Kinetic Energy and Work Chapter 7 Slide 2 Introduction to Energy
The concept of energy is one of the most important topics in
science Every physical process that occurs in the Universe involves
energy and energy transfers or transformations Slide 3 Work and
Energy Energy: scalar quantity associated with a state (or
condition) of one or more objects. Work and energy are scalars,
measured in Nm or Joules, J, 1J=kgm 2 /s 2 Energy can exist in many
forms - mechanical, electrical, nuclear, thermal, chemical. Slide 4
Energy Approach to Problems The energy approach to describing
motion is particularly useful when the force is not constant An
approach will involve Conservation of Energy This could be extended
to biological organisms, technological systems and engineering
situations This could be extended to biological organisms,
technological systems and engineering situations Slide 5 Work and
Energy Energy is a conserved quantity - the total amount of energy
in the universe is constant. Energy can be converted from one type
to another but never destroyed. Work and energy concepts can
simplify solutions of mechanical problems - they can be used in an
alternative analysis Slide 6 Systems A system is a small portion of
the Universe We will ignore the details of the rest of the Universe
We will ignore the details of the rest of the Universe A critical
skill is to identify the system Slide 7 Valid System A valid system
may be a single object or particlebe a single object or particle be
a collection of objects or particlesbe a collection of objects or
particles be a region of spacebe a region of space vary in size and
shapevary in size and shape Slide 8 Environment There is a system
boundary around the system The boundary is an imaginary surfaceThe
boundary is an imaginary surface It does not necessarily correspond
to a physical boundaryIt does not necessarily correspond to a
physical boundary The boundary divides the system from the
environment The environment is the rest of the UniverseThe
environment is the rest of the Universe Slide 9 Work The work, W,
done on a system by an agent exerting a constant force on the
system is the product of the magnitude, F, of the force, the
magnitude r of the displacement of the point of application of the
force, and cos , where is the angle between the force and the
displacement vectors Slide 10 Work W = F r cos The displacement is
that of the point of application of the forceThe displacement is
that of the point of application of the force A force does no work
on the object if the force does not move through a displacementA
force does no work on the object if the force does not move through
a displacement The work done by a force on a moving object is zero
when the force applied is perpendicular to the displacement of its
point of applicationThe work done by a force on a moving object is
zero when the force applied is perpendicular to the displacement of
its point of application Slide 11 Work Example The normal force, n,
and the gravitational force, mg, do no work on the object cos = cos
90 = 0 The force F does do work on the object Slide 12 More About
Work The system and the environment must be determined when dealing
with work The environment does work on the system (Work by the
environment on the system) The sign of the work depends on the
direction of F relative to r Work is positive when projection of F
onto r is in the same direction as the displacement Work is
negative when the projection is in the opposite direction Slide 13
Work Is An Energy Transfer This is important for a system approach
to solving a problemThis is important for a system approach to
solving a problem If the work is done on a system and it is
positive, energy is transferred to the systemIf the work is done on
a system and it is positive, energy is transferred to the system If
the work done on the system is negative, energy is transferred from
the systemIf the work done on the system is negative, energy is
transferred from the system Slide 14 Work done by a constant force
Work done on an object by a constant force is defined to be the
product of the magnitude of the displacement and the component of
the force parallel to the displacement Where F II is the component
of the force F parallel to the displacement d W = F II d Slide 15
Work In other words - Where is the angle between F and d If is >
90 o, work is negative. A decelerating car has negative work done
on it by its engine. The unit of work is called Joule (J), 1 J = 1
Nm 1J=kgm 2 /s 2 d F W = F d cos Slide 16 Scalar Product of Two
Vectors The scalar product of two vectors is written as A. B It is
also called the dot product A. B = A B cos is the angle between A
and B Slide 17 Scalar Product The scalar product is commutative A.
B = B. A The scalar product obeys the distributive law of
multiplication A. (B + C) = A. B + A. C Slide 18 Dot Products of
Unit Vectors Using component form with A and B: Slide 19 Work - on
and by A person pushes block 30 m along the ground by exerting
force of 25 N on the trolley. How much work does the person do on
the trolley? W = F d = 25N x 30m = 750 Nm Trolley does -750 Nm work
on the person FpFptFpFpt F tp d Slide 20 A force acts on an object
as the object moves in the x direction from the origin to x = 5.00
m. Find the work W = F dr done on the object by the force. Slide 21
Mechanical Energy Mechanical energy (energy associated with masses)
can be thought of as having two components: kinetic and potential
Kinetic energy is energy of motion Potential energy is energy of
position Slide 22 Kinetic Energy In many situations, energy can be
considered as the ability to do work Energy can exist in different
forms Kinetic energy is the energy associated with the motion of an
object A moving object can do work on another object E.g hammer
does work on nail. Slide 23 Kinetic Energy Consider an object with
mass m moving in a straight line with initial velocity v i. To
accelerate it uniformly to a speed v f a constant net force F is
exerted on it parallel to motion over a distance d. Work done on
object W = F d = m a d W = F d = m a d (NII)So Slide 24 Kinetic
Energy If we rearrange this we obtain We define the quantity mv 2
to be the translational kinetic energy (KE) of the object This is
the Work-Energy Theorem: The net work done on an object is equal to
its change in kinetic energy W = KE Slide 25 The Work-Energy
Theorem NoteNote The net work done on an object is the work done by
the net force. Units of energy and work must be the same (J) W = KE
Slide 26 I.Kinetic energy I. Kinetic energy Energy associated with
the state of motion of an object. Units: 1 Joule = 1J = 1 kgm 2 /s
2 II. Work Energy transferred to or from an object by means of a
force acting on the object. To +W From -W Slide 27 - Constant force
- Constant force: Work done by the force = Energy transfer due to
the force. Slide 28 -To calculate the work done on an object by a
force during a displacement, we use only the force component along
a displacement, we use only the force component along the objects
displacement. The force component perpendicular the objects
displacement. The force component perpendicular to the displacement
does zero work. to the displacement does zero work. Assumptions:1)
F = constant force Assumptions: 1) F = constant force 2) Object is
particle-like (rigid object, all parts of the object must move
together). 2) Object is particle-like (rigid object, all parts of
the object must move together). A force does +W when it has a
vector component in the same direction as the displacement, and W
when it has a vector component in the opposite direction. W=0 when
it has no such vector component. Slide 29 Net work done by several
forces = Sum of works done by individual forces. Calculation: 1) W
net = W 1 +W 2 +W 3 + Calculation: 1) W net = W 1 +W 2 +W 3 + 2) F
net W net =F net d 2) F net W net =F net d Slide 30 II.
Work-Kinetic Energy Theorem Change in the kinetic energy of the
particle = Net work done on the particle III. Work done by a
constant force - Gravitational force: Rising object:W= mgd cos180 =
-mgd Rising object: W= mgd cos180 = -mgd F g transfers mgd energy
from the objects kinetic energy. Falling down object:W= mgd cos 0 =
+mgd Falling down object: W= mgd cos 0 = +mgd transfers energy to
the objects kinetic energy F g transfers mgd energy to the objects
kinetic energy. Slide 31 External applied force + Gravitational
force: Object stationary before and after the lift: Object
stationary before and after the lift: W a +W g =0 The applied force
transfers the same amount of energy to the object as the
gravitational force transfers from the object. Slide 32 IV. Work
done by a variable force - Spring force: Hookes law = spring
constant measures springs stiffness. k = spring constant measures
springs stiffness. Units: N/m Slide 33 Hookes Law When x is
positive (spring is stretched), F is negative When x is 0 (at the
equilibrium position), F is 0 When x is negative (spring is
compressed), F is positive Slide 34 Hookes Law The force exerted by
the spring is always directed opposite to the displacement from
equilibrium F is called the restoring force If the block is
released it will oscillate back and forth between x and x Slide 35
Work done by a spring force: Hookes law: Assumptions: Spring is
massless Spring is massless m spring 1)The block displacement must
be divided into many segments of infinitesimal width, into many
segments of infinitesimal width, x. 2)within each shortsegment. 2)
F(x) cte within each short x segment. - Calculation: x FxFxFxFx
xixixixi xfxfxfxf xxxx FjFjFjFj W s >0 Block ends up closer to
the relaxed position (x=0) than it was initially. W s
