Physics 151: Lecture 16, Pg 1 Physics 151: Lecture 16 Today’s Agenda l Today’s Topics: çConservation of mechanical energy çNonconservative forces and loss of energy from a system çOther relationships for potential energy

Physics 151: Lecture 16, Pg 2 See text: 8-4 Conservation of Energy l If only conservative forces are present, the total energy (sum of potential and kinetic energies) of a system is conserved.  E =  K +  U = W +  U using  K = W = W + (-W) = 0 using  U = -W E = K + U Both K and U can change, but E = K + U remains constant. constant!!! E = K + U is constant !!!

Physics 151: Lecture 16, Pg 3 Example: The simple pendulum. l Suppose we release a mass m from rest a distance h 1 above its lowest possible point. ç What is the maximum speed of the mass and where does this happen ? ç To what height h 2 does it rise on the other side ? v h1h1 h2h2 m See text: Example 8.3

Physics 151: Lecture 16, Pg 4 Example: The simple pendulum. l Energy is conserved since gravity is a conservative force (E = K + U is constant) l Choose y = 0 at the bottom of the swing, and U = 0 at y = 0 (arbitrary choice). E = 1 / 2 mv 2 + mgy. v h1h1 h2h2 y y=0 See text: Example 8.3

Physics 151: Lecture 16, Pg 5 Lecture 16, ACT 1 The Roller Coaster l I have built a Roller Coaster. A motor tugs the cars to the top and then they are let go and are in the hands of gravity. To make the following loop, how high do I have to let the release the car ?? h ? R Car has mass m A) 2RB) 5RC) 5/2 RD) none of the above

Physics 151: Lecture 16, Pg 6 Another Problem Involving a Spring A large spring is used to stop the cars after they come down the last hill of a roller coaster. The cars start at rest at the top of the hill and are caught by a mechanism at the instant their velocities at the bottom are zero. Compare the compression of the spring, x A, for a fully loaded car with that, x B, for a lightly loaded car when m A = 2m B. a. x A = 1/2 x B. b. x A = x B. c. x A = (2) 1/2 x B. d. x A = 2 x B. e. x A = 4 x B.

Physics 151: Lecture 16, Pg 7 Non-conservative Forces: l If the work done does not depend on the path taken, the force involved is said to be conservative. l If the work done does depend on the path taken, the force involved is said to be non-conservative. l An example of a non-conservative force is friction: l Pushing a box across the floor, the amount of work that is done by friction depends on the path taken. ç Work done is proportional to the length of the path ! See text: 8.5

Physics 151: Lecture 16, Pg 8 Generalized Work Energy Theorem: W NC =  K +  U =  E

Physics 151: Lecture 16, Pg 9 Lecture 16, ACT 2 Stones and Friction l I throw a stone into the air. While in flight it feels the force of gravity and the frictional force of the air resistance. The time the stone takes to reach the top of its flight path (i.e. go up) is, A) larger than B) equal to C) less than The time it takes to return from the top (i.e. go down).

Physics 151: Lecture 16, Pg 10 Lecture 16, ACT 2 Solution We know that W NC =  K +  U =  E l Thus as time progresses the amount of energy in the earth-rock system is continually decreasing l If I consider the way up versus the way down at a given height, then E up = K up + U up > E down = K down + U down l But, at a given height U is always the same. So, Kup > Kdown l v at a given height is always less on the way down The answer is (C) it takes less time to go up

Physics 151: Lecture 16, Pg 11 Same problem but with numbers: A 2.0-kg mass is projected vertically upward from ground level with an initial speed of 30 m/s. The mass rises to a maximum height of 35 m above ground level. How much work is done on the mass by air resistance between the point of projection and the point of maximum height? a. –0.21 kJ b kJ c. –0.40 kJ d kJ e. –0.69 kJ

Physics 151: Lecture 16, Pg 12 Question l As an object moves from point A to point B only two forces act on it: one force is nonconservative and does –30 J of work, the other force is conservative and does +50 J of work. Between A and B, 1. the kinetic energy of object increases, mechanical energy decreases. 2. the kinetic energy of object decreases, mechanical energy decreases. 3. the kinetic energy of object decreases, mechanical energy increases. 4. the kinetic energy of object increases, mechanical energy increases. 5. None of the above.

Physics 151: Lecture 16, Pg 13 Question - 2 l As an object moves from point A to point B only two forces act on it: one force is conservative and does –70 J of work, the other force is nonconservative and does +50 J of work. Between A and B, 1. the kinetic energy of object increases, mechanical energy increases. 2. the kinetic energy of object decreases, mechanical energy increases. 3. the kinetic energy of object decreases, mechanical energy decreases. 4. the kinetic energy of object increases, mechanical energy decreases. 5. None of the above.

Physics 151: Lecture 16, Pg 14 ACT- 2 l Objects A and B, of mass M and 2M respectively, are each pushed a distance d straight up an inclined plane by a force F parallel to the plane. The coefficient of kinetic friction between each mass and the plane has the same value. At the highest point, 1.K A > K B. 2.K A = K B. 3.K A < K B. 4.The work done by F on A is greater than the work done on B. 5.The work done by F on A is less than the work done on B.

Physics 151: Lecture 16, Pg 15 l Let’s now suppose that the surface is not frictionless and the same skateboarder reach the speed of 7.0 m/s at bottom of the hill. What was the work done by friction on the skateboarder ? R=3 m.. m = 25 kg Conservation of Total Energy : Lecture 16, Example Skateboard K 1 + U 1 = K 2 + U 2 W f mgR = 1/2mv W f = 1/2mv 2 - mgR W f = (1/2 x25 kg x (7.0 m/s 2 ) kg x 10m/s 2 3 m) W f = J = J Total mechanical energy decreased by 122 J !.. W f +

Physics 151: Lecture 16, Pg 16 Conservative Forces and Potential Energy l We have defined potential energy for conservative forces  U = -W l But we also now that W = F x  x l Combining these two,  U = - F x  x l Letting small quantities go to infinitesimals, dU = - F x dx l Or, F x = -dU/dx See Text: section 8.6

Physics 151: Lecture 16, Pg 17 Examples of the U - F relationship l Remember the spring, çU = (1/2)kx 2 l Do the derivative çF x = - d ( (1/2)kx 2 ) / dx çF x = - 2 (1/2) kx çF x = -kx See Text: section 8.6

Physics 151: Lecture 16, Pg 18 Examples of the U - F relationship l Remember gravity, çF y = mg l Do the integral çU = -  F dy çU = -  (-mg) dy çU = mgy + C   U = U 2 – U 1 = (mgy 2 + C) – (mgy 1 + C) = mg  y See Text: section 8.6

Physics 151: Lecture 16, Pg 19 Recap of today’s lecture l Conservation of Mechanical Energy l Non-conservative forces and loss of energy from a system l Other relationships for potential energy