Physics 207, Lecture 13, Oct. 15EnergyExample of a conservative system: The simple pendulum.Example: The simple pendulum.Slide 5Slide 6Example The Loop-the-Loop … againSlide 8Example SkateboardSlide 10Potential Energy, Energy Transfer and PathSlide 12Slide 13Elastic vs. Inelastic CollisionsSlide 15Inelastic collision in 1-D: Example 1Slide 17Variable force devices: Hooke’s Law SpringsHome Exercise Hooke’s LawF-s relation for a single DNA moleculeMeasurement technique: optical tweezersForce vs. Energy for a Hooke’s Law springEnergy for a Hooke’s Law springEnergy diagramsSlide 27EquilibriumComment on Energy ConservationSlide 30Physics 207: Lecture 13, Pg 1Physics 207, Lecture 13, Oct. 15Goals:Goals:Assignment: Assignment: HW6 due Wednesday, Oct. 22HW6 due Wednesday, Oct. 22For Monday: Read all of Chapter 11For Monday: Read all of Chapter 11•Chapter 10Chapter 10 Understand the relationship between motion and energy Define Potential Energy in a Hooke’s Law spring Develop and exploit conservation of energy principlein problem solving•Chapter 11Chapter 11 Understand the relationship between force, displacement and workPhysics 207: Lecture 13, Pg 2EnergyIf only “conservative” forces are present, the total energy If only “conservative” forces are present, the total energy ((sum of potential, U, and kinetic energies, K) of a system) of a system is is conservedconservedFor an object in a gravitational “field” Emech = K + UK and U may change, but Emech = K + U remains a fixed value.Emech = K + U = constant Emech is called “mechanical energy”K ≡ ½ mv2U ≡ mgy ½ m vyi2 + mgyi = ½ m vyf2 + mgyfPhysics 207: Lecture 13, Pg 3Example of a conservative system: The simple pendulum.Suppose we release a mass m from rest a distance h1 above its lowest possible point.What is the maximum speed of the mass and where does this happen ?To what height h2 does it rise on the other side ?vh1h2mPhysics 207: Lecture 13, Pg 4Example: The simple pendulum.yy=0y=h1 What is the maximum speed of the mass and where does this happen ?E = K + U = constant and so K is maximum when U is a minimum.Physics 207: Lecture 13, Pg 5Example: The simple pendulum.vh1yy=h1y=0 What is the maximum speed of the mass and where does this happen ?E = K + U = constant and so K is maximum when U is a minimumE = mgh1 at topE = mgh1 = ½ mv2 at bottom of the swingPhysics 207: Lecture 13, Pg 6Example: The simple pendulum.yy=h1=h2y=0 To what height h2 does it rise on the other side?E = K + U = constant and so when U is maximum again (when K = 0) it will be at its highest point.E = mgh1 = mgh2 or h1 = h2Physics 207: Lecture 13, Pg 7ExampleThe Loop-the-Loop … againTo complete the loop the loop, how high do we have to let the release the car?Condition for completing the loop the loop: Circular motion at the top of the loop (ac = v2 / R) Use fact that E = U + K = constant !h ?RCar has mass mRecall that “g” is the source of the centripetal acceleration and N just goes to zero is the limiting case. Also recall the minimum speed at the top isgRvUb=mghU=mg2Ry=0Physics 207: Lecture 13, Pg 8ExampleThe Loop-the-Loop … againUse E = K + U = constantmgh + 0 = mg 2R + ½ mv2 mgh = mg 2R + ½ mgR = 5/2 mgRh = 5/2 RRgRvh ?Physics 207: Lecture 13, Pg 9What speed will the skateboarder reach halfway down the hill if there is no friction and the skateboarder starts at rest? Assume we can treat the skateboarder as a “point”Assume zero of gravitational U is at bottom of the hillR=10 m..m = 25 kgExampleSkateboard..R=10 m30°y=0Physics 207: Lecture 13, Pg 10What speed will the skateboarder reach halfway down the hill if there is no friction and the skateboarder starts at rest? Assume we can treat the skateboarder as “point”Assume zero of gravitational U is at bottom of the hillR=10 m..m = 25 kgExampleSkateboard..R=10 m30°Use E = K + U = constant Ebefore = Eafter0 + m g R = ½ mv2 + mgR (1-sin 30°) mgR/2 = ½ mv2 gR = v2 v= (gR)½ v = (10 x 10)½ = 10 m/sPhysics 207: Lecture 13, Pg 11Potential Energy, Energy Transfer and PathA ball of mass m, initially at rest, is released and follows three difference paths. All surfaces are frictionless 1. Ball is dropped2. Ball slides down a straight incline3. Ball slides down a curved inclineAfter traveling a vertical distance h, how do the three speeds compare?h(A) 1 > 2 > 3 (B) 3 > 2 > 1 (C) 3 = 2 = 1 (D) Can’t tell1 32Physics 207: Lecture 13, Pg 12Potential Energy, Energy Transfer and PathA ball of mass m, initially at rest, is released and follows three difference paths. All surfaces are frictionless 1. The ball is dropped2. The ball slides down a straight incline3. The ball slides down a curved inclineAfter traveling a vertical distance h, how do the three speeds compare?(A) 1 > 2 > 3 (B) 3 > 2 > 1 (C) 3 = 2 = 1 (D) Can’t tellh1 32Physics 207: Lecture 13, Pg 13Now what is the normal force on the skate boarder? R=10 m..m = 25 kgExampleSkateboard..R=10 m30° Fr = mar = m v2 / R = N – mg cos 60°N = m v2 /R + mg cos 60°N = 25 100 / 10 + 25 10 (0.87)N = 250 + 220 =470 Newtons..Nmg60°Physics 207: Lecture 13, Pg 14Elastic vs. Inelastic CollisionsA collision is said to be elastic when energy as well as momentum is conserved before and after the collision. Kbefore = Kafter Carts colliding with a perfect spring, billiard balls, etc.vviPhysics 207: Lecture 13, Pg 15Elastic vs. Inelastic CollisionsA collision is said to be inelastic when energy is not conserved before and after the collision, but momentum is conserved. Kbefore Kafter Car crashes, collisions where objects stick together, etc.Physics 207: Lecture 13, Pg 16Inelastic collision in 1-D: Example 1A block of mass M is initially at rest on a frictionless horizontal surface. A bullet of mass m is fired at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a speed V. What is the initial energy of the system ? What is the final energy of the system ? Is energy conserved?vVbefore afterxPhysics 207: Lecture 13, Pg 17Inelastic collision in 1-D: Example 1What is the momentum of the bullet with speed v ? What is the initial energy of the system ? What is the final energy of the system ? Is momentum conserved (yes)? Is energy conserved? Examine Ebefore-EaftervVbefore
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