Transcript Potential Energy

Physics
Conservative and
Non-Conservative
Forces
Teacher: Luiz Izola
Chapter Preview
1.
2.
3.
4.
5.
6.
7.
Conservative Forces
Non-Conservative Forces
Potential Energy
Work Done by Conservative Forces
Conservation of Mechanical Energy
Work Done by Non-Conservative Forces
Potential Energy Curves and Equipotentials
Introduction
 One of the greatest physics concepts is the
conservation of energy.
 Energy has several forms (mechanical,
thermal). The universe has a constant amount
of energy that flows from one form to another.
 In this chapter we will focus on the
conservation of energy, the first “conservation
law”.
Conservative Forces
 The work done by a conservative force can be
stored in the form of energy and released at a later
time.
 Simplest case of conservative force  Gravity
Conservative Forces
 Definition1: A conservative force does zero
work on a closed path.
 Definition2: Work done by a conservative force
in going from points A to B is independent of the
path taken.
Conservative Forces
 W1 + W 2 = 0
 W1 + W3
=0
 Therefore: W2 = W3
Non-Conservative Forces
 The work done by a non-conservative force
cannot be recovered later as kinetic energy.
 It is converted to other forms of energy such as
heat. See the work done by friction below.
Non-Conservative Forces
 Considering the same closed path but analyzing
the effect of friction on the total work, we would
have: Wtotal = -4μkmgd
Example
 A 4.57kg box is moved with constant speed from
A to B along the two paths below. Calculate the
work done by gravity on each one of these paths.
Also, calculate the work done by friction (μk = 0.63)
along the two paths.
Potential Energy (U)
 Potential energy is a storage system for energy.
 Energy is never lost as long as the separation
remains the same. For example, when we lift a ball
we produce an amount of potential energy related
to the height we lift it. The ball can rest on a shelf
for a million years and when it falls, it will gain the
same amount of kinetic energy.
 Work done against friction is not stored as
potential energy. It dissipates as heat or sound.
 Only conservative forces have the potentialenergy storage system.
Work by Conservative Forces
 When a conservative force does an amount of
work Wc (c = conservative), the corresponding
potential energy U is changed according to:
Wc = Ui – Uf = -ΔU
 The work done by a conservative force is equal to
the negative of the change in potential energy.
 For example, when an object falls, gravity does
positive work on it and its potential energy
decreases.
Work by Conservative Forces
 Gravity: Let’s apply our definition of potential
energy to the force of gravity near the Earth’s
surface.
Work by Conservative Forces
 Gravitational Potential Energy
U = mgy
y = height
m = mass
g = acceleration of gravity
Example: Find the gravitational potential energy of
a 65-kg person on a 3.0meter diving board. Let
U = 0 be the water level.
Work by Conservative Forces
 Example: An 82kg mountain climber is in the final
stage of a 4301-meter-high peak. What is the change
in potential energy as the climber gains the last 100
meters of altitude? Let U = 0 be (a) at sea level (b) at
the top of the peak.
Work by Conservative Forces
 Springs: Consider a spring that is stretched from
its equilibrium position a distance x. The work
required to cause this stretch is W = 1/2Kx2.
 From our definition of potential energy, we have:
Wc = 1/2Kx2 = Ui – Uf
 Assuming that at x=0 (equlibirum position), U = 0,
we can simplify the formula: Ui = 1/2Kx2 .
 Spring Potential Energy: U = 1/2Kx2
Work by Conservative Forces
Example: Find the potential energy of a spring with
force constant k=680N/m if it is (a) stretched 5cm and
(b) compressed 7cm.
Example: When a force of 12.0N is applied to a
certain spring, it causes a stretch of 2.25cm. What is
the potential energy of this spring when it is
compressed by 3.50cm?
Practice Session
1) Calculate the work done by the gravity as a
2.6kg object is moved from point A to point B,
along paths 1,2, and 3.
Practice Session
2) Calculate the work done by friction as a 2.6kg box
is slid along a floor from point A to point B along
paths 1,2, and 3. Assume the kinetic coefficient of
friction is 0.23. Based on previous picture.
3) A 4.1kg block is attached to a spring with force
constant of 550N/m. Find the work done by the
spring on the block as the block moves from A to B
along paths 1 and 2.
Practice Session
4) Calculate the work done by gravity as a 5.2kg
object is moved from A to B along paths 1 and
2. How does the mass of the object affects the
results?
Practice Session
5) As an Acapulco cliff diver drops to the water
from a height of 40.0meters, his gravitational
potential energy decreases by 25,000J. How
much does the diver weighs?
6) Find the gravitational potential energy of an
80.0kg person standing atop Mt. Everest, at an
altitude of 8848meters. Use the sea level as
the location y = 0.
7) Compressing a spring 0.50cm produces a
potential energy equals 0.0035J. Which
compression is required to generate a energy
equals 0.080J?
Practice Session
8) A force of 4.7N is required to stretch a certain
spring by 1.30cm. (a) How far must the spring
be stretched for its potential energy to be
0.020J? (b) How much stretch is required for
the spring potential energy to be 0.080J?
9) A 0.33kg pendulum bob is attached to a string
1.2meters long. What is the bob’s potential
energy change from points A to B?
Conservation of Mechanical Energy
 Mechanical Energy is the sum of potential and
kinetic energies of an object.
E=U+K
 Mechanical energy is conserved on systems
involving ONLY conservative forces. E is constant
in this case.
 Proving that E is constant for conservative forces
Wtotal = ΔK = Kf – Ki and Wtotal = Wc
We know that: Wc = Ui – Uf . Then, replacing both,
we get: Kf – Ki = Ui – Uf .
Rearranging, we get: Ef = Ei.
Conservation of Mechanical Energy
 In terms of physical systems, conservation of
mechanical energy means that energy can be
converted between potential and kinetic forms,
but the SUM REMAINS THE SAME.
 In systems with conservative forces only, the
mechanical energy E is conserved, that is:
E = U + K = constant
Example
Example
Ei = Ef
Ui + Ki = Uf + Kf
From left side of previous picture, we have:
mgh + 0 = 0 + 1/2mv2
Therefore:
v = (2gh)1/2
Now: Prove for right side of previous picture.
Practice
At the end of a graduation ceremony, graduates fling
their caps into the air. Suppose a 0.12kg cap is
thrown straight upward with an initial speed of
7.85m/s and there is no friction. (a) Use kinematics
to find the speed of the cap when is 1.18m above
the release point. (b) Show that mechanical energy
is the same at the release and at 1.18m
Practice
In the bottom of the ninth inning, a player hits a 0.15kg
baseball over the outfield fence. The ball leaves the
bat with a speed of 36m/s, and a fan in the
bleachers catches it 7.2m above the point where it
was hit. Find (a) Kinetic Energy when the ball is
caught. (b) The speed when it is caught.
Practice
A 55kg skateboarder enters a ramp
horizontally with a speed of 6.5m/s, and
leaves the ramp vertically with a speed of
4.1m/s. Find the height of the ramp.
Practice
A 1.7kg block slides on a horizontal, frictionless
surface until it encounters a spring with a force
constant k=955n/m. The block comes to rest after
compressing the spring a distance of 4.60m. Find
the initial speed of the block.
Practice
Suppose the spring and block are oriented vertically.
Initially, the spring is compressed 4.60cm and the
block is at rest. When the block is released, it
accelerates upward. Find the speed of the block
when the spring returns to the equilibrium position.
Homework
1. If a 30.0J of work is required to stretch a spring
from 4.00cm to 5.00cm., how much work is
necessary to stretch it from 5.00cm to 6.00cm?
2. A spring scale has a spring with a force constant of
250N/M and a weighing pan with a mass of
0.075kg. During the first weighing, the spring is
stretched a distance of 12cm from equilibrium. The
second time is stretched 18cm. (a) How much
greater is the elastic potential energy of the spring
during the second than the first weighing? (b) If the
spring is released after each weighing. What is the
ratio of the pan’s maximum speed between second
and first weighing?
Homework
3. An 80.0N box of clothes is pulled 20.0m up a
30o ramp by a force of 115N that points along
the ramp. If the coefficient of kinetic friction
between the box and the ramp is 0.22,
calculate the change in the boy’s kinetic
energy.
4. A 0.60kg rubber ball has speed of 2.0m/s at
point A and kinetic energy of 7.5J at point B.
Determine the following: (a) Ball’s kinetic
energy at A. (b) Ball’s speed at B. (c) Work
done by ball from A to B.
Homework
5. Starting from rest, a 5.0kg block slides 2.5m
down a rough 30o incline in 2.0s. Determine
the following: (a) The work done by gravity.
(b) The mechanical energy lost due to friction
6. At a park, a swimmer uses a water slide to
enter the main pool. If the swimmer starts at
rest, slides without friction, and falls through a
vertical height of 2.61m, what is her speed at
the bottom of the slide?
7. A player passes a 0.60kg ball. The ball leaves
the player’s hands with a 8.30m/s speed and
slows to 7.10m/s at its highest point. How high
is the ball at the highest point from release?
Homework
8. An 18kg child plays on a slide that drops
through a height of 2.2m. The child starts at
rest. On the way down a non-conservative
work of -373J is done on the child. What is the
child’s speed at the bottom of the slide?
9. A 17,000kg airplane lands with a speed of
82m/s on a 115m long carried deck. Find the
work done by non-conservative forces in
stopping the plane.
10. A 5.0kg rock is dropped and allowed to fall
freely. Find the initial kinetic energy, the final
kinetic energy, and the change in kinetic
energy for (a) first 2 meters of the fall. (b) next
two meters of the fall