.5m.
Estimate the jogger power output in watts.
𝑊 𝑚𝑔𝑦
𝑃=
=
= 661.5𝑊
𝑡
𝑡
Efficiency in a system can be
defined as:
𝐏𝐨𝐮𝐭
=𝐞
𝐏𝐢𝐧
It can never be greater than 1.0
because no engine can create
energy, and can not transform one
type of energy into another without
some energy going to friction,
thermal energy and other non useful
forms of energy. For example in car
engines efficiency is close to15%
with the rest wasted as thermal
energy and friction between parts.
• Giancoli , Douglas C. Physics Sixth Edition. USA Pearson 2005
• Serway, Raymond A. Essentials of College Physics. USA Thomson 2007