Download Conservative Forces and Potential Energy

Conservative Forces and Potential Energy
Conservative Forces
Definition: A force is called conservative if the work it does on an object depends only on the
initial and final positions of the object and is independent of the path taken between those
positions.
Gravity is one such conservative force. Near Earth’s surface, the work done by gravity on an
object of mass m depends only on the change in the object’s height h.
In the case of conservative forces, we assign a number--the potential energy, U--to each
configuration of the system. The zero level of the potential energy is arbitrary; it can be assigned
to any position. If the place with zero potential energy is chosen, the potential energy at any point
A is the work done by the force when the body moves from the point A to the point 0 with zero
potential energy. So, U(A) = WA0.
The work is additive. If the body moves from A to B and then to point 0, then
WA0 = WAB + WB 0 ,
U ( A) = WAB + U ( B ) or WAB = U ( A) − U ( B ) .
Examples of Potential Energy
The Gravitational Potential Energy
Gravity is an example of a conservative force. The work done on an object of mass m when it is
raised height h at constant velocity by an external force is mgh. Ug = mgh is called the
gravitational potential energy of the object.
The only physically significant quantity is the change in potential energy, and not the absolute
value of the potential energy. For this reason, in problem solving, we usually choose a convenient
point (table top level, ground level, sea level, etc.) and set the potential energy at this level to be
zero. All heights are measured from this level. Then, if the object is below the “zero” level, its
potential energy is negative. It is important to stress again that only the change in potential
energy is physically significant.
Gravitational Potential Energy: Ug = mgy
Change in Gravitational Potential Energy: ΔU g = U g - U go = mgh - mgho
In the more general case of universal gravitation, the work done by gravity depends only on the
distance r between the centers of the two attracting masses m and M, so that we have
K
GMm
GMm
FG = − 2 rˆ ⇔ U G (r ) = −
.
r
r
Note that this formula sets U G = 0 when the masses are infinitely far apart ( r → ∞ ).
The Elastic Potential Energy
Springs, rubber bands, and other “springy” objects also exert conservative forces. At relatively
small elongations, the force exerted by a spring is proportional to the amount x by which the
spring has been stretched,
Fspr = −k x ,
(The “-“ sign indicates that the force is directed in the opposite direction to the stretch.)
k is the constant of proportionality and is called the spring’s spring constant. This proportionality
between the force F and the amount of stretching (or compression) x is known as Hooke’s Law.
The work done by a spring with a given spring constant depends only on x, so an associated
potential energy is
U spr ( x ) = 12 k x 2 .
Note that “x = 0” is at the equilibrium (unstretched) position of the spring.
If the elongation goes from zero to x, the work done by the applied force FA
x
x
x
x
0
0
0
0
W = ∫ FA dx = ∫ (− F )dx = ∫ k xdx = 12 k x 2 = 12 k x 2
Suppose the spring is stretched a distance xi initially. Then the work we have to stretch it to a
greater elongation xf is
xf
xf
xi
xi
W = ∫ Fdx = ∫ k xdx = 12 k x f 2 − 12 k xi 2
Nonconservative forces
Unlike conservative forces, the work done by nonconservative forces depends on the path the
object moves and not just on the initial and final points. Friction is an example of a
nonconservative force. A nonconservative force is often called a dissipative force.
Work done is the dot product of the force and the displacement.
Dot product:
G G
A ⋅ B = AB cos θ = Ax Bx + Ay By + Az Bz
(h, h)
G
G
If F or d is not a constant, then
G G
Δwi = Fi ⋅ Δsi
h
G G
W = ∑ Fi ⋅ Δ si
mg
i
(0, 0)
C
G
where Δsi is extremely small. The work done by gravity on the ball’s falling down is given by
G G
0
0
W = ∫ F ⋅ d s = ∫ -Fˆj ⋅ (dx iˆ + dy ˆj ) = − F ∫ dy = −mg ∫ dy = mgh
C
C
h
h
A force is conservative if it depends only on the position of the body (neither on time nor
on the velocity of the body). The work done by the force on the body moving between any
two points depends only on the initial and final positions, and so is independent of the
particular path taken.
The work done by a conservative force along a closed path is zero.
K
K
Every conservative force F has associated with it a potential energy function, U F (r ) , such that
the work done by the force is just the negative of the difference of this potential energy between
the final and the initial positions:
K
K
WF = −ΔU F = −[U F (rf ) − U F (ri )] .
Note that only changes in potential energy have physical significance. The reference level or
“zero” of the potential energy scale can always be chosen arbitrarily.
y
x