Potential Energy, Conservation of Energy, and Energy Diagrams 8.01 W06D Today s Reading ssignment: Chapter 14 Potential Energy and Conservation of Energy, Sections 14.1-14.7 nnouncements Problem Set 5 due Oct 16 at 9 pm in box outside 6-15 Review: Conservative Forces Definition: Conservative Force If the work done by a force in moving an object from point to point is independent of the path (1 or ), Wc F c d r (path independent) then the force is called a conservative force which we denote by F c. Then the work done only depends on the location of the points and. 1
Potential Energy Difference Definition: Potential Energy Difference between the points and associated with a conservative force is the negative of the work done by the conservative force in moving the body along any path connecting the points and. ΔU F c d r = W c F c Force: Work: Potential Energy Difference: Constant Gravity F = mg = F y ĵ = mg ĵ Potential Energy: W = F y Δy = mgδy Choice of Zero Point: Choose U (0) 0. ( ) ΔU = W = mgδy = mg y f y i Potential Energy Function: Let y f = y then U ( y) = mgy; U (0) = 0 Worked Example: Change in Potential Energy for Inverse Square Gravitational Force Consider an object of mass m 1 moving towards the sun (mass m ). Initially the object is at a distance r 0 from the center of the sun. The object moves to a final distance r f from the center of the sun. For the object-sun system, what is the change in potential energy during this motion?
Force: Work done: Worked Example Solution: Inverse Square Gravity F Potential Energy Change: Gm m r 1 ˆ m1, m = r r f W = F dr r f = Gm m 1 r dr = Gm m r f 1 1 = Gm r 1 m 1 r f r i r i r i 1 ΔU = W = Gm 1 m 1 r f r i r i Zero Point: Potential Energy Function U ( ) = 0 U (r) = Gm m 1 ; U ( ) = 0 r Concept Question: Potential Energy and Inverse Square Gravity comet is speeding along a hyperbolic orbit toward the Sun. While the comet is moving away from the Sun, the change in potential energy of the Sun-comet system is: (1) positive () zero (3) negative Table Problem: Change in Potential Energy Spring Force Connect one end of a spring of length l 0 with spring constant k to an object resting on a smooth table and fix the other end of the spring to a wall. Stretch the spring until it has length l and release the object. Consider the object-spring as the system. When the spring returns to its equilibrium length what is the change in potential energy of the system? 3
Force: Work done: Potential Energy Difference: Spring Force Potential Energy Change: F = F x î = kx î W = x=x f ( kx)dx = 1 k x f x i x=x i x = displacement from equilibrium ( ) ΔU = W = 1 k x ( x f i ) Zero Point: Potential Energy U (0) = 0 U (x) = 1 kx ; U (0) = 0 Work-Energy Theorem: Conservative Forces The work done by the force in moving an object from to is equal to the change in kinetic energy W When the only forces acting on the object are conservative forces F = F c then the change in potential energy is Therefore F dr = 1 mv 1 mv ΔK ΔU = W ΔU = ΔK Forms of Energy kinetic energy gravitational potential energy elastic potential energy thermal energy electrical energy chemical energy electromagnetic energy nuclear energy mass energy 4
Energy Transformations Falling water releases stored gravitational potential energy turning into a kinetic energy of motion. Human beings transform the stored chemical energy of food into catabolic energy urning gasoline in car engines converts chemical energy stored in the atomic bonds of the constituent atoms of gasoline into heat Stretching or compressing a spring stores elastic potential energy that can be released as kinetic energy Mechanical Energy When a sum of conservative forces are acting on an object, the potential energy function is the sum of the individual potential energy functions with an appropriate choice of zero point potential energy for each function U = U 1 + U + Definition: Mechanical Energy The mechanical energy function is the sum of the kinetic and potential energy function E mech K + U Conservation of Mechanical Energy When the only forces acting on an object are conservative ΔK + ΔU = 0 (K f K i ) (U f U i ) = 0 Equivalently, the mechanical energy remains constant in time E mech mech f = K f + U f = K i + U i = E i ΔE mech = E f mech E i mech = 0 5
Demonstration: Experiment Cart- Spring on an Inclined Plane cart of mass m slides down a plane that is inclined at an angle θ from the horizontal. The cart starts out at rest. The center of mass of the cart is a distance d from an unstretched spring with spring constant k that lies at the bottom of the plane. plot of distance vs. time shows the change in the mechanical energy of the system. Identify system Energy Diagrams Identify state of system before interaction and state of final states. Draw energy diagrams for these states including: 1) Choice of zero point P for potential energy for each interaction in which potential energy difference is welldefined. ) Draw speed of all objects in system 3) Identify initial and final mechanical energy E mech = K+U Demo slide: Loop-the-Loop 95 ball rolls down an inclined track and around a vertical circle. This demonstration offers opportunity for the discussion of dynamic equilibrium and the minimum speed for safe passage of the top point of the circle. http://tsgphysics.mit.edu/front/?page=demo.php&letnum= 95&show=0 6
Table Problem: Loop-the-Loop n object of mass m is released from rest at a height h above the surface of a table. The object slides along the inside of the loop-the-loop track consisting of a ramp and a circular loop of radius R shown in the figure. ssume that the track is frictionless. When the object is at the top of the track (point a) it just loses contact with the track. What height was the object dropped from? Table Problem: Sliding on Slipping lock small cube of mass m 1 slides down a circular track of radius R cut into a large block of mass m as shown in the figure below. The large block rests on a, and both blocks move without friction. The blocks are initially at rest, and the cube starts from the top of the path. Find the velocity of the cube as it leaves the block. 0 Review: Non-Conservative Forces Definition: Non-conservative force Whenever the work done by a force in moving an object from an initial point to a final point depends on the path, then the force is called a non-conservative force and the work done is called non-conservative work W nc F nc dr F nc 7
Non-Conservative Forces Work done on the object by the force depends on the path taken by the object Example: friction on an object moving on a level surface F = µ N friction k friction = frictionδ = µ k Δ < 0 W F x N x Change in Energy for Conservative and Non-conservative Forces Force decomposition: F = F c + F nc Work done is change in kinetic energy: W = F dr = ( F c + F nc ) dr = ΔU + W nc = ΔK Mechanical energy change: ΔK + ΔU = ΔE mech = W nc Modeling the Motion Energy Concepts Change in Mechanical Energy: Identify non-conservative forces. Calculate non-conservative work: final W = F d r. pply Energy Law: W nc = ΔK + ΔU = ΔE mech nc initial nc 8
Modeling the Motion: Newton s Second Law Define system, choose coordinate system. Draw free body force diagrams. Newton s Second Law for each direction. Example: x-direction Example: Circular motion d x m dt ˆ : total i Fx =. v m R total rˆ : Fr =. Table Problem: Sliding Friction lock of mass m is moving horizontally with speed V along a frictionless surface. It collides elastically with block of mass m that is initially at rest. fter the collision block enters a rough surface at x = 0 with a coefficient of kinetic friction that increases linearly with distance, µ k (x) = bx, 0 x d where b is a positive constant. t x = d, block collides with an unstretched spring with spring constant k. For x>d the surface is frictionless. The downward gravitational acceleration has magnitude g. What is the distance the spring is compressed when block first comes to rest? Express your answer in terms of V, m, m, b, d, g, and k. 9