Page 1Physics 207 – Lecture 13Physics 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 If only ““conservativeconservative””forces are present, the total energy forces are present, the total energy ((sum of potential, U, and kinetic energies, K) of a system) of a systemis is conservedconservedFor an object in a gravitational “field”Emech= K + U K and U may change, but Emech= K + U remains a fixed value.Emech= K + U = constantEmechis called “mechanical energy”K ≡½mv2U ≡mgy½ m vyi2+ mgyi= ½ m vyf2 + mgyfPage 2Physics 207 – Lecture 13Physics 207: Lecture 13, Pg 3Example of a conservative system: The simple pendulum. Suppose we release a mass m from rest a distance h1above its lowest possible point. What is the maximum speed of the mass and where does this happen ? To what height h2does 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.Page 3Physics 207 – Lecture 13Physics 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 = mgh1at topE = mgh1= ½ mv2at bottom of the swingPhysics 207: Lecture 13, Pg 6Example: The simple pendulum.yy=h1=h2y=0To what height h2does 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 = h2Page 4Physics 207 – Lecture 13Physics 207: Lecture 13, Pg 7ExampleThe Loop-the-Loop … again To 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 isgR=vUb=mghU=mg2Ry=0Physics 207: Lecture 13, Pg 8ExampleThe Loop-the-Loop … again Use E = K + U = constant mgh + 0 = mg 2R + ½ mv2 mgh = mg 2R + ½ mgR = 5/2 mgRh = 5/2 RRgR=vh ?Page 5Physics 207 – Lecture 13Physics 207: Lecture 13, Pg 9 What 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 10 What 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 = constantEbefore= Eafter0 + m g R = ½ mv2 + mgR (1-sin 30°)mgR/2 = ½ mv2gR = v2 v= (gR)½v = (10 x 10)½= 10 m/sPage 6Physics 207 – Lecture 13Physics 207: Lecture 13, Pg 11Potential Energy, Energy Transfer and Path A 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 Path A ball of mass m, initially at rest, is released and follows three difference paths. All surfaces are frictionless1. 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 32Page 7Physics 207 – Lecture 13Physics 207: Lecture 13, Pg 13 Now what is the normal force on the skate boarder? R=10 m..m = 25 kgExampleSkateboard..R=10 m30° Σ 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 Collisions A 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.vviPage 8Physics 207 – Lecture 13Physics 207: Lecture 13, Pg 15Elastic vs. Inelastic Collisions A 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 1 A 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 afterxPage 9Physics 207 – Lecture 13Physics 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 afterxvrm v21 vv212mm =⋅rr V)(212Mm+ V)( 0 M vMmm+=+)(1v21 vv)(21 v21 V]V)[(21 v21222MmmmMmmmmMmm+−=+−=+−No!Physics 207: Lecture 13, Pg 18Example – Fully Elastic Collision Suppose I have 2 identical bumper cars. One is motionless and the other is approaching it with velocity v1. If they collide elastically, what is the final velocity
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