Problems Involving Mechanical Energy and Linear
Momentum
There are groups of problems that on first examination look like they
could be solved using only the work-energy principle:
+ Ekinetic ++ E potential ++ Espring = Wmechanical
What are the limitations stated in
class (or in the notes) for applying
However, upon closer examination we discover that additional laws are
required. Why is this the case?
If we re-examine the work-energy principle, we see that the left-hand
side involves only mechanical energy—kinetic energy, gravitational
potential energy, and spring (elastic) energy. Furthermore, the righthand side involves mechanical work—a mechanism to transport energy.
In the framework of the accounting concept, we might say from this
equation “Mechanical energy is conserved for a closed system.”
Unfortunately, mechanical energy is conserved only under very restricted
conditions. The work-energy principle is not a fundamental law of physics
like conservation of mass or momentum. In general, mechanical energy is
not conserved and in most real situations it is destroyed (or consumed).
As we will demonstrate shortly, mechanical energy is a subset of a
larger more fundamental property called energy (or total energy). In the
context of conservation of energy, we will demonstrate that during many
mechanical processes a destruction of mechanical energy is balanced by a
creation of internal energy within the system. In almost all cases, these
problems involve impacts between two different bodies.
For example, consider the problem of two masses colliding on a
frictionless table as shown on the figure at right. Further, assume that
the blocks stick together (perfectly inelastic collision). Writing the
conservation of linear momentum for a closed system and over finite-time
we have in the x-direction :
G
G
dPx
→+
= ∑ Fx
dt
+ Px = 0
the Work-Energy Principle
correctly?
VB
VA
B
A
x
( mA + mB ) V final − [ mAVA + mB (−VB )] = 0
V final =
mAVA − m BVB
mA + mB
A negative number for Vfinal would indicate that the combined masses
would move to the left.
If we also wrote the work energy principle shown above for this
collision, we would have the following result:
C:\Documents and Settings\richards\My Documents\ES Courses\ES201\Best of the
Rest\Chapter Stuff\Mechanical Energy\LM_&_MEB_Problems_02.doc
CA_Block_01.dot
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+ Ekinetic + + E potential + + Espring = Wmechanical
( mA + mB )
⎡ V2
V2⎤
− ⎢ mA A + mB B ⎥ = 0
2
2
2 ⎦
⎣
2
V final
V final =
mAVA2 + mBVB2
mA + mB
This clearly contradicts the result from applying the conservation of
linear momentum. What gives? Which is correct? THE LINEAR
MOMENTUM RESULT IS CORRECT!
So what happened? An inelastic collision process involves internal
friction as the two bodies deform and stick together; thus "mechanical
energy is not conserved" as required by the work energy principle. From a
linear momentum standpoint the equation fails because there is internal
friction within the system. In this specific case, the internal friction is
inside the blocks as they deform and stick together.
In fact if you just apply the full conservation of energy equation to the
process assuming no work, finite-time, and adiabatic conditions you
discover that:
+E = Q + W
+U ++ Ekinetic + + E potential + + Espring = Q + W
+U ++ Ekinetic = 0
+U = −+ Ekinetic
m
⎡m
⎤
2
2
− VA2 ) + B (V final
− VB2 ) ⎥
+U = − ⎢ A (V final
2
⎣ 2
⎦
2
⎡
V ⎤ ⎡ V2
V2⎤
+U = ⎢( mA + mB ) final ⎥ − ⎢ mA A + mB B ⎥
2 ⎦⎥ ⎣
2
2 ⎦
⎣⎢
Thus you see that the change in velocities correctly predicted by
conservation of linear momentum results in an increase in the internal
energy of the system for this process.
If you are faced with a problem that appears to satisfy the constraints
of the Work-Energy Principle except for an impact or collision between
systems you can frequently solve the problem by dividing the process into
three sub-processes:
1. Pre-impact: apply work-energy principle (or mechanical energy
balance).
2. Impact: apply conservation of linear momentum to find final
velocities after impact assuming that only impulsive force is
significant.
3. Post-impact: apply work-energy principle (or mechanical energy
balance).
The key idea is to recognize the internal energy changes that occur
during an impact process.
We will revisit this discussion
after developing the complete
conservation of energy
equation.