Download WORK AND ENERGY

Introductory Physics I, phy231, Spring 1995, page 7
WORK AND ENERGY
WORK
Work is the product of the force and the displacement. Only the force component parallel to the
displacement counts: W = Fxcos, where is the angle between the direction of the force and the
displacement. This formula is valid for constant forces.
An example of non-constant forces is the spring: F = kx. k is called the spring constant. The work
for a spring is: W = 12 kx2
ENERGY
Energy is the ability to do work.
Spring potential energy: W = 21 kx2 = PE
(Gravitational) Potential energy: W = ,MmG=R = PE
S pring
The convention is that the potential energy is zero when the distance reaches innity.
Special case: Near the surface of the earth:
Potential energy: W = Fh = mgh = PE
Potential energy is a relative quantity. Always dene your reference point. The convention is to
dene the lowest point in the problem as height zero, so that this point has a PE of zero.
Kinetic Energy: W = 12 mv2 = KE
Again, this is assuming that the initial velocity was zero. If the initial velocity is not zero, the change
of KE is given by: KE = 12 mv2 , 12 mv2
f
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CONSERVATIVE AND NONCONSERVATIVE FORCES
If the work done by a force depends only on the initial and nal position and not on the path taken,
then the force is conservative.
The sum of kinetic and potential energy is constant for conservative forces and is called total mechanical energy:
Total mechanicalenergy = PE + KE , or
PE
+ KE
= PE + KE
initial
initial
f inal
f inal
If the work done by a force depends on the path taken, then the force is nonconservative. Friction is
a nonconservative force.
For friction: The initial mechanical energy is the sum of the nal mechanical energy and the work
done by friction:
PE
+ KE
= PE + KE + W
initial
initial
f inal
f inal
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f riction
Introductory Physics I, phy231, Spring 1995, page 8
POWER, ENERGY AND MOMENTUM CONSERVATION
POWER
Power is the capacity to do work quickly: P = W=t
If the work is done at a constant rate: P = W=t
The unit of power is Watt: 1W = 1J=s
If the work is done by a constant force: P = W=t = Fd=t = Fv
LINEAR MOMENTUM
The linear momentum of an object is the product of the mass and the velocity: ,!
p = m,!
v.
Momentum is a vector!
Newton's second law yields: ,!
F = ,!
p =,!
t.
The impuls is the change in momentum: F t = ,!
p.
When the net force on a system is zero, the total momentum is constant!
COLLISIONS
Collisions in one dimension: The momentum is conserved: p
=p
Elastic collision: In addition to the momentum, the kinetic energy is conserved: KE
= KE
General equations to be solved (for elastic collisions): 1 m1v1 +1 m2v2 = 1m1v1 + m12v2
bef ore
af ter
bef ore
0
0
m v + 2 m v = 2 m v + 2 m2v22
Inelastic collision: Kinetic energy is not conserved: KE
> KE .
In a perfectly inelastic collision, the objects stick together: m1v1 + m2v2 = (m1 + m2)v
Collisions in two dimension: The momentum in each direction (x and y) is conserved:
p
=p
and p
=p
Changing mass: F = (mv)=t. For a rocket with exhaust speed v : F = v m=t
Hints: Look for simplifying initial and/or nal conditions:
{ Perfectly inelastic or objects stick together: v1 = v2 = v
{ Body '2' initially at rest: v2 = 0
{ Body '1' comes to rest after the collision: v1 = 0
{ Head-on collision with both bodies having the same speed: v1 = ,v2
2
1 1
2
2
2 2
bef ore
2
1 1
0
0
af ter
0
x;bef ore
af ter
x;af ter
y;bef ore
y;af ter
r
0
0
R
r
0
0
MOTION IN A GRAVITATIONAL POTENTIAL
The initial vertical velocity v0 necessary to reach a height hqcan be calculated by energy conservation:
E = PE + KE = PE , (KE = 0) ! v0 = 2GM (1=R , 1=(R + h))
The escape velocity is dened as the minimum velocity
q to leave the gravitational eld. This condition
is fullled when the total energy is zero ! v = 2GM =R
T otal
ini
ini
f inal
f inal
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