lecture on theoretical mechanics - 4
DESCRIPTION
Lecture Note Series prepared for 1st year physics (hons. and engineering) students. Comments and criticisms are welcome.TRANSCRIPT
PBC Lecture Notes Series in Physics: Classical Mechanics -I Lecture 4
Prepared by Dr. Abhijit Kar Gupta, e-mail: [email protected]
1
Particle Dynamics: Conservative Force, work done, Power, Impulse
Work done by a force:
Work done to move a particle from point 1 to point 2
∫
Similarly, the work done to move the particle from point 2 back to point 1 is
∫
So the total work done on the closed path from 1 to 2 and back to 1,
∫
∫
= ∮
Closed Curve:
Conservative Force:
If the work done by the force over a closed path is zero,
∮ , (1)
the force is called conservative force.
[Note: the integral ∮ is called cyclic integral.]
Now we apply Stoke’s law for a closed curve:
∮ ∫( ) , (2)
where = is the vector surface area, meaning be the surface area enclosed by the closed curve and the direction is upward if the curve is traversed anticlockwise and it
PBC Lecture Notes Series in Physics: Classical Mechanics -I Lecture 4
Prepared by Dr. Abhijit Kar Gupta, e-mail: [email protected]
2
is downward if clockwise according to convention.
Therefore, for a conservative force, we can write
∫( ) . (3)
Now since the surface element is arbitrary, we must have
= 0 (4)
The above equation (4) is a necessary and sufficient condition that has to be satisfied if
the fore is conservative. It is true for any vector.
Equation (4) implies that we can write the force in the following way:
,
where is a scalar potential and the negative sign is by choice. We call as the
potential energy.
For work done to move a particle from a point 1 to point 2 by a conservative force:
∫
∫
∫
(A)
[Note, (
) ( )
]
We can also calculate the work done in terms of Newton’s 2nd law of motion,
Consider, =
( )
(
)
[Considering constant force, ]
(
)
∫
∫ (
)
(B)
Now, from (A) and (B),
PBC Lecture Notes Series in Physics: Classical Mechanics -I Lecture 4
Prepared by Dr. Abhijit Kar Gupta, e-mail: [email protected]
3
So, we can say Total energy, = constant. [Conservation of energy]
Note the following:
(1) The potential is a function of position coordinates only.
( ) ( )
(2) A system can not be conservative if friction or other kinds of dissipative forces
are present. In that case, is always positive, there is always some work
done to overcome friction!
(3) For dissipative or non-conservative forces, we can NOT write .
Power:
Power is rate of doing work.
Consider,
Thus,
( )
(
)
Rate of change of Kinetic energy.
Impulse:
If a force is applied, momentum of the particle changes according to Newton’s 2nd law of
motion,
.
If the force is applied during a certain time, we can write,
∫
∫
Change in momentum, which is called the
impulse of the force .
H.W. Problem:
#1 A particle of mass moves in the XY-plane such that its position vector is given by,
, where = const.
(i) Find the force on the particle
(ii) Is the force conservative?
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