Chapter 9Potential Energy & Conservation of Energy

DEHS 2011-12Physics 1

Conservative vs. Nonconservative Forces

• A conservative force is a force whose work is stored as potential energy and can later be turned back into kinetic energy

• A nonconservative force is a force whose work cannot be recovered at kinetic energy and is converted into another form of energy (usually light, sound, or especially heat)

Examples of Conservative and Nonconservative Forces

• Examples of conservative forces include:– Gravity– Springs– Electric and Magnetic forces (not seen in this course)

• Examples of nonconservative forces include:– Friction from surfaces– Internal friction (seen mostly during collisions)– Tension in a rope, cable, etc.– Forces exerted by a motor– Forces exerted by muscles

Definitions of a Conservative ForceDefinition 1: A conservative force is a force that

does zero total work on any closed path

Wtotal = WAB + WBC + WCD + WDA

Wtotal = 0 + (-mgh) + 0 + (mgh)

Wtotal = 0 Means gravity is a conservative force

Wtotal = WAB + WBC + WCD + WDA

Wtotal = (-μmgd) + (-μmgd) + (-μmgd) + (-μmgd) = -4μmgd

Wtotal ≠ 0 means friction is a nonconservative force

Definitions of a Conservative Force

Definition 2: The work done by a force in going from an arbitrary point A to an arbitrary point B is independent of the path from A to B, the force is conservative

Example 9-1A 4.57 kg box is moved with a constant speed along two paths as shown

below. Calculate the work done by gravity on each of these paths.

Example 9-2The same box as in example 9-1 (mass = 4.57 kg) is pushed across a floor

from A to B along paths 1 and 2 as shown below. If the coefficient of kinetic friction between the box and the surface is μk = 0.63, how much work is done by friction along each path?

Potential Energy• When a conservative force does an amount of

work there will be corresponding change in the amount of potential energy, U, as defined as:

• Scalar quantity with units of Joules (J)• IMPORTANT: the definition determines only a

difference of potential energy so we are free to choose the place where U = 0

€

Wc = −ΔU

Gravitational Potential EnergyFor an object of mass m that is dropped a distance

of y, the work done by gravity is:

And since gravity is a conservative force, we apply our definition of potential energy:

So we can say:€

Wg = mgy

€

Wg =Wc

€

=−ΔU

€

=U i −Uf

€

=mgy

€

U i = mgy +Uf

Gravitational Potential EnergyYou can set U = 0 anywhere, but if you set Uf = 0

then Ui = Ug (potential energy from gravity) then the potential energy at a point y above U = 0 is

• IMPORTANT: this assumes that g is constant, which it is not! (it decreases with height) But when you are close to the surface of the Earth (h << RE), g can be approximated as being a constant

€

Ug = mgy

Example 9-3An 82-kg mountain climber is in the final stage of the ascent of 4301-m

Pikes Peak. What is the change in the gravitational potential energy as the climber gains the last 100.0 m of altitude? Let U = 0 be (a) at sea level or (b) at the top of the peak.

Example 9-4A Snickers© Bar has a calorie content of 271 Calories. A 50-kg (~110 lbs)

girl eats a snickers bar, but wishes to keep her girlish figure. If her body converts all of the Snickers bar to potential energy, how many stairs does she have to climb to work off the Snickers? Assume that each stair is 20 cm tall.

Potential Energy & SpringsLast chapter we calculated the work done to stretch

or compress a spring is

And since the work that the spring will do to bring the spring back its equilibrium position, will be the same, the force of a spring is conservative, so:

€

Ws =12 kx

2

€

Wc = −ΔU

€

=U i −Uf

€

=12 kx

2

Spring Potential EnergyYou can set U = 0 anywhere, but if you set Uf = 0 (at

the spring’s equilibrium position, x = 0) then the potential energy of the spring for a spring stretched/compressed a distance x will be:

€

Us =12 kx

2

Example 9-5When a force of 120.0 N is applied to a certain spring, it causes

a stretch of 2.25 cm. What is the potential energy when it is compressed 3.50 cm?

Conservation of EnergyFor an object that is acted on by ONLY

conservative forces:

Work-KE theoremDefinition of PE

Combining the two gives€

W total =Wc

€

W total = ΔK

€

Wc = −ΔU

€

ΔK = −ΔUThe increase in KE is equal to the decrease in PE!

Conservation of Energy

Expanding out the equation:

A quick rearrangement gives:

€

K f −K i =U i −Uf

€

U i +K i =Uf +K f

Conservation of Energy• It is useful to think the sum of an object’s KE and

PE when in a particular state as a single quantity E that we call mechanical energy

• The result on the last slide tells us that

• When a quantity does not change as a system is evolves, that quantity is said to be conserved

€

E =U +K

€

E i = E f

Example 9-6At the end of a graduation ceremony, graduates fling their caps

into the air. Suppose a 0.120-kg cap is thrown straight upward with an initial speed of 7.85 m/s, and that frictional forces can be ignored. Find the speed of the cap when it is 1.18m above the release point.

Example 9-7A 55-kg skateboarder enters a ramp moving horizontally with a

speed of 6.5 m/s and leaves the ramp moving vertically with a speed of 4.1 m/s. (a) Find the height of the ramp, assuming no energy loss to frictional forces. (b) Find the skater’s maximum height above the ground.

Example 9-8A block of mass m = 2.0 kg is

dropped from a height h = 40 cm onto a spring of force constant 1960 N/m as shown on the right. Find the maximum distance that the spring is compressed.

Work done by Nonconservative Force

• Nonconservative forces can either add to (+W) or can subtract from (-W) the total energy of a system

• For any nonconservative force acting on an object:

• For friction (VERY common force) the work done is:

€

W fr = − fkd

€

Wnc = ΔK + ΔU = ΔE

€

W fr = −μkmgdor

common result (only valid for objects moving across a horizontal surface with no component of another Fnc in the vertical direction

Conservation of Energy w/ NC Forces• When one or more nonconservative forces acts

on a system, energy is no longer conserved

• The result on the last slide tells us that

• The most-useful version of the C of E eqn is:

€

E i + ΔE = E f

€

Wnc = ΔE

€

E i +Wnc = E f

Example 9-9A block of mass m1 = 2.40 kg is connected to a second block of

mass m2 = 1.80 kg, as pictured below. When the blocks are released from rest, they move through a distance d = 0.500 m at which point m2 hits the floor. Given that the coefficient of kinetic friction between m1 and the horizontal surface is μk=0.450 m, find the speed of the blocks just before m2 lands.

Example 9-10 (conceptual)A golfer badly misjudges a putt, sending the ball only one-

quarter of the distance to the hole. The original putt gave the ball an initial speed of v0. If the force of resistance due to the grass is constant, what initial speed should the golfer give the ball so that the ball just reaches the hole ?

Example 9-11A 1.2-kg block is held at rest against a spring with a force constant k =730 N/m.

Initially, the spring is compressed a distance d. When the block is released, it slides across a surface that is frictionless except for a rough patch of width 5.0 cm that has a coefficient of kinetic friction μk=0.44. Find d such that the block’s speed after crossing the rough patch is 2.3 m/s.

Example 9-11A 1.2-kg block is held at rest against a spring with a force constant k =730 N/m.

Initially, the spring is compressed a distance d. When the block is released, it slides across a surface that is frictionless except for a rough patch of width 5.0 cm that has a coefficient of kinetic friction μk=0.44. Find d such that the block’s speed after crossing the rough patch is 2.3 m/s.