Recall: Gravitational Potential Energy
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1 Welcome back to Physics 15 Today s agenda: Work Power Physics 15 Spring 017 Lecture Recall: Gravitational Potential Energy For an object of mass m near the surface of the earth: U g = mgh h is height above arbitrary reference line Measured in Joules -- J (like kinetic energy) Physics 15 Spring 017 Lecture
2 Recall: If the velocity at B is v, then what is the velocity at C? A. v B. v C. sqrt() v D. v/sqrt() E. None of the above h A h/ B C Physics 15 Spring 017 Lecture Recall: Springs -- Elastic potential energy Force F = -kx (Hooke s law) frictionless table Area of triangle lying under straight line graph of F vs. x = (1/)(+/-x)(-/+kx) F U s = (1/)kx F = -kx x Physics 15 Spring 017 Lecture
3 (Horizontal) Spring frictionless table x = displacement from relaxed state of spring Elastic potential energy stored in spring: U s = (1/)kx (1/)kx + (1/)mv = constant Physics 15 Spring 017 Lecture SG A 0.5 kg mass is attached to a spring on a horizontal frictionless table. The mass is pulled to stretch the spring 5.0 cm and is released from rest. When the mass crosses the point at which the spring is not stretched, x = 0, its speed is 0 cm/s. If the experiment is repeated with a 10.0 cm initial stretch, what speed will the mass have when it crosses x = 0? cm/s. 0 cm/s 3. 0 cm/s cm/s Physics 15 Spring 017 Lecture
4 Work, Energy Newton s Laws are vector equations Sometimes easier to relate speed of a particle to how far it moves under a force a single equation can be used need to introduce concept of work Physics 15 Spring 017 Lecture What is work? Assume constant force in 1D Consider: v F = v I + a D x Multiply by m/ à (1/)mv F - (1/)mv I = mad x But: F = ma à (1/)mv F - (1/)mv I = FD x Physics 15 Spring 017 Lecture
5 Work-Kinetic Energy theorem (1) (1/)mv F - (1/)mv I = Fs Points: s = D x = displacement (for D) W = Fs à defines work done on particle = force times displacement K = (1/)mv à defines kinetic energy =1/ mass times square of v Physics 15 Spring 017 Lecture Improved definition of work For forces, write F à F AB Thus W = Fs à W AB = F AB D s B is work done by A on B as B undergoes displacement D s B Work-kinetic energy theorem: W net,b = S A W AB = D K Physics 15 Spring 017 Lecture
6 The Work - Kinetic Energy Theorem W net = D K = K f - K i The net work done on an object is equal to the change in kinetic energy of the object. Physics 15 Spring 017 Lecture SG Suppose a tennis ball and a bowling ball are rolling toward you. The tennis ball is moving much faster, but both have the same momentum (mv), and you exert the same force to stop each. Which of the following statements is correct? A. It takes equal distances to stop each ball. B. It takes equal time intervals to stop each ball. C. Both of the above. D. Neither of the above. Physics 15 Spring 017 Lecture
7 SG Suppose a tennis ball and a bowling ball are rolling toward you. Both have the same momentum (mv), and you exert the same force to stop each. It takes equal time intervals to stop each ball. The distance taken for the bowling ball to stop is A. less than. B. equal to C. greater than the distance taken for the tennis ball to stop. Physics 15 Spring 017 Lecture SG Two carts of different mass are accelerated from rest on a low-friction track by the same force for the same time interval. Cart B has greater mass than cart A (m B > m A ). The final speed of cart A is greater than that of cart B (v A > v B ). After the force has stopped acting on the carts, the kinetic energy of cart B is A. less than the kinetic energy of cart A (K B < K A ). B. equal to the kinetic energy of cart A (K B = K A ). C. greater than the kinetic energy of cart A (K B > K A ). D. Can t tell. Physics 15 Spring 017 Lecture
8 Work and Kinetic Energy in D Work done by object on object 1: Kinetic energy of an object: K = 1 mv [or: 1 m( v v)] Physics 15 Spring 017 Lecture Scalar (or dot ) product of vectors The scalar product is a way to combine two vectors to obtain a number (or scalar). It is indicated by a dot ( ) between the two vectors. (Note: component of A in direction n is just A n) A B = ABcosθ = A x B x + A y B y + A z B z Physics 15 Spring 017 Lecture
9 SG Is the scalar ( dot ) product of the two vectors 1. positive. negative 3. equal to zero 4. Can t tell. Physics 15 Spring 017 Lecture SG Is the scalar ( dot ) product of the two vectors 1. positive. negative 3. equal to zero 4. Can t tell. Physics 15 Spring 017 Lecture
10 SG A person lifts a book at constant speed. Since the force exerted on the book by the person s hand is in the same direction as the displacement of the book, the work (W 1 ) done on the book by the person s hand is positive. The work done on the book by the earth is: A. negative and equal in absolute value to W 1 B. negative and less in absolute value than W 1 C. positive and equal in absolute value to W 1 D. positive and less in absolute value than W 1 Physics 15 Spring 017 Lecture A person lifts a book at constant speed. The work (W 1 ) done on the book by the person s hand is positive. Work done on the book by the earth: Net work done on the book: Change in kinetic energy of the book: Physics 15 Spring 017 Lecture
11 SG A person carries a book horizontally at constant speed. The work done on the book by the person s hand is A. positive B. negative C. equal to zero D. Can t tell. Physics 15 Spring 017 Lecture Sample problem: The two ropes seen in the figure are used to lower a 55 kg piano 5.1 m from a second-story window to the ground. How much work is done by each of the three forces? Physics 15 Spring 017 Lecture
12 SG Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B. Using the work-kinetic energy theorem, the speed of block A at the bottom of its ramp is A. less than the speed of block B. B. equal to the speed of block B. C. greater than the speed of block B. D. Can t tell. Physics 15 Spring 017 Lecture Solution Which forces do work on block? Which, if any, are constant? What is F D s for motion? Physics 15 Spring 017 Lecture
13 Work done by gravity N Work W = -mg j D s j mg D s Therefore, W = -mgd h N does no work! i Physics 15 Spring 017 Lecture Work done on an object by gravity W (on object by earth) = m g D h, where D h = h final h initial is the change in height. Object moves Change in height Δh Work done on object by earth upward positive negative downward negative positive Physics 15 Spring 017 Lecture
14 Defining gravitational potential energy W (on obj. by earth) = ΔK 0 = ΔK W (on obj. by earth) 0 = ΔK + ΔUg The change in gravitational potential energy of the objectearth system is just another name for the negative value of the work done on an object by the earth. Physics 15 Spring 017 Lecture Curved ramp D s = W = F D s = Work done by gravity between fixed pts does not depend on path taken! Work done by gravitational force in moving some object along any path is independent of the path depending only on the change in vertical height Physics 15 Spring 017 Lecture
15 Demo: 4 tracks Physics 15 Spring 017 Lecture Conservative forces If the work done by some force (e.g. gravity) does not depend on path the force is called conservative. Then gravitational potential energy U g only depends on (vertical) position of object U g = U g (h) Elastic forces also conservative elastic potential energy U = (1/)kx... Physics 15 Spring 017 Lecture
16 Many forces For a particle which is subject to several (conservative) forces F 1, F D K= W 1 +W + D K+D U 1 +D U +.=0 or equivalently: i.e E = (1/)mv + U 1 + U + is constant Principle called Conservation of total mechanical energy Physics 15 Spring 017 Lecture
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