Download Gravitational Potential Energy Consider a ball falling from a height of

Gravitational Potential Energy
Consider a ball falling from a height of y0=h to the floor at height
y=0. A net force of gravity has been acting on the ball as it drops.
So the total work done on the ball is:
Wtot = Fy ∆y = (-mg) (y-y0) = (-mg)(-h) =mgh
but
Wtot = ∆Ek = ½ mv2 – 0
so
mgh = ½ mv2
and the work done by gravity equals the kinetic energy gained by
the ball. Call the gravitational potential energy Ep and define it to
be:
Ep = mgy
Realize that only changes in potential energy can be measured.
Above is valid so long as g remains constant over the range of
distance represented by y. (Really ∆y with y the distance dropped)
From your text page 6-15:
A few comments should be made here about the selection of
the y=0 position. Gravitational potential energy is a relative
quantity, always measured with respect to some arbitrary position
where y=0. In most cases, it does not matter where y=0 is chosen,
but it is usually easiest to choose y=0 at the lowest possible
position in the diagram (for example at floor level). This ensures
that all y-values and all gravitational potential energies will be
positive (or zero) and will be easy to work with. We shall see that
what is usually important is the change in the gravitational
potential energy, which does not depend on where we have chosen
y=0. In the defining equation for Ep, mg is positive (because it is
the magnitude of a vector not the vector itself) and the potential
energy increases with elevation. Remember, energy is not a force
and is not a vector.
Conservation of Energy
Energy can be converted into different forms but it can be neither
created nor destroyed.
When you burn a log to create heat, you are changing chemical
energy into heat energy. The only seeming exception to the above
law relates to nuclear energy whereby some mass is converted into
energy. However, at a fundamental level, mass and energy are two
manifestations of the same thing (it has no name!) and this thing is
conserved. It is convenient and correct to call this thing energy.
Note that “conservation of energy” differs from “energy
conservation,” which refers to the careful use of energy resources.
Conservation of energy is a law of nature; energy conservation is
something that we should all practise. “Energy conservation” is a
bit of a misnomer as nature does that for us. What we are
conserving is energy in a particularly useful and concentrated
form.
The law of conservation of energy can be used to solve problems
that kinematics alone could not solve.
Kinetic plus potential energies at the start of the problem must be
the same at the end of the problem. Or
Total energy (start) = Total energy (end)
KE1 + PE1 = KE 2 + PE2
½ mv12 + mgy1 = ½ mv22 + mgy2
Consider sample problem 6-5.
The units for energy can be found from either the kinetic or
potential energy formulas.
 m2 
m

kg  2  = kg  2 m = Newton m ≡ Joule
s 
s 
Work done by Friction
The work done by friction is the kinetic friction force multiplied by
the distance moved. This work always produces heat or thermal
energy.
Wf = -Fk ∆r
Eth = Fk ∆r
The friction force is always opposite to the velocity and will also
be opposite to the displacement. Hence the work is done by the
body not on the body and the work is negative. The body loses
energy because of friction. The lost energy goes into heating the
surroundings.
Efficiency of Energy Conversions
useful energy out
efficiency =
total energy in
Power
Power is the rate of doing work. It is also the rate at which energy
is being consumed. It is simply
P = E/t
The unit for power is watt which is a joule per second. Kilowatt,
or kW, is a common power unit for electricity. Because your
house is metred to measure kilowatts, your rate of energy
consumption, the amount of energy you use is given by the
kilowatts multiplied by the time. Rather than use joules which is a
rather small unit of energy, the power company uses kilowatthours. There are 3600 seconds in an hour and 1000 watts in a
kilowatt so there are 3.6 million joules in a kilowatt-hour.
Chapter 7
Momentum
Momentum is defined as the product of the object’s mass and
velocity. It is a vector quantity, one that is useful for problem
solving. The symbol for momentum is p.
p = mv
Of course this means that
px = mvx
and
py = mv y
Even as work is defined as the force times a distance, so impulse is
defined as force times time. An impulse applied to an object
changes its momentum. We write:
F∆t = ∆p
Consider motion in the x-direction:
Fx∆t = mv2x – mv1x
max∆t = mv2x – mv1x
or
ax∆t = v2x – v1x
You recognize this as vx = v0x + axt
Note that the dimensions of Force × time is Newton-second. The
dimensions of momentum are mass × velocity or kg m/s. But a
Newton-second is a:
m
m
kg 2 s = kg
s
s
and we get the same either way as we must.
Forces involved in collisions are usually very large. Consider a
baseball bat striking a ball, as in sample problem 7-1.