Physic 31 Lecture 11 Main points o last lecture: Work: W F s cos( θ ) F x F v Kinetic energy: θ 1 s v mv Work-energy theorem: W total Potential energy: PE PE PE mg h Main points o today s lecture: Potential energy PE PE PE mg y y Conservative orces and the Conservation o energy: E + PE + PE Varying orces potential energy o a spring PE 1 kx ( ).
Example A 1 kg block is thrown downward with an initial velocity o 1 m/s. What is the kinetic energy and speed o the block when it strikes the ground 1 m below. Would it matter whether the block were thrown horizontally as in trajectory 1 or at other angles such as trajectories or 3? 1 m/s 1 m 1 mv Fig. 5.15, p. 17 Slide 16 148J; Work g + ( mg)(1m) 1 (1kg)(1m / s) v ( mg)( 1m) + (1kg)(9.8N )(1m) (148J) 1 kg 17.m / s 148J The inal speed and kinetic energy are independent o the initial angle.
Potential energy For certain orces, the work done by the orce in going rom position (x,y ) to position (x,y) depends only the displacement and not on the path taken. Such a orce is called conservative. gravity is such a orce. Consider the work on a mass m under a displacement s at an angle θ with respect to the vertical as shown below: s v (x,y ) θ (x,y) W gravity v F g mg downwards ( ) s mg( y y) Fgravity cos θ From work - energy theorem : mg ( y y ) From this we can see that being higher initially means that you can have a higher inal kinetic energy. Thus, mg(y -y) is the part o the stored potential energy, which was changed into kinetic energy as the object moves rom its initial to its inal position. The potential energy only depends on the dierence in height between the initial and inal positions, i.e. on the vertical component o the displacement.
Example What dierence between the PE o a kg car raised 1 m in the air and that o the same car on the ground? How much work would it require to lit it to that height? PE mgh PE 1.96x1 W PE 1.96x1 (kg)(9.8m / s 5 J 5 J )(1m)
Gravitational potential energy The most important point o a potential energy is that is only a unction o the position and not o the path taken to get there. I we break the path o the pail in to vertical sections the potential energy change is just mg y and horizontal sections where the potential energy remains constant, we can see that the potential energy depends on the total vertical displacement and is independent o the path over which it is achieved. PE-PE mg(y-y ) (x,y) (x,y ) Thus we can deine PE or some y and then PEmg(y-y ) thereater. PE is W grav. It is the work one would need to do to move the pail rom A to B.
Conceptual problem At the bowling alley, the ball-eeder mechanism must exert a orce to push the bowling balls up a 1.-m long ramp.the ramp leads the balls to a chute.5 m above the base o the ramp. Approximately how much orce must be exerted on a 5.-kg bowling ball? a) N b) 5 N c) 5 N d) 5. N e) impossible to determine work PE F s F PE s mg y s J 1m ()( 5 9.8)(.5) 5N
Conservation o energy Up to an additive constant, we can deine PEmgy. It is equal in magnitude but opposition in sign to the work being done by gravity.. Then PE PE We can deine the total energy E as the sum o kinetic and potential energy. Then we have E + PE + PE Thus, when all work being done by orces in a problem can be expressed in terms o a potential energy, the total energy is conserved (i.e. remains constant). This is true or gravity and many other orces, but not or riction, or example.
Example A water slide is constructed so that swimmers, starting rom rest at the top o the slide, leave the end o the slide traveling horizontally. As the drawing shows, one person is observed to hit the water 5. m rom the end o the slide in a time o.5 s ater leaving the slid. Ignoring riction and air resistance, ind the height H in the drawing. x 5m x v t; v 1m / s t.5s h height at bottom o slide : 1 - h gt h (4.9m / s) Conservation o energy : 1 mv mg (.5s) v g 1.3m ( H h) ; H h + 6.33m
quiz A.4-kg bead slides on a curved wire, starting rom rest at point A in the igure below. I the wire is rictionless, ind the speed o the bead at C. A C B a) 5. m/s b) 1.4 m/s c) 3 m/s d) 7.7 m/s 1 mv v + PE PE PE mg(y g(y y y + PE mg(y ) y ) ) 7.7m / s
Work and PE or non-constant orces When the orce is not constant, the work can still be computed or small displacements x over which orce is approximately constant: W Fx x To get the total work, we add the contributions or the steps: W Fx x W is just the area under the curve, i the orce is conservative the change in potential energy is W. It is equal to the work one would need to do against the orce to move the object over the chosen path.