Potential Energy and Energy Conservation

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Goals for Chapter 7 • Learn the following concepts: conservative forces, Chapter 7 potential energy, and conservation of mechanical energy. • Learn when and how to apply conservation of mechanical energy to solve physics problems. Potential Energy and • Examine situations when mechanical energy is not conserved, that is, when there are non-conservative Energy Conservation forces such as kinetic friction and air resistance. • Learn to relate the conservative force to the potential PowerPoint® Lectures for energy curve. University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman

Lectures by James Pazun Modified by P. Lam 7_14_2010

Introduction/Summary Work done by a conservative force - e.g. gravity • Start with W-KE theorem: • Work done by gravity: f ! " ! ! y f y f K f ! Ki = W = " F • d# (or F • s for constant force) i W = ! Fdy = ! "mgdy = mgyi " mgy f • There are two type of forces: yi yi Conservative forces (such as gravity and spring force) Define gravitational potential energy : Non-conservative forces (such as kinetic friction and air resistance) U(y) ! mgy ! K " K = W + W f i conservative non"conservative " W = Ui #U f • If there are only conservative forces then “mechanical energy” is If only gravity is doing work, then by conserved. Mechanical energy=kinetic energy (K) + potential energy (U) combining with the W - KE Theorem K ! K = W = U !U " (K + U ) ! (K + U ) = 0 f i conservative i f f f i i K ! K = W = U !U • If there are also non-conservative forces then “mechanical energy” is f i i f

NOT conserved, but total energy is still conserved " K f + U f = Ki + Ui

! (K f + U f ) " (Ki + Ui ) = Wnon"conservative; ""Conservation of mechanical energy"

(Wnon"conservative ) = other form of energy, e.g. "heat" Mechanical energy # K +U

1 Apply conservation of mechanical energy - Example 7.1 Property of conservative force • Assume there is no air resistance. After the ball leave your hand, the only force acting on the ball is gravity. • Work done by conservative force depends only on the initial and • How high can the ball reach? final position, independent of the path.

W = Ui !U f (independent of path)

e.g. W = mgyi ! mgy f

1 1 mv 2 + mgy = mv 2 + mgy 2 1 1 2 2 2

Put in the values for v1, y1, and v2; find y2. => Initially, the ball has only KE and no PE. At final position, the ball has no KE and only PE.

Consider projectile motion Application: What’s the speed in a vertical circle? Given: R=3 m, find the skater’s speed at the bottom (assume no friction or air-resistance)

• If the initial SPEED and initial height are the same, then the SPEED at a given height will be the same, even though the velocity vectors point at different directions. What about the “normal” force, does it do any work on the skater?

2 Another conservative force - idealize spring Example: Conservation of mechanical energy with spring

See figure below, find the speed at x2.

Given:

x f 1 2 1 2 W = " !kxdx = kxi ! kx f = Ui !U f xi 2 2 1 # Spring potential energy U(x) = kx 2 Find v2 at x2. 2

Also true for compression 1 1 1 1 mv 2 + kx 2 = mv 2 + kx 2; Subtitue values and find v . 2 1 2 1 2 2 2 2 2

Visualize KE and PE in a potential energy graph Combine gravity and spring • Elevator’s cable broke. Suppose the glider is initially stretched to position A. • Given: v1=4 m/s; spring As the glider slides back and forth, is compressed 2 m. Find potential energy is converted to the spring constant. kinetic energy and back to potential energy and so forth.

K+U=constant can be visualized in the potential energy graph.

When does the glider speed up? 1 1 1 1 When does it slow down? mv 2 + mgy + ky 2 = mv 2 + mgy + ky 2 2 1 1 2 1 2 2 2 2 2

What are the appropriate values for y1 and y2 ? Solve for k.

3 Non-conservative force - e.g. kinetic friction Combine gravity, spring force and friction • Work done by the non-conservative force going • DO NOT neglect from point 1 to point 2 depends on the path, hence friction (f), find the spring constant.

Wnon!conservative " Ui !U f

1 1 1 1 ( mv 2 + mgy + ky 2 ) + W = ( mv 2 + mgy + ky 2 ) 2 1 1 2 1 Friction 2 2 2 2 2

WFriction = ! fs = !(17,000)(2)Joule

y1 = 0, y2 = 2m Solve for k.

Conservative Force as the derivative of potential energy The potential energy curve for motion of a particle

• Refer to Figure 7.22 as it examines spring and • Refer the potential energy function and its corresponding components of force. gravitational energies.

What does negative force mean?

Stable equilibrium position