Transcript work

• Lesson 1: Basic Terminology and
Concepts
• Work
– Definition and Mathematics of Work
– Calculating the Amount of Work Done by
Forces
•
•
•
•
Potential Energy
Kinetic Energy
Mechanical Energy
Power
Definition and Mathematics of Work
• In physics, work is defined as a
force
_________
acting upon an object to
cause
displacement
____________
a __________________.
Work is being done
Work
Workisisnot
notbeing
beingdone
done
Let’s practice – work or no work
1. A student applies a force to a wall and
becomes exhausted.
2. A calculator falls off a table and free falls
to the ground.
3. A waiter carries a tray full of beverages
above his head by one arm across the
room
4. A rocket accelerates through space.
Calculating the Amount of Work Done by Forces
W = F∙d∙cosθ
F
θ
• F - is the force in Newton, which
causes the displacement of the
object.
• d - is the displacement in meters
• θ = angle between force and
displacement
• W - is work in N∙m or Joule (J). 1 J
= 1 N∙m = 1 kg∙m2/s2
d
Fy
θ
d
F
scalar
• Work is a _____________
quantity
Fx
• Work is independent of time the
force acts on the object.
Only the horizontal component of
the force (Fcosθ) causes a
horizontal displacement.
W = F∙d∙cosθ
positive, negative or
zero work
Positive work
negative work - force acts in
the direction opposite the
objects motion in order to
slow it down.
no work
To Do Work, Forces Must Cause Displacements
W = F∙d∙cosθ = 0
The angle in work equation
W = F∙d∙cosθ
• The angle in the equation is the angle between
the force and the displacement vectors.
F & d are in the same direction, θ is 0o.
d
F
example
• A 20.0 N force is used to push a 2.00 kg cart a
distance of 5.00 meters. Determine the amount of
work done on the cart by the force.
20.0 N
example
• How much work is done in lifting a 5.0 kg
box from the floor to a height of 1.2 m
above the floor?
example
• A 2.3 kg block rests on a horizontal surface. A constant
force of 5.0 N is applied to the block at an angle of 30.o to
the horizontal; determine the work done on the block a
distance of 2.0 meters along the surface.
5.0 N
30o
2.3 kg
practice
• Matt pulls block along a horizontal surface at constant
velocity. The diagram show the components of the force
exerted on the block by Matt. Determine how much work
is done against friction.
6.0 N
F
8.0 N
3.0 m
example
• A neighbor pushes a lawnmower four
times as far as you do but exert only half
the force, which one of you does more
work and by how much?
Force vs. displacement graph
Example: a block is pulled along
a table with 10. N over a distance
of 1.0 m.
W = Fd = (10. N)(1.0 m) = 10. J
height
base
area
Force (N)
• The area under a force versus
displacement graph is the work done by
the force.
work
Displacement (m)
Potential energy
• An object can store energy as the result of
Potential energy
its position. ________________________
is the stored energy of position possessed
by an object.
• Two form:
– Gravitational
– Elastic
Gravitational potential energy
• Gravitational potential energy is the energy stored in an
object as the result of its _________________________
vertical position (height).
gravitational
• The energy is stored as the result of the _____________
attraction of the Earth for the object.
• The work done in raising an object must result in an
gravitational potential energy
increase in the object's _______________________
• The gravitational potential energy of an object is
dependent on three variables:
– The mass of the object
– The height of the object
– The gravitational field strength
PEgrav = m∙g∙h
• Equation: ______________________
– m: mass, in kilograms
– h: height, in meters
– g: acceleration of gravity = 9.81 m/s2
GPE
• GPE = mgh
• The equation shows that . . .
• the more mass a body has
• or the stronger the gravitational field it’s in
• or the higher up it is
• . . . the more gravitational potential energy
it’s got.
GPE and work done by gravity
• When an object falls, gravity does positive
work. Object loses GPE.
hi
• Wgrav = mg(hi – hf)
• Wgrav = - mg(hf – hi) = - mg∆h
As long as the falling height is the
same, gravity did The same
amount of work regardless of
which path is taken.
hf
GPE and work against gravity
• When an object is raised against gravity at
constant speed (no change in kinetic energy),
gravity does negative work. Object gains GPE.
hf
• Work done against gravity = mg∆h
As long as the object is raised to
the same height, work done
against gravity is the same
regardless of which path is
taken.
hi
The increase in an object's potential energy equals the
work done in raising an object
Each path up to the seat top requires the same amount of
work. The amount of work done by a force on any object is
given by the equation W = F∙d∙cosθ
where F is the force, d is the displacement and θ is the angle
between the force and the displacement vector. In all three
cases, θ equals to 0o
example
• The diagram shows points A, B, and C at or near Earth’s
surface. As a mass is moved from A to B, 100. joules of
work are done against gravity. What is the amount of
work done against gravity as an identical mass is moved
from A to C?
Unit of energy
• The unit of energy is the same as work:
Joules
_______
• 1 joule = 1 (kg)∙(m/s2)∙(m) = 1 Newton ∙
meter
• 1 joule = 1 (kg)∙(m2/s2)
Work and energy has the same unit
Gravitational potential energy is
relative
• To determine the gravitational potential energy
of an object, a _______
zero height position must first
be assigned.
ground
• Typically, the ___________
is considered to be a
position of zero height.
• But, it doesn’t have to be:
– It could be relative to the height above the lab table.
– It could be relative to the bottom of a mountain
– It could be the lowest position on a roller coaster
example
•
How much potential energy is gained by
an object with a mass of 2.00 kg that is
lifted from the floor to the top of 0.92 m
high table?
•
The graph of
gravitational
potential energy
vs. vertical height
for an object near
Earth's surface
gives the weight
of the object.
The weight of the
object is the slope
of the line.
Weight = __________
Elastic potential energy
• Elastic potential energy is the energy
elastic
stored in ______________
materials as
the result of their stretching or
compressing.
• Elastic potential energy can be stored in
– Rubber bands
– Bungee cores
– Springs
– trampolines
Hooke’s Law
F = kx
Spring force = spring constant x displacement
• F in the force needed to displace (by stretching or
compressing) a spring x meters from the equilibrium
(relaxed) position. The SI unit of F is Newton.
• k is spring constant. It is a measure of stiffness of the
spring. The greater value of k means a stiffer spring
because more force is needed to stretch or compress it
that spring. The Si units of k are N/m. depends on the
material made up of the spring. k is in N/m
• x the distance difference between the length of
stretched/compressed spring and its relaxed
(equilibrium) spring.
example
• A spring has a spring constant of 25 N/m. What is the
minimum force required to stretch the spring 0.25 meter
from its equilibrium position?
example
•
The graph below shows elongation as a
function of the applied force for two springs, A
and B. Compared to the spring constant for
spring A, the spring constant for spring B is
1. smaller
2. larger
3. the same
Elastic potential energy in a spring
•
Elastic potential energy is the Work done on the spring.
PEs = Favg∙d = Favg∙x = (½ k∙x)∙x = ½ kx2
Note: F is the average force
PEs = ½ k∙x2
– k: spring constant
– x: amount of compression or extension relative
to equilibrium position
Elastic potential energy
Elastic potential energy is directly
proportional to x2
elongation
example
• A spring has a spring constant of 120
N/m. How much potential energy is stored
in the spring as it is stretched 0.20 meter?
example
• The unstretched spring in the diagram has a length of
0.40 meter and a spring constant k. A weight is hung
from the spring, causing it to stretch to a length of 0.60
meter. In terms of k, how many joules of elastic potential
energy are stored in this stretched spring?
example
• Determine the potential energy stored in the spring with
a spring constant of 25.0 N/m when a force of 2.50 N is
applied to it.
example
• As shown in the diagram, a 0.50-meter-long spring is
stretched from its equilibrium position to a length of 1.00
meter by a weight. If 15 joules of energy are stored in the
stretched spring, what is the value of the spring
constant?
example
• A 10.-newton force is required to hold a
stretched spring 0.20 meter from its rest
position. What is the potential energy
stored in the stretched spring?
• A force of 0.2 N is needed to compress a
spring a distance of 0.02 meter. What is
the potential energy stored in this
compressed spring?
Kinetic energy
motion
• Kinetic energy is the energy of _______.
• An object which has motion - whether it be
vertical or horizontal motion - has kinetic energy.
• The equation for kinetic energy is:
2
KE
=
½
mv
__________________
– Where KE is kinetic energy, in joules
– v is the speed of the object, in m/s
– m is the mass of the object, in kg
Kinetic Energy
• KE = ½ m v 2
• The equation shows that . . .
• the more mass a body has
• or the faster it’s moving
• . . . the more kinetic energy it has.
• KE is directly proportional to m, so doubling the mass
doubles kinetic energy, and tripling the mass makes it three
times greater.
Kinetic energy
Kinetic energy
• KE is proportional to v 2, so doubling the speed quadruples
kinetic energy, and tripling the speed makes it nine times
greater.
speed
mass
Example
• A 55 kg toy sailboat is cruising at 3
m/s. What is its kinetic energy?
Note: Kinetic energy (along with every
other type of energy) is a scalar, not a
vector!
example
•
An object moving at a constant speed of
25 meters per second possesses 450
joules of kinetic energy. What is the
object's mass?
example
•
a.
b.
c.
d.
A cart of mass m traveling at a speed v
has kinetic energy KE. If the mass of the
cart is doubled and its speed is halved,
the kinetic energy of the cart will be
half as great
twice as great
one-fourth as great
four times as great
example
•
Which graph best represents the relationship
between the kinetic energy, KE, and the
velocity of an object accelerating in a straight
line?
a
b
c
d
Mechanical Energy
• Mechanical energy is the energy that is
possessed by an object due to its motion or due
to its position. Mechanical energy can be either
kinetic energy (energy of motion) or potential
energy (stored energy of position) or both.
The total amount of mechanical energy is
merely the sum of the potential energy and the
kinetic energy
TME = KE + PEg + PEs
Mechanical Energy as the Ability
to Do Work
• Any object that possesses mechanical energy - whether
it is in the form of potential energy or kinetic energy - is
able to do work.
• The diagram shows the motion of Brie as she glides
down the hill and makes one of her record-setting jumps.
TME =
TME =
TME =
TME =
TME =
Power
• Power is the rate at which work is done. It is the
work/time ratio. Mathematically, it is computed using the
following equation.
• The standard metric unit of power is the Watt.
• All machines are typically described by a power
rating. The power rating indicates the rate at
which that machine can do work upon other
objects.
• The power rating of a car relates to how rapidly
the car can be accelerated.
• Some people are more power-full than others.
That is, some people are capable of doing the
same amount of work in less time or more work
in the same amount of time
example
• Ben Pumpiniron elevates his 80-kg body up the
2.0-meter stairwell in 1.8 seconds. What is his
power?
It can be assumed that Ben must apply an (80 kg x 9.81
m/s2) -Newton downward force upon the stairs to elevate his
body.
Another equation for power
example
•
Two physics students, Will N. Andable and Ben
Pumpiniron, are in the weightlifting room. Will lifts the
100-pound barbell over his head 10 times in one
minute; Ben lifts the 100-pound barbell over his head
10 times in 10 seconds. Which student does the most
work? ______________ Which student delivers the
most power? ______________ Explain your answers.
example
• When doing a chin-up, a physics student lifts her 42.0-kg
body a distance of 0.25 meters in 2 seconds. What is the
power delivered by the student's biceps?
kilowatt-hour is unit for energy
• Your household's monthly electric bill is often
expressed in kilowatt-hours. One kilowatt-hour
is the amount of energy delivered by the flow of l
kilowatt of electricity for one hour. Use
conversion factors to show how many joules of
energy you get when you buy 1 kilowatt-hour of
electricity.
• Lesson 2: The Work-Energy Theorem
• Internal vs. External Forces
• The Work-Energy Connection
– Analysis of Situations Involving External
Forces
– Analysis of Situations in Which
Mechanical Energy is Conserved
– Application and Practice Questions
• Bar Chart Illustrations
What is energy?
ability
• Energy is the __________to
do work.
scalar
• Energy is a _________quantity.
• When work is done on or by a system,
the total energy of the system is
changed.
Work-Energy theorem
• The net work done (work done by net force) on
an object equals to the change in the object’s
kinetic energy
Wnet = ∆KE = KE2 – KE1
Fnetd = KE2 – KE1
Practice Problem #1
• A 1000-kg car traveling with a speed of 25 m/s skids to a
stop. The car experiences an 8000 N force of friction.
Determine the stopping distance of the car.
Practice Problem #2
• At the end of the Shock Wave roller coaster ride, the
6000-kg train of cars (includes passengers) is slowed
from a speed of 20 m/s to a speed of 5 m/s over a
distance of 20 meters. Determine the braking force
required to slow the train of cars by this amount.
• The above problems have one thing in common: there is a
force which does work over a distance in order to remove
mechanical energy from an object.
• The force acts opposite the object's motion and thus does
negative work which results in a loss of the object's total
amount of mechanical energy. In each situation, the work
is related to the kinetic energy change.
TMEi + Wext = TMEf
KEi + Wext = 0 J
½ •m•vi2 + F•d•cos(180o) = 0 J
F•d = ½ •m•vi2
d ~ v i2
Stopping distance is dependent upon the square
of the velocity.
Stopping distance and initial
velocity
Wnet = ∆KE
Wnet = 0 - ½ •m•vi2
Ff•d = - ½ •m•vi2
• Ff = μFnorm = μmg
• - μmg•d = - ½ mvi2
• d = vi2 / 2μg
d ~
v i2
practice
(m/s)
Stopping Distance (m)
0 m/s
0
5 m/s
4m
10 m/s
15 m/s
20 m/s
25 m/s
Conservative vs. non-conservative Forces
• There are a variety of ways to categorize all the
types of forces.
1. Contact force: Forces that arise from the
physical contact of two objects.
2. Field force exist between objects, even in the
absence of physical contact between the objects.
• We can also categorize forces based upon
whether or not their presence is capable of
changing an object's total mechanical energy.
1. Conservative force can never change the total
mechanical energy of an object
2. Non-conservative forces will change the total
mechanical energy of the object
• The conservative forces include the gravity
forces, spring force, magnetic force, electrical
force.
• We will simply say all the other forces are nonconservative forces, such as applied force,
normal force, tension force, friction force, and air
resistance force.
conservative
forces
Non-conservative
forces
Fgrav
Fspring
Fapp
Ffrict
Ften
FNorm
Work energy theorem
Wnet = ∆KE
Wgrav + Wspring + Wother = ∆KE
Wgrav = work done by Gravity
Wgrav = mg(hi – hf)
Wspring = work done by spring
Wspring = ½ k(xi2 – f
mghi – mghf + ½ kxi2 – ½kxf2 + Wother = ½ kvf2 – ½kvi2
mghi + ½ kxi2 + ½kvi2 + Wother = ½ kvf2 + mghf + ½kxf2
PEgi + PEsi + KEi + Wother = PEgf + PEsf + KEf
PEgi + PEsi + KEi + Wother = PEgf + PEsf + KEf
TMEi + Wother = TMEf
Wother = TMEf - TMEi
• When net work is done upon an object by an
non-conservative force, the total mechanical
energy (KE + PE) of that object is changed.
– If the work is positive work, then the object will gain
energy.
– If the work is negative work, then the object will lose
energy.
– The gain or loss in energy can be in the form of
potential energy, kinetic energy, or both.
– The work done will be equal to the change in
mechanical energy of the object.
TMEi + Wother = TMEf
Wother = TMEf - TMEi
• When the only type of force doing net work upon
an object is conservative force (Wother = 0), the
total mechanical energy (KE + PE) of that object
remains constant. TMEf = TMEi. In such cases,
the object's energy changes form.
• For example, as an object is "forced" from a high
elevation to a lower elevation by gravity, some of
the potential energy of that object is transformed
into kinetic energy. Yet, the sum of the kinetic
and potential energies remain constant.
• When Wother = 0, TMEi = TMEf, energy is transformed or
changes its form from kinetic energy to potential energy
(or vice versa); the total amount present is conserved i.e., always the same.
In Summary
• When only conservative forces do work
– TME is conserved,
– TME1 = TME2
• When non-conservative forces do work
– TME changes,
– Wother = TME2 – TME1
example
example
example
example
example
• A block weighing 15 N is pulled to the top of an incline
that is 0.20 meter above the ground, as shown below. If
5.0 joules of work are needed to pull the block the full
length of the incline, how much work is done against
friction?
example
• A shopping cart full of groceries is sitting at the top of a 2.0-m
hill. The cart begins to roll until it hits a stump at the bottom of
the hill. Upon impact, a 0.25-kg can of peaches flies
horizontally out of the shopping cart and hits a parked car with
an average force of 500 N. How deep a dent is made in the
car (i.e., over what distance does the 500 N force act upon the
can of peaches before bringing it to a stop)?
Example
In the diagram below, 450. joules of work is
done raising a 72-newton weight a vertical
distance of 5.0 meters. How much work is
done to overcome friction as the weight is
raised?
Example
•
A box with a mass of 0.04 kg starts from rest at point A
and travels 5.00 meters along a uniform track until
coming to rest at point B, as shown in the picture.
Determine the magnitude of the frictional force acting on
the box. (assume the frictional force is constant.)
A
0.80 m
B
0.50 m
example
• A block weighing 40. newtons is released from rest on an
incline 8.0 meters above the horizontal, as shown in the
diagram below. If 50. joules of heat is generated as the
block slides down the incline, what is the maximum
kinetic energy of the block at the bottom of the incline?
The Example of Pendulum Motion
• Consider a pendulum bob swinging to and fro on the end
of a string. There are only two forces acting upon the
pendulum bob. Gravity (an internal force) acts downward
and the tensional force (an external force) pulls upwards
towards the pivot point. The external force does not do
work since at all times it is directed at a 90-degree angle
to the motion. The only force doing work is gravity, which
is a conservative force.
The pendulum
• The sum of the kinetic and potential energies in system
is called the total ______________________________.
mechanical energy
• In the case of a pendulum, the total mechanical energy
constant
(KE + PE) is _________________:
at the highest point,
all the energy is potential energy, at the lowest point, all
the energy is kinetic energy.
• As the 2.0-kg pendulum bob in the above diagram swings to
and fro, its height and speed change. Use energy equations
and the above data to determine the blanks in the above
diagram.
use the heights and the speeds given in the table below to
fill in the remaining cells at the various locations in a 0.200kg bob's trajectory.
h
(m)
Speed
(m/s)
2.000
0.0
1.490
1.128
0.897
0.357
PE
(J)
KE
(J)
TME
(J)
0
3.920
Example
•
1.
2.
3.
4.
As the pendulum swings from position A to position C
as shown in the diagram, what is the relationship of
kinetic energy to potential energy? [Neglect friction.]
The kinetic energy decreases more than the potential
energy increases.
The kinetic energy increases more than the potential
energy decreases.
The kinetic energy decrease is equal to the potential
energy increase.
The kinetic energy increase is equal to the potential
energy decrease.
example
• A pendulum is pulled to
the side and released
from rest. Sketch a
graph best represents
the relationship
between the
gravitational potential
energy of the pendulum
and its displacement
from its point of release.
PE
pos
Example
•
1.
2.
3.
4.
In the diagram, an ideal pendulum
released from point A swings freely
through point B. Compared to the
pendulum's kinetic energy at A, its
potential energy at B is
half as great
twice as great
the same
four times as great
Roller coaster – friction is ignored,
Wother = 0
• A roller coaster operates on the principle of energy
transformation. Work is initially done on a roller coaster
car to lift the car to the first and highest hill. The roller
coaster car has a large quantity of potential energy and
virtually no kinetic energy as it begins the trip down the
first hill. As the car descents hills and loops, it potential
energy is transformed into kinetic energy; as the car
ascends hills and loops, its kinetic energy is transformed
into potential energy. The total mechanical energy of the
conserved
car is _______________
when friction is ignored.
• The total mechanical energy of the roller
coaster car is a constant value of 40 000
Joules.
The skier
• Transformation of energy from the potential to the kinetic
also occurs for a ski jumper. As a ski jumper glides down
the hill towards the jump ramp and off the jump ramp
towards the ground, potential energy is transformed into
kinetic energy. If friction can be ignored, the total
mechanical energy is ______________________.
A free falling object
• If a stationary object having mass m is located a vertical
distance h above Earth’s surface, the object has initial
PE = mgh and KE = 0. as object falls, its PE
___________ and KE ________________. The total
mechanical energy is conserved.
Energy conversion of a free falling object
The graph shows as a ball is dropped, how its energy is
transformed.
• The total mechanical energy remains _____________.
• GPE decreases as KE increases
example
• A 3.0-kilogram object is placed on a frictionless track at
point A and released from rest. (Assume the gravitational
potential energy of the system to be zero at point C.)
Calculate the kinetic energy of the object at point B.
example
•
a.
b.
c.
d.
A 250.-kilogram car is initially at rest at point A on a
roller coaster track. The car carries a 75-kilogram
passenger and is 20. meters above the ground at point
A. [Neglect friction.] Compare the total mechanical
energy of the car and passenger at points A, B, and C.
The total mechanical energy is less at point C than it is
at points A or B.
The total mechanical energy is greatest at point A.
The total mechanical energy is the same at all three
points.
The total mechanical energy is greatest at point B.
example
• The diagram represents a 0.20-kilogram sphere moving
to the right along a section of a frictionless surface. The
speed of the sphere at point A is 3.0 meters per second.
• Approximately how much kinetic energy does the sphere
gain as it goes from point A to point B?
example
• A 1.0 kg mass falls freely for 20. meters near the
surface of Earth. What is the total KE gained by
the object during its free fall?
example
• Base your answer to the question on the information and
diagram. A 250.-kilogram car is initially at rest at point A
on a roller coaster track. The car carries a 75-kilogram
passenger and is 20. meters above the ground at point
A. [Neglect friction.] Calculate the speed of the car and
passenger at point B.
example
• A 20.-kilogram object strikes the ground with
1960 joules of kinetic energy after falling freely
from rest. How far above the ground was the
object when it was released?
Example
• A 55.0-kilogram diver falls freely from a diving platform
that is 3.00 meters above the surface of the water in a
pool. When she is 1.00 meter above the water, what are
her gravitational potential energy and kinetic energy with
respect to the water's surface?
When work is done by elastic force,
energy is conserved
• A person does 100 joules
of work in pulling back the
string of a bow. What will
be the initial speed of a
0.5-kilogram arrow when it
is fired from the bow?
example
• A vertically hung spring has a spring constant of 150.
Newton per meter. A 2.00-kilogram mass is
suspended from the spring and allowed to come to
rest. Calculate the total elastic potential energy stored
in the spring due to the suspended 2.00-kilogram
mass.
example
• The diagram shows a 0.1-kilogram apple attached to a
branch of a tree 2 meters above a spring on the ground
below. The apple falls and hits the spring, compressing it
0.1 meter from its rest position. If all of the gravitational
potential energy of the apple on the tree is transferred to
the spring when it is compressed, what is the spring
constant of this spring?
example
• A person does 64 joules of work in pulling
back the string of a bow. What will be the
initial speed of a 0.5-kilogram arrow when
it is fired from the bow?
example
• A spring in a toy car is compressed a distance, x. When
released, the spring returns to its original length,
transferring its energy to the car. Consequently, the car
having mass m moves with speed v. Derive the spring
constant, k, of the car’s spring in terms of m, x, and v.
[Assume an ideal mechanical system with no loss of
energy.]
example
• A vertically hung spring has a spring constant of 150.
Newton per meter. A 2.00-kilogram mass is
suspended from the spring and allowed to come to
rest. Calculate the total elastic potential energy stored
in the spring due to the suspended 2.00-kilogram
mass.
example
•
1.
2.
3.
4.
Which pair of quantities can be
expressed using the same units?
work and kinetic energy
power and momentum
impulse and potential energy
acceleration and weight
example
• Which of the following statements are true about work?
Include all that apply.
1. Work is a form of energy.
2. Units of work would be equivalent to a Newton times a
meter.
3. A kg•m2/s2 would be a unit of work.
4. Work is a time-based quantity; it is dependent upon how fast
a force displaces an object.
5. Superman applies a force on a truck to prevent it from
moving down a hill. This is an example of work being done.
6. An upward force is applied to a bucket as it is carried 20 m
across the yard. This is an example of work being done.
7. A force is applied by a chain to a roller coaster car to carry it
up the hill of the first drop of the Shockwave ride. This is an
example of work being done.
example
A
B
elongation
force
• Determine the meaning of slope in each graph
force
weight
Gravitational
potential
energy
elongation
C
D
mass
height