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 W c = F  x (if force is constant) W c =  Fdx = -  dU = -  U (if force varies)  Fdx = -  dU Fdx = -dU F = -dU/dx

Review Springs

Springs Elastic Potential Energy Energy stored in a compressed or extended spring. U s = ½ kx 2 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 = -kx 3 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 E i = E f When nonconservative forces act on a system Our Energy formula becomes E i – W nc = E f **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 20 o 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?

© 2013 Pearson Education, Inc. Power  The rate at which energy is transferred or transformed is called the power P.  The SI unit of power is the watt, which is defined as: 1 watt = 1 W = 1 J/s Slide Highly trained athletes have a tremendous power output. If the particle moves at velocity while acted on by force, the power delivered to the particle is:

© 2013 Pearson Education, Inc. Example Choosing a Motor Slide ,800 W

© 2013 Pearson Education, Inc. Example Power Output of a Motor Slide W

© 2013 Pearson Education, Inc. Four students run up the stairs in the time shown. Which student has the largest power output? QuickCheck Slide

© 2013 Pearson Education, Inc. Four students run up the stairs in the time shown. Which student has the largest power output? QuickCheck Slide