Chapter 7 - Potential Energy and Energy Conservation
I.The Work-Energy Theorem is modified by introducing a different type of energy, but we first need to define the difference between Conservative forces and Nonconservative forces.
II.Conservative Forces
A.A force is conservative if
1.the work done by the force around a closed path is zero:
,
which can be written as
.
When rearranged
2.This equation says that the work done by a conservative force is independent of the path, i.e., the work done depends only on the end points (i.e., where you start and where you end).
B.Look at various forces to determine whether or not they are conservative.
1.Gravitational force, mg
Move an object of mass m from an initial position (x1, y1) to a final position (x2, y2) along an arbitrary path, and calculate the work by mg done.
,
which becomes
.
Since the work done by the gravitational force depends only on the end points, we can define a function for the gravitational force mg that only depends on these end positions. We label the function Ugrav and call it "potential energy" – specifically, the “gravitational potential energy. It is the energy the object of mass m has because of its position. So,
,
where the gravitational potential energy, Ugrav = mgy.
2.Spring force, kx
Remember that the work done by a spring force when the end of the spring moves from x1 to x2 is
.
This work depends only on the end points, so again we can say that this force is conservative. Again a potential energy function can be defined called the "spring potential energy" or the “elastic” potential energy of the spring
,
where the spring potential energy, .
3.Friction force, f
Different paths will give different amounts of work. Therefore, the friction force is not conservative.
III.Note in all the cases of conservative forces, the work done by the conservative forces is
.
So, modify the work-energy theorem to:
(the total amount of work done by all the forces) = (work done by nonconservative forces) + (work done by conservative forces)
`According to the work-energy theorem:
Wtot = K , and since Wtot = Wnon + Wcon , then
Wtot = Wnon + Wcon = K . Rearranging, we get
Wnon = K - Wcon , then if Wcon = -U ,
Wnon = K + U .
Alternative Form:
, which we call the "Energy Equation," or
.
A special case: Suppose Wnon = 0, then . This is referred to as the "Conservation of Mechanical Energy." Or, we say “mechanical energy” is conserved.
IV.Using the Energy Equation
A.Can sometimes be used to solve dynamic problems where the force is complicated.
B.Procedure for using the energy equation:
1.Choose initial and final positions. These are positions where you know something and where you want to find something.
2.Identify the conservative and nonconservative forces. Draw a free body diagram showing the forces that act on the object as it moves from its initial position to its final position. Is mechanical energy “conserved?”
3.If mechanical energy is conserved, then . If not, then .
C.Examples:
1.Throw a ball straight upward with an initial velocity vo. How high does the ball travel if air resistance is ignored?
2.If a mass m of a simple pendulum of length L is released from rest at the angle , then
a.find the speed of the mass at the bottom of the arc.
b.To what height or angular position will the pendulum return?
3.A 2 kg block is dropped from rest from a height of 1 meter onto a spring in its unstretched position. How far will the spring compress? Take k = 500 N/m.
4.A 500 kg roller coaster starts from rest from a height of 40 m. It travels down and around a loop-the-loop whose radius is 10 m. What is the value of the normal force at the top of the loop? Assume the surfaces are smooth.
5.From what height must the roller coaster start in order to barely make it around the loop-the-loop?
6.A 5 kg block is traveling at 2 m/s. It slides on a horizontal surface and comes to rest in 3 m. What is the coefficient of friction between the block and the surface?
7.A 5 kg block is traveling at 2 m/s on a horizontal smooth surface. It travels 1 meter and hits an unstretched spring (k = 8000 N/m). By how much does the spring compress?
8.Same as question 8 except find the value of the coefficient of friction if the spring compresses a maximum distance of 0.02 m.
9.A light string passing over a massless, frictionless pulley connects masses m1 and m2. The mass m2 is on the horizontal surface and the coefficient of friction is = 0.20. The system is released from rest. What is the velocity of m1 after it has dropped 10 cm? Take m1 = 8 kg, m2 = 10 kg, and = 0.20.
10.A light string passing over a massless, frictionless pulley connects masses m1 and m2. The mass m2 is on the horizontal surface and has a spring attached to it and the wall. Friction is present between m2 and the surface. The system is released from rest. Take m1 = 8 kg, m2 = 10 kg, = 0.20, and k = 150 N/m.
a.What is the velocity of m1 after it has dropped 10 cm?
b.What is the maximum distance m1 drops?
V.Given the potential energy can we find the conservative force?
A.Derivation:
B.Examples with corresponding energy diagrams
1.Gravitational potential energy, Ugrav = mgy
2.Spring potential or elastic potential energy is.
3.The potential energy of one atom relative to the other in a diatomic molecule: Given the potential energy function shown, graph the force.
4.General potential energy
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