The most important thing about energy is that it is *conserved*, which means that
energy cannot be created or destroyed, it can only be converted into other kinds of
energy. The total amount of energy remains constant. How many different kinds of energy
are there? We know that mechanical energy consists of kinetic and potential energy, but
energy can also appear in the form of heat, light, electric fields, magnetic fields, or
even nuclear energy.
When we are talking about mechanical systems we are only concerned with kinetic and
potential energy. Friction converts kinetic energy into heat, and so it represents a net
loss of mechanical energy. Once energy is converted into heat it is for all practical
purposes lost forever, because the heat will just drift off into the environment. This is
what the brakes on a car do. In order to stop the car, the friction produced by the brake
pads must generate a quantity of heat equal to the kinetic energy of the car, and as a
result the brakes get very, very hot.
The law of conservation of energy can be stated in three (equivalent) ways:
- Energy cannot be created or destroyed, only changed into different kinds of energy.
- The total energy of an isolated system is constant. (An
*isolated system* has no
energy or mass entering or leaving it.)
- The Energy of an non-isolated system changes only by the amount added or removed. If the
only energy involved is mechanical, this can be stated as
* W* =
D*K* + D*PE*,
because the only way to change mechanical energy is to do work on the system. Doing work
will either change the kinetic energy or the potential energy (or both) of the system.
Example: Rock falling off
cliff of height *h* (ignoring air friction)
Initially, the rock has a *PE* of *mgh* (the work it would require to raise
it that high from ground level). At first this is also the total energy of the system,
because potential energy is the only kind of energy the system has at that time.
As it falls it loses *PE* and gains *KE*, but always the total energy remains
the same. Since it started with a total energy of *mgh*, this will always be the
total energy. On the way down, its energy will be a mixture of *PE* and *KE*,
but will still add up to that original value of *mgh*:
This means that at a given height *y* we can calculate the velocity. At a height *y*
its *PE* is *mgy* and its *KE* is *mv*2, but their total is still the original amount of energy *mgh*, so
*mgh* = *mgy* + *mv*^{2},
from which it follows that
.
Notice that we can find *v* at any point without knowing any details of the path!
At the very bottom, just before it hits, all the *PE* will be converted to *KE*,
so
*mgh* = *mv*^{2}
or
When the rock hits the bottom, all of the *KE* will be lost. Some will be spent
breaking the rock and ground into pieces, some will go off as sound waves, and some will
go into doing work on the ground by compressing it. All this moving matter will experience
friction, which converts KE into heat. Wherever all the energy goes, it must still add up
to the original amount *mgh*.
Example: Block sliding down ramp with or
without friction
Suppose a block is released form rest at the top of the ramp and allowed to slide down.
How fast will it be going when it reaches the bottom? You could solve this problem by
using the equations for constant acceleration, but it is much simpler to solve using
conservation of energy.
### Without friction
The initial energy, when the block is at rest at the top of the ramp, is purely
gravitational potential energy. If we measure the height *h* from the bottom of the
ramp, the initial energy is
*E*^{i} = *mgh*
*
*
When the block reaches the bottom of the ramp. All of this potential energy will have
been converted into kinetic energy, which is given by
*E*_{f} = *mv*^{2}
But because energy is conserved, and we are assuming that we are not losing any
mechanical energy through friction, the initial and final energies must be equal. Thus
*E*_{i} = *E*_{f
}*mgh *= *mv*^{2}
which can be solved to give the final velocity as
### With friction
If we allow for friction while the block is sliding down the ramp, we have to take into
account the amount of mechanical energy that will be lost due to friction. The work done
against friction will be
*W*_{f} = *F*_{k}*d*,
where *F*_{k} is the force of kinetic friction,
*F*_{k} = *Nm*_{k},
And *d* is the distance traveled along the ramp. The normal force is
*N* = *mg* cos *q*
*
* *
*And the distance along the ramp, which is the hypotenuse of the triangle, is
*d* = *h* sin *q*
*
* *
*Putting all this together, we can get an expression for the work done against
friction:
*W*f = *F*k*d*
= *m*_{k}(*mg* cos
*q*)(*h*/sin *q*)
*W*_{f }= *mgm*_{k}*h*
cot *q*
*
*
Now we can return to conservation of energy. The initial energy of the block is still
*E*_{i} = *mgh*,
and the final energy is still
*E*_{f }= *mv*^{2},
But the initial and final energies are no longer equal because along the way we lost an
amount *W*_{f} to kinetic friction. Thus
*E*_{f} = *E*_{i} – *W*_{k}
or
*mv*^{2} = *mgh* – *mgm*_{k}*h*
cot *q *
*
*
which gives a final velocity of
Example: Height to do loop-the-loop
Suppose you were designing a roller coaster ride that featured a loop, as illustrated
above. For simplicity we will assume that it is a frictionless roller coaster, even though
that is not very realistic. You will need to figure out how high the car needs to start so
that it has enough velocity to make it around the loop. The key observation is that the
centrifugal force must be just enough to keep the car on the track when it is at the top
of the loop. The centrifugal force is given by. From the car’s reference frame when
it is at the top of the loop, this is an upward force which must counter the downward
force of the car’s weight *mg*. Therefore we set *mv*^{2}/*r* = *mg*, or
This is the minimun velocity that the car must have at the top of the loop. Now that we
know the required velocity at this point, we can use conservation of energy. The initial
energy, when the car starts from rest at the very top of the track, is all potential:
*E*_{i}
= *mgh*
*
*
The energy at the top of the loop is partly kinetic and partly potential, but now we
know the velocity there, and we know that the height is twice the radius, so we can write
an expression for the total energy:
*E*_{f} = *mg*(2*r*) + *mv*^{2}
*E*_{f} = 2*mgr* +
*mgr*
*
**E*_{f} =
Now, because we are ignoring friction, we can set the initial energy equal to the final
energy and solve for *h*:
or
So, you need to start the track at a height that is 2.5 times the radius of the loop.
Of course a real roller coaster will lose some energy to friction, and so you would have
to start the track even higher |