Download Nonconservative Forces

Conservative and
Nonconservative Forces
A Force is “Conservative” if:
• The work it does on an object is available for kinetic energy.
These forces store energy
• The work done by conservative forces is the negative of the
potential energy change.
W = -ΔU
• Force of gravity and Spring force are
examples of conservative forces
• Wc = FDx (if force is constant)
• Wc =  Fdx = - dU = -DU (if force varies)
•  Fdx = - dU
• Fdx = -dU
• F = -dU/dx
Review Springs
Springs have a restoring force
 always directed opposite to the stretch or
compression of the spring
 Fs=kx
Springs can oscillate

period of a spring (Ts) = 2π
𝑚
𝑘
Springs
Elastic Potential Energy
Energy stored in a compressed or extended
spring.
Us = ½ kx2
k is the spring constant.
x is the distance the spring is compressed or
stretched.
A special spring is constructed in which the restoring force is in the
opposite direction to the displacement, but is proportional to the cube
of the displacement; i.e., F = -kx3
This spring is placed on a horizontal frictionless surface. One end of the
spring is fixed, and the other end is fastened to a mass M. The mass is
moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
a. The potential energy in the spring at the instant the mass is released
b. The maximum speed of the mass
c. The displacement of the mass at the point where the potential
energy of the spring and the kinetic energy of the mass are equal
A 2 kg block is dropped from a height of 0.45 m above an uncompressed
spring, as shown above. The spring has an elastic constant of 200 n/m and
negligible mass. The block strikes the end of the spring and sticks to it.
(a) Determine the speed of the block at the instant it hits the end of the
spring
(b) Determine the period of the simple harmonic motion that ensues
(c) Determine the distance that the spring is compressed at the instant the speed of the block
is maximum
(d) Determine the maximum compression of the spring
(e) Determine the amplitude of the simple harmonic motion
Block A of mass 4.0 kg is on a horizontal, frictionless tabletop and is placed
against a spring of negligible mass and spring constant 650 N/m. The other
end of the spring is attached to a wall. The block is pushed toward the wall
until the spring has been compressed a distance x, as shown. The block is
released and follows the trajectory shown, falling 0.80 m vertically and
striking a target on the floor that is a horizontal distance of 1.2 m from the
edge of the table.
(a) Calculate the time elapsed from the instant block A leaves the table to
the instant it strikes the floor.
(b) Calculate the speed of the block as it leaves the table
(c) Calculate the distance x the spring was compressed
A Force is “Non-Conservative” if:
• The work done by these forces causes energy to be
lost to the system. They “steal” energy from the
system
• Examples of this type of force is friction and air
resistance
• The energy they remove from the system is no longer
available for kinetic energy
Conservation of energy and
conservative and nonconservative
forces
When conservative forces act on a system Ei = Ef
When nonconservative forces act on a system
Our Energy formula becomes
Ei – Wnc = Ef
**Work done is negative. It removes energy from
the system. It is in the form of heat lost to the
system.
Example: One of the tallest and fastest roller coasters in the world is the Steel
Dragon in Mie, Japan. The ride includes a vertical drop of 93.5 m. The coaster has
a speed of 3.0 m/s at the top of the drop. The speed at the bottom is 41.0 m/s.
Find the work done by the nonconservative forces on a 55.0 kg rider during the
descent.
Example: A 60 kg sledder starts from rest and slides down a 20o incline 100 m
long. (a) If the coefficient of friction is 0.090, what is the sledder’s speed at the
base of the incline.
(b) If the snow is level at the foot of the incline and has the same coefficient
of friction, how far will the sledder travel along the level snow?
Power
 The rate at which energy is transferred or transformed
is called the power P.
Highly trained athletes have a tremendous
power output.
 The SI unit of power is the
watt, which is defined as:
1 watt = 1 W = 1 J/s
If the particle moves at velocity while acted on by force, the
power delivered to the particle is:
© 2013 Pearson Education, Inc.
Slide 11-98
Example 11.13 Choosing a Motor
58,800 W
© 2013 Pearson Education, Inc.
Slide 11-99
Example 11.14 Power Output of a Motor
882 W
© 2013 Pearson Education, Inc.
Slide 11-105
QuickCheck 11.11
Four students run up the stairs in the time shown.
Which student has the largest power output?
© 2013 Pearson Education, Inc.
Slide 11-103
QuickCheck 11.11
Four students run up the stairs in the time shown.
Which student has the largest power output?
© 2013 Pearson Education, Inc.
Slide 11-104