Potential Energy and Conservation of Energy Chapter 8 Potential Energy and Conservation of Energy

Units of Chapter 8 Conservative and Nonconservative Forces Potential Energy and the Work Done by Conservative Forces Conservation of Mechanical Energy Work Done by Nonconservative Forces Potential Energy Curves and Equipotentials

8-1 Conservative and Nonconservative Forces Conservative force: the work it does is stored in the form of energy that can be released at a later time Example of a conservative force: gravity Example of a nonconservative force: friction Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force

8-1 Conservative and Nonconservative Forces Work done by gravity on a closed path is zero:

8-1 Conservative and Nonconservative Forces Work done by friction on a closed path is not zero:

8-1 Conservative and Nonconservative Forces The work done by a conservative force is zero on any closed path:

Example Calculate the work one by gravity as a 3.2 kg object is moved from point A to point B on the figure along paths 1, 2, and 3

8-2 The Work Done by Conservative Forces (8-1)

8-2 The Work Done by Conservative Forces Gravitational potential energy:

8-2 The Work Done by Conservative Forces Springs: (8-4)

ConcepTest 8.4 Elastic Potential Energy How does the work required to stretch a spring 2 cm compare with the work required to stretch it 1 cm? 1) same amount of work 2) twice the work 3) 4 times the work 4) 8 times the work

ConcepTest 8.4 Elastic Potential Energy How does the work required to stretch a spring 2 cm compare with the work required to stretch it 1 cm? 1) same amount of work 2) twice the work 3) 4 times the work 4) 8 times the work The elastic potential energy is 1/2 kx2. So in the second case, the elastic PE is 4 times greater than in the first case. Thus, the work required to stretch the spring is also 4 times greater.

Pre- and Post- Tests

8-3 Conservation of Mechanical Energy Definition of mechanical energy: (8-6) Using this definition and considering only conservative forces, we find: Or equivalently:

8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve:

ConcepTest 8.2 KE and PE You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on? 1) only B 2) only C 3) A, B and C 4) only A and C 5) only B and C A) skier’s PE B) skier’s change in PE C) skier’s final KE

ConcepTest 8.2 KE and PE You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on? 1) only B 2) only C 3) A, B and C 4) only A and C 5) only B and C A) skier’s PE B) skier’s change in PE C) skier’s final KE The gravitational PE depends upon the reference level, but the difference DPE does not! The work done by gravity must be the same in the two solutions, so DPE and DKE should be the same. Follow-up: Does anything change physically by the choice of y = 0?

ConcepTest 8.6 Down the Hill Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp? 4) same speed for all balls 1 2 3

ConcepTest 8.6 Down the Hill Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp? 4) same speed for all balls 1 2 3 All of the balls have the same initial gravitational PE, since they are all at the same height (PE = mgh). Thus, when they get to the bottom, they all have the same final KE, and hence the same speed (KE = 1/2 mv2). Follow-up: Which ball takes longer to get down the ramp?

ConcepTest 8.8a Water Slide I Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. At the bottom, whose velocity is greater? 1) Paul 2) Kathleen 3) both the same

ConcepTest 8.8a Water Slide I Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. At the bottom, whose velocity is greater? 1) Paul 2) Kathleen 3) both the same Conservation of Energy: Ei = mgH = Ef = 1/2 mv2 therefore: gH = 1/2 v2 Since they both start from the same height, they have the same velocity at the bottom.

Example In the figure below, the water slide ends at a height of 1.50 m above the pool. If the person starts from rest at A and lands in the water at B, what is the height h of the slide?

8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

ConcepTest 8.10a Falling Leaves You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare? 1) PEi + KEi > PEf + KEf 2) PEi + KEi = PEf + KEf 3) PEi + KEi < PEf + KEf 4) impossible to tell from the information provided

ConcepTest 8.10a Falling Leaves You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare? 1) PEi + KEi > PEf + KEf 2) PEi + KEi = PEf + KEf 3) PEi + KEi < PEf + KEf 4) impossible to tell from the information provided As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a non-conservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy. Follow-up: What happens to leaf’s KE as it falls? What net work is done?

8-5 Potential Energy Curves and Equipotentials The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring:

Summary of Chapter 8 Conservative forces conserve mechanical energy Nonconservative forces convert mechanical energy into other forms Conservative force does zero work on any closed path Work done by a conservative force is independent of path Conservative forces: gravity, spring

Summary of Chapter 8 Work done by nonconservative force on closed path is not zero, and depends on the path Nonconservative forces: friction, air resistance, tension Energy in the form of potential energy can be converted to kinetic or other forms Work done by a conservative force is the negative of the change in the potential energy Gravity: U = mgy Spring: U = ½ kx2

Summary of Chapter 8 Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces Nonconservative forces change a system’s mechanical energy Work done by nonconservative forces equals change in a system’s mechanical energy Potential energy curve: U vs. position