Phys 207. Announcements. Hwk 6 is posted online; submission deadline = April 4 Exam 2 on Friday, April 8th. Today s Agenda

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1 Phs 07 Announcements Hwk 6 is posted online; submission deadline = April 4 Exam on Frida, April 8th Toda s Agenda Freshman Interim Grades eview Work done b variable force in 3-D Newton s gravitational force Conservative forces & potential energ Conservation of total mechanical energ Example: pendulum Work b variable force in 3-D: Work dw F of a force F acting through an infinitesimal displacement r is: dw = F. r F r The work of a big displacement through a variable force will be the integral of a set of infinitesimal displacements: W TOT = F. r Page

2 Work b variable force in 3-D: Newton s Gravitational Force Work dw g done on an object b gravit in a displacement dr is given b: dw g = F g. dr = (-GMm / r). ^ (d r ^ + dθ θ) ^ ^ ^ ^^ dw g = (-GMm / ) d (since r. θ = 0, r. r = ) F g d dθ dr m ^ θ r^ dθ M 3 Work b variable force in 3-D: Newton s Gravitational Force Integrate dw g to find the total work done b gravit in a big displacement: W g = dw g = (-GMm / ) d = GMm (/ -/ ) F g ( ) m F g ( ) M 4 Page

3 Work b variable force in 3-D: Newton s Gravitational Force Work done depends onl on and, not on the path taken. W g = GMm m M 5 Lecture 6, Act Work & Energ A rock is dropped from a distance E above the surface of the earth, and is observed to have kinetic energ K when it hits the ground. An identical rock is dropped from twice the height ( E ) above the earth s surface and has kinetic energ K when it hits. E is the radius of the earth. What is K / K? (a) (b) 3 E E (c) 4 3 E 6 Page 3

4 Lecture 6, Act Solution Since energ is conserved, K = W G. W G = GMm K = c Where c = GMm is the same for both rocks E E E 7 For the first rock: Lecture 6, Act Solution K = c E E = c E K = c For the second rock: K = c E 3 E = c 3 E So: K K = 3 = 4 3 E E E 8 Page 4

5 Newton s Gravitational Force Near the Earth s Surface: Suppose = E and = E + ( + ) ( ) E E ( + )( ) GM W = GMm GMm m g = E E E but we have learned that GM = g E So: W g = -mg m E + E M 9 Conservative Forces: We have seen that the work done b gravit does not depend on the path taken. m W g = GMm M m h W g = -mgh 0 Page 5

6 Conservative Forces: In general, if the work done does not depend on the path taken (onl depends the initial and final distances between objects), the force involved is said to be conservative. Gravit is a conservative force: W g = GMm Gravit near the Earth s surface: W g = mg A spring produces a conservative force: W = k( x x ) s Conservative Forces: We have seen that the work done b a conservative force does not depend on the path taken. W W = W Therefore the work done in a closed path is 0. W W W NET = W -W = W -W = 0 W Page 6

7 Lecture 6, Act Conservative Forces The pictures below show force vectors at different points in space for two forces. Which one is conservative? (a) (b) (c) both x () x () 3 Lecture 6, Act Solution Consider the work done b force when moving along different paths in each case: W A = W B W A > W B () () 4 Page 7

8 Lecture 6, Act In fact, ou could make mone on tpe () if it ever existed: Work done b this force in a round trip is > 0! Free kinetic energ!! W NET = 0 J = K W = 0 Note: NO EAL FOCES OF THIS TYPE EXIST, SO FA AS WE KNOW W = 5 J W = 0 W = -5 J 5 Potential Energ For an conservative force F we can define a potential energ function U in the following wa: W = F. dr = - U The work done b a conservative force is equal and opposite to the change in the potential energ function. This can be written as: r U U = U -U = -W =- r F. dr r r U 6 Page 8

9 Gravitational Potential Energ We have seen that the work done b gravit near the Earth s surface when an object of mass m is lifted a distance is W g = -mg The change in potential energ of this object is therefore: U = -W g = mg j m W g = -mg 7 Gravitational Potential Energ So we see that the change in U near the Earth s surface is: U = -W g = mg = mg( - ). So U = mg + U 0 where U 0 is an arbitrar constant. Having an arbitrar constant U 0 is equivalent to saing that we can choose the location where U = 0 to be anwhere we want to. j m W g = -mg 8 Page 9

10 Potential Energ ecap: For an conservative force we can define a potential energ function U such that: S U = U -U = -W =- F. dr S The potential energ function U is alwas defined onl up to an additive constant. You can choose the location where U = 0 to be anwhere convenient. 9 Conservative Forces & Potential Energies (stuff ou should know): Force F Work W(-) Change in P.E U = U -U P.E. function U ^ F g = -mg j -mg( - ) mg( - ) mg + C GMm F g = r ^ GMm GMm GMm + C F s = -kx k ( x x ) k ( x x ) kx + C ( is the center-to-center distance, x is the spring stretch) 0 Page 0

11 Lecture 6, Act 3 Potential Energ All springs and masses are identical. (Gravit acts down). Which of the sstems below has the most potential energ stored in its spring(s), relative to the relaxed position? (a) (b) (c) same () () Lecture 6, Act 3 Solution The displacement of () from equilibrium will be half of that of () (each spring exerts half of the force needed to balance mg) 0 d d () () Page

12 Lecture 6, Act 3 Solution The potential energ stored in () is kd = kd The potential energ stored in () is ( ) kd = kd The spring P.E. is twice as big in ()! 0 d d () () 3 Conservation of Energ If onl conservative forces are present, the total kinetic plus potential energ of a sstem is conserved, i.e. the total mechanical energ is conserved. (note: E=E mechanical throughout this discussion) E = K + U E = K + U = W + U using K = W = W + (-W) = 0 using U = -W E = K + U is constant!!! Both K and U can change, but E = K + U remains constant. But we ll see that if non-conservative forces act then energ can be dissipated into other modes (thermal,sound) 4 Page

13 Example: The simple pendulum Suppose we release a mass m from rest a distance h above its lowest possible point. What is the maximum speed of the mass and where does this happen? To what height h does it rise on the other side? m h h v 5 Example: The simple pendulum Kinetic+potential energ is conserved since gravit is a conservative force (E = K + U is constant) Choose = 0 at the bottom of the swing, and U = 0 at = 0 (arbitrar choice) E = / mv + mg = 0 h h v 6 Page 3

14 Example: The simple pendulum E = / mv + mg. Initiall, = h and v = 0, so E = mgh. Since E = mgh initiall, E = mgh alwas since energ is conserved. = 0 7 Example: The simple pendulum / mv will be maximum at the bottom of the swing. So at = 0 / mv = mgh v = gh v = gh = h = 0 h v 8 Page 4

15 Example: The simple pendulum Since E = mgh = / mv + mg it is clear that the maximum height on the other side will be at = h = h and v = 0. The ball returns to its original height. = h = h = 0 9 Example: The simple pendulum The ball will oscillate back and forth. The limits on its height and speed are a consequence of the sharing of energ between K and U. E = / mv + mg = K + U = constant. 30 Page 5

16 Example: The simple pendulum We can also solve this b choosing = 0 to be at the original position of the mass, and U = 0 at = 0. E = / mv + mg. = 0 h h v 3 Example: The simple pendulum E = / mv + mg. Initiall, = 0 and v = 0, so E = 0. Since E = 0 initiall, E = 0 alwas since energ is conserved. = 0 3 Page 6

17 Example: The simple pendulum / mv will be maximum at the bottom of the swing. So at = -h / mv = mgh v = gh v = gh Same as before! = 0 = -h h v 33 Example: The simple pendulum Since / mv -mgh= 0it is clear that the maximum height on the other side will be at = 0 and v = 0. The ball returns to its original height. = 0 Same as before! 34 Page 7