Download Energy - Nathan Dawson

Energy
Kinds of energy:
•Energy of motion (kinetic energy)
•Energy of position (potential energy)
–Position with respect to earthGravitational potential energy
–Position with respect to a spring Elastic potential energy
•Energy of chemical bonds (charge position energy)
•Energy of disordered random motion of molecules
(thermal energy-kinetic)
•Next semester Energy in electric and magnetic fields
Work
The work done on a object from a constant force is given by
In words, the above equation is stated, “the work from a constant applied
force is equal to the magnitude of the force vector on the object
multiplied by the magnitude of the displacement vector of the object and
the cosine of the angle between them.”
The total work done on an object subject to constant forces is the sum of
all the work
(a) Determine the work a hiker must do on a 15.0kg backpack to
carry it up a hill of height 10.0m with an incline of 30o from
the vertical.
(b) Determine the work done by gravity on the backpack.
(c) Determine the net work done on the backpack.
Assume the motion is smooth and at constant velocity (i.e., acceleration is zero).
Circular orbit review
- Remember that centripetal acceleration is
- The magnitude of the force is
which gives
- Newtons law of gravity is
- Determine the sum of the forces,
- The only acceleration is
, so
A
- We know the circumference of a circle and that
constant
velocity, therefore
is a
B
- Find the radius of orbit by plugging in B into A and solving for
radius of orbit,
gives the
Work & circular orbit
How much work is done on a
satellite to have it orbit 1 period?
NO WORK DONE!!!
and therefore
Thus,
Translational kinetic energy
The translational kinetic energy of a moving object is one-half the
mass multiplied by the velocity squared. Symbolically, the kinetic
energy may be written as
Another important quantity that we will often use is the change in
translational kinetic energy of an object. We may write the change
in translational kinetic energy equation as
Consider two carts, of masses m and 2m, at rest on an air
track. If you push first one cart for 3 s and then the other
for the same length of time, exerting equal force on each,
the kinetic energy of the light cart is
1. larger than
2. equal to
3. smaller than
the kinetic energy of the heavy car.
Work-kinetic energy theorem
We previously discovered that the total work done on an object is given by
For simplicity, let us assume that the sum of the forces are along the direction
of motion for an object traveling in one-dimension, or
Our time-independent one-dimensional kinematic equation for constant
acceleration was given as
Solving for the acceleration and substituting into the expression for Wtot gives
Therefore,
• Identical constant forces
vo = 0
push two identical masses A
and B continuously from a
starting line to a finish line.
If A is initially at rest and B
vo = 0
is initially moving to the
right, which mass has the
larger change in kinetic
energy?
1. A
2. B
3. The kinetic energy changes are the same.
What if the force acts at an angle?
A 1000 kg runaway car is rolling slowly across a
level surface at 1 m/s, heading toward a small group
of children. The doors are locked so that you can’t
get inside. So, you run in front of the car and push on
the hood at an angle of 30o below the horizontal.
a) How much work must you do to stop the car?
b) How hard must you push to stop the car if there
is only one meter left between the car and the
kids?
Change in the gravitational potential
energy near Earth’s surface
The change in potential energy of gravity (for constant g),
For objects initially at a height, h, and falling to a zero, we have
(the work done on the object by gravity is (
)
For objects initially at zero, and being raised to a final height h, we have
(the work done on the object by gravity is (
)
Potential Energy, Conservation of Energy
Consider a block sliding down each of the following frictionless ramps.
In which case did gravity do more work?
Work done by
gravity: = mg (hf-hi)
d1
Does your answer
change if kinetic
friction is present?
d2
More gravity
• Work done by gravity on a projectile:
• Only the vertical component matters.
• Only the difference in the initial and final heights matters.
Define Potential Energy due to gravity:
Ug= mgh
Wg = -DUg
(We know that W = DK, and without any friction, W = Wg which gives DK = DU)
Change in a spring’s potential energy
The change in potential energy of a spring
Typically, we set the equilibrium distance of the a spring to zero,
where
or the change in the distance of the spring, from compressed/stretched
position to the equilibrium (neutral) position, has a potential energy
difference of
Internal energy
The change in the internal energy of a system is given as
Note that the internal energy can be changed by several phenomena
Examples are from dissipative forces such as friction
and explosions
where
is the energy released from an explosion
Non-Conservative Forces
Work done on the object by the force depends on the path
taken by the object
Example: friction on an object moving on a level surface
Ffriction   k N
Wfriction   Ffriction x    k N x  0
Work done by constant frictional force
The work done by kinetic friction
We know that the magnitude of the force of kinetic friction from
the model presented in class is equal to the coefficient of kinetic
friction multiplied by the normal force.
A person moves a 5.00kg box 10.0m by pulling on a rope making
an angle of 30.0o above the horizontal. The floor and box have a
coefficient of kinetic friction of 0.500 and she pulls with a force
of 30.0N.
(a)
(b)
(c)
(d)
How much work does she do?
How much work is done by gravity?
How much work is done by friction?
What is the change in kinetic energy after the box has
traveled the 10.0m?
A cart on an air track is moving at 0.5 m/s when the air is suddenly
turned off. The cart comes to rest after traveling 1 m. The experiment is
repeated, but now the cart is moving at 1 m/s when the air is turned off.
How far does the cart travel before coming to rest?
(Hint: What force causes it to slow in each case?)
1. 1 m
2. 2 m
3. 3 m
4. 4 m
5. 5 m
6. impossible to determine
Complete the following statement: A force that acts on an object
is said to be conservative if
1. it obeys Newton's laws of motion.
2 it results in a change in the object's kinetic energy.
3 it always acts in the direction of motion of the object.
4 the work it does on the object is independent of the path of the
motion.
5 the work it does on the object is equal to the increase in the
object's kinetic energy.
Some forms of energy
Mechanical Energy
K
Thermal Energy
Ug
Us
Other forms include
Eth
Echem
Enuclear
The Basic Energy Model
Energy Transformations
Kinetic energy K = energy of motion
Potential energy U = energy of
position
Thermal energy Eth = energy
associated with temperature
System energy E = K + U + Eth +
Echem + ...
Energy can be transformed within
the system without loss.
Energy is a property of a system.
Some Energy Transformations
Echem  Ug
K  Eth
Us  K  Ug
Which of the following is an energy transfer?
A.
B.
C.
D.
E.
Kinetic energy
Heat
Potential energy
Chemical energy
Thermal energy
If you raise an object to a greater height, you are increasing
A.
B.
C.
D.
E.
kinetic energy.
heat.
potential energy.
chemical energy.
thermal energy.
A skier is moving down a slope at a constant speed. What energy
transformation is taking place?
A.
B.
C.
D.
E.
K  Ug
Ug Eth
Us  Ug
Ug  K
K  Eth
A child is on a playground swing, motionless at the highest point of
his arc. As he swings back down to the lowest point of his motion,
what energy transformation is taking place?
A.
B.
C.
D.
E.
K  Ug
Ug Eth
Us  Ug
Ug  K
K  Eth
Conservation of energy
The change in the energy of a system is given by
In words, this is stated as “The total change in energy of a system
is equal to the sum of the changes in the internal energy, the
kinetic energy, and the potential energy, where this sum is equal to
the total external work done on the system.
You put fresh batteries into a flashlight. Then you turn it on and leave
it on until the bulb gradually dims and finally goes out. Which
statement best describes the involvement of energy in this process?
1. The energy has been used up by the bulb and no longer exists
anywhere.
2. All the energy that the batteries originally had when new still exists
somewhere.
3. The energy of the batteries was converted to heat by the bulb, so it
no longer exists.
4. The amount of energy in the flashlight, batteries, and bulb remain
the same, because energy is conserved.
5. The energy has moved from one end of the batteries to the other.
Conservation of mechanical energy
When there is no net work done on a system and no
change in the internal energy, then
In words, this is stated as “when mechanical energy is conserved,
the total change in energy is zero. Therefore, the sum of the
changes in the kinetic energy and the potential energies is zero.”
Use conservation of mechanical energy when:
1. You know or want to know information about initial
and final positions (potential energy, gravitational or
spring) and speeds (kinetic energy).
2. You are not concerned with details of the motion
between the initial and final points, such as acceleration
and time.
3. You do not want vector details of the initial or final
motion, such as the direction of the velocity vector
(Kinetic energy only involves the speed -- magnitude of
the velocity vector.).
A car goes down a hill a distance d. The hill is inclined at an
angle q with respect to the horizontal. Thus the car’s vertical
drop is h = d sinq , the component of gravity parallel to the hill
in the downward direction is F|| = +mgsin , and the
component of the displacement parallel to gravity and
downward is d|| = + d sin
Which of the following is NOT a correct expression for the work done
BY gravity on the car during the descent?
1) +mgd
3) +mgsin d
2) +mg d || 4) +mgh
Three balls are thrown off a cliff with the same speed, but in different
directions. Which ball has the greatest speed just before it hits the ground?
A.
B.
C.
D.
Ball A
Ball B
Ball C
All balls have
the same speed
What is the system? What energy transformations are taking place?
A block slides down a frictionless ramp of height h. It
reaches velocity v at the bottom. To reach a velocity of 2v,
the block would need to slide down a ramp of height
1.
2.
3.
4.
5.
1.41h
2h
3h
4h
6h
Note that
unless
4m
finish line
m
Two pucks on level ice are shown above, pressed back by
equal amounts against identical springs. The pucks are the same size
and shape, but one has four times the mass of the other. The pucks
are released and the springs propel them to the finish line. At the
finish line how does the speed of the less massive puck compare to
the speed of the more massive puck?
1. It is the same as the maximum speed of the more massive puck.
2. It is four times the speed of the more massive puck.
3. It is twice the speed of the more massive puck.
4. It is half the speed of the more massive puck.
5. It is one-fourth the speed of the more massive puck.
Solving Conservation of Energy Problems:
1.Draw a rough diagram. Label your arbitrary reference position for
the potential energy (example, y = 0 for gravity). To avoid negative
numbers, this is often taken to be at the lowest position you’ll be
interested in.... Check to see that your system is isolated (no external
forces doing work).
2. Designate the initial and final points (points where you either know
or want to know information about the object).
3. Apply the conservation of energy:
Ki + Ui + W = Kf + Uf
where W is the work done by all of the non-conservative forces acting
on the object between the initial and final points, such as friction.
(Note that friction does negative work. Other forces, such as nonconservative forces from explosions, for example, can do positive
work – that is, they give energy to the object.)
4. Solve for the desired unknown quantity. (You should still be using
symbols at this point! Only plug in numbers at the end.)
Example problem (classical: loop-the-loop)
A object of mass 2.50 kg on a frictionless track is released from rest at a height of 12.0 m.
There is a loop of radius 1.50 m afer the track becomes level as shown in the below diagram.
(a)
(b)
(c)
(d)
What is the magnitude of the velocity of the object at the top of the loop-the-loop?
What is the normal force (effective weight) of the object at the top of the loop-the-loop?
What is the magnitude of the velocity of the object at the bottom of the loop-the-loop?
What is the normal force of the object at the bottom of the loop-the-loop?
Example problem
A spring connected to a wall releases and transfers all of its
potential energy to the kinetic energy of a ball of mass 1.5 kg. The
spring is initially compressed from equilibrium by Dx = 10.0 cm.
The final velocity is 5.0 m/s. What is the spring constant?
Known: m, Dx, vf
Unknown: k
Dx
Example problem (similar to question 90 in Giancoli)
A pendulum made from a mass connected to the ceiling by a string
of length r = 2.00 m is released from rest at an angle q0 = 30.0 as
shown in the below diagram. At the lowest point the string comes
into contact with a beam that is a distance d = 1.00 m below the
ceiling. What is the angle, qf, where the kinetic energy is zero?
Known: r, q0, d
Unknown: qf
A dart of mass 0.100kg is pressed against the spring of a toy dart gun.
The spring, with spring stiffness constant 250N/m and negligible
mass, is compressed 6.0cm and released. If the dart detaches from the
spring when the spring reaches its natural length, then what speed
does the dart acquire?
Two atoms interacting with
each other have the
potential energy graph
shown. Assume the big
atom’s mass is much
greater than the little
atom’s and that all the
motion is along x.
Imagine that the little atom approaches the big atom from infinity
with an initial K= 5 x 10-21 J. How close to the big atom does it get?
(nm=nanometer=10-9 m)
(i) x = 0
(ii) x = 0.04 nm
(iii) x = 0.2 nm
(iv) x = 0.3 nm
(v) other (specify)
Two atoms interacting
with each other have
the potential energy
graph shown. Assume
the big atom’s mass is
much greater than the
little atom’s and that all
the motion is along x.
Imagine that at t=0, the little atom is at position x = 0.11
nm and has a kinetic energy of 5 x 10-21 J. About how
much energy would we have to add to break the bond?
(1) 1 x 10-21 J
(2) 6 x 10-21 J
(3) 12 x 10-21
(4) none, the bond is already broken.
Two atoms interacting
with each other have the
potential energy graph
shown. Assume the big
atom’s mass is much
greater than the little
atom’s and that all the
motion is along x.
Imagine that at t = 0, the little atom is at rest at x = 0.20 nm. What is
the closest that it will ever get to the big atom subsequently (in the
absence of any external forces)?
(1) 0.03 nm
(2) 0.05 nm (3) 0.20 nm (4) none of these.
A bead of mass 0.150 kg
slides without friction on
a track as shown in the
diagram.
The bead initially starts
from rest at point A.
(a) Determine the change in the gravitational potential energy from
point A to point B.
(b) Determine the work done by gravity from point A to point B.
(c) Determine the change in kinetic energy from point C to Point D.
(d) Determine the kinetic energy at point E.
(e) Determine the velocity at point E.
• Analyze the following in terms of forces
and then in terms of energy conservation:
q
A ball on a string of length L is moving in a vertical circle,
with only gravity and the string tension acting on it. If it has a
speed v at the top of the circle. What is its speed when it is at
an angle q from the top?
Power = rate of doing work
= W/Dt
• 1 J/s=1 Watt (W) [We’ve run out of symbols!]
• 1 horsepower = 746 W
• There are no negative powers.
• Electrical Power kW, But what is a kW-hr?
• For work done by a force on an object moving at
constant speed:
P= W/Dt = F Dx/Dt = F v cosq
Power example
If you have a mass of 90 kg and you can
run upstairs to the next floor (3 m above)
in 2 seconds, what is your power output?