Download Ch 7 Conservation of Energy 7-1 The Conservation of Mechanical

Ch 7 Conservation of Energy
7-1 The Conservation of Mechanical Energy
Q: What is mechanical energy?
Q: Is energy conserved? Is mechanical energy conserved?
Q: How many different types of energy can you name?
Q: What does it mean to say that energy is quantized?
We already know that Wtotal = ∑ ∆ Ksystem
Each internal force is either conservative or nonconservative.
Also, the negative of the total work done by all of the conservative forces = the change in
PE of the system
- Wc = ∆Usys
The total work done is the sum of all of the external forces + the work done by all of the
internal nonconservative forces + all internal conservative forces
Wtotal = Wext + Wnc + Wc
Wext + Wnc = Wtotal - Wc
From above, then,
Wext + Wnc = ∆Ksys + ∆Usys = ∆(Ksys + Usys)
DEF: Total Mechanical Energy = Ksys + Usys = Emech
Recombining and substituting gives us
This “conservation of mechanical energy” is the basis for what we mean by a conservative
force
Applications
As we saw in ex 6-12, for a conservative force, like gravity, where the work done is
internal to the system, your end speed, v = (2g (h0 – h))1/2 is the same no matter how you
get down the hill; even considering free-fall! What changes is that you have a longer
distance
taking morr time.
See Example 7-1, p 186
Standing near the edge of the roof of a 12-m-high building, you kick a ball with an initial
speed of vi = 16 m/s at an angle of 600 above the horizontal. Neglecting air resistance, find:
a) How high above the height of the building the ball rises
b) Its speed just before it hits the ground
See Example 7-2, p 186-187
A pendulum consists of a bob of mass m attached to a string of length L. The bob is pulled
aside so that the string makes an angle θ0 with the vertical, and is released from rest.
Assuming no air resistance, as it passes through the bottom of the arc, find the expressions
for
a) the speed
b) the tension
NOTE: The tension at the bottom has to be greater than the weight because the bob
accelerates upward. Force causes acceleration. If there is no “extra” force, the bob won’t
be able to accelerate back upward. Make sure you read the remarks below before you try
the problem!
TRY IT NOW!! Do examples 7-3 – 7-6. Make sure you are reading the commentary
below each problem, and looking at the plausibility checks!
7-2 The Conservation of Energy
Q: If a block is pushed across a table, and encounters friction, what happens to the
a) motion of the block
b) energy of the block (therefore mechanical energy)
c) energy of the system
A: The block slows down due to friction, a responsive force. The mechanical energy of
the block is lost to thermal energy. The energy of the system is conserved!
Q: If mechanical energy is lost as thermal energy is gained, is thermal energy conserved
individually?
A: Nope. Neither is conserved individually. There are lots of potential (excuse the pun!
☺) energy transfers going on. Some could be converted to light, sound, chemical potential,
et cetera.
To really consider conservation of energy, we have to what is going on with respect to all
forms of energy contributing to a system.
The Work-Energy Theorem
If you want to transfer energy either in or out of a defined system, you can do work on the
system from the outside. From chemistry, you might remember that whenever energy is
transferred, it is accompanied either by a gain or loss of heat.
See example 7-7, p 193
A ball of modeling clay with a mass m is released from rest from a height h and falls to a
perfectly rigid floor. Discuss the application of the law of conservation of energy to
a) the system consisting of the ball alone
b) the system consisting of the earth, the floor, and the ball
DEMO: Happy-Sad Balls (be nice!)
Which ball experiences the greatest ∆E? Why?
Problems Involving Kinetic Friction
As we stated before, if kinetic friction is involved, the work-energy theorem gives us
0 = ∆Emech + ∆Etherm
Since the ME lost = KE of the block, ∆Emech = - ½ mv2
What really causes a loss in ME has to be a force (friction); a response force that opposes
the motion. So, from N2, we get - f = ma
IF you multiply both sides of the equation by ∆s, the displacement, and you use the
formula 2a∆s = vf2 – vi2
If vf = 0, then what we really have is
f ∆s = - ∆Emech = ∆Etherm = the energy dissipated by friction
Substituting back into the work-energy theorem with Echem = Eother = 0 gives us
YOU TRY IT! Do ex’s 7-8 – 7-11. Make sure to read the remarks underneath each
example!!
Systems with Chemical Energy
Sometimes the conversion from chemical to mechanical and / or thermal energy is done
internally, so no work is being done on the system by an outside force.
See Example 7-12, p 199
Suppose you have a mass m and you walk up a flight of stairs to a height h. Discuss the
application of energy conservation to a system consisting of you alone.
Write the work-energy equation for “you alone” Wext = ∆Esys = ∆Emech + ∆Etherm + ∆Echem.
There are two forces acting on the system: gravity and the force of the stairs on your feet
So, Wext = - mgh. Why is mgh negative? Why not include the force of the stairs on your
feet?
If the system consists of you alone, because your configuration does not change, any
change in ME is due to a change in your KE, which is the same initially and finally.
So… ∆Emech = 0
If you substitute back, -mgh = ∆Etherm + ∆Echem.
If there were no ∆Etherm, then your chemical energy would decrease by mgh. Because of
the inefficiency of our bodies, this increase in thermal energy would be > mgh. The
decrease in stored chemical energy = mgh + any thermal energy, which is eventually
transferred from your body to your surroundings via heat. Kinda gives new meaning to
being a “hottie!” ☺
YOU TRY IT! See example 7-13, p 200. ☺