Power and Gravitational Potential Energy

Transcription

1 Power and Gravitational Potential Energ REVIE of Last eek s Lecture Scalar Product A B AB cos A B x x A B A z B B cos B z A ork Fs F s constant force parallel to displacement force not parallel to displacement ork-energ Theorem: F tot net s where F net F i i and is the "kinetic energ"

2 i-clicker In each case below, an arrow is shot from the top of a building either up at a 45 o angle, straight out horizontall, or down at a 45 o angle. All arrows are identical and are shot at the same speed. Ignore air resistance. Rank these arrows on the basis of their speeds when hitting the ground. A) C = F > A = E > B = D B) C = F > A = E > B > D C) F > C > A > E > B > D D) C > E > B > D > F > A tot mgh

3 OR DONE BY VARYING FORCE For varing force: F x F(x i ) x x x i F(x) x lim F( x ) x tot x0 tot i x x i i F( x) dx EXAMPLE: ork done to stretch spring F x x F(x) X kx x F X F ( x) dx o x o X kxdx X kx 0 kx hat is the work done b the spring?

4 i-clicker cow A pumpkin falls straight down. As it falls, the gravitational force: cow s F F s A. Does positive work B. Does negative work C. Does no work D. First does negative work, then positive work E. First does positive work, then negative work

5 POER Rate at which work is done (Rate at which energ changes) P AV t d P lim t0 t dt F x P F t t v Units: att = Joule/sec EXAMPLE: A 5 kg mass falls 3. meters. hat is the average power generated b gravit? s Fs mgs s (3. m) gt t 0.64 s 0.8 s g (0 m/s ) P AV (5 kg)(0 m/s (60 J) (0.8 s) 00 The can stops the mass in /0 the time it takes to fall. hat is the average power transferred to the can? 60 t J 0.08 s P AV (60 J) (0.08 s) )(3. m) 60 J 000 k

6 GRAVITATIONAL POTENTIAL ENERGY Suppose ou lift an object of mass m in the presence of gravit: F other mg tot Fnetd ( Fother mg ) d F otherd other mgd grav tot other mg ( ) Clearl mg has units of energ mg = U g () = gravitational potential energ of mass m at height. other U g g U Let U g E = Total (Mechanical) Energ Then: other E E = Change in Total Energ

7 Suppose: Then: F other 0 0 E E other TOTAL ENERGY IS CONSERVED ( if other 0) EXAMPLE: A baseball is thrown straight up. After traveling 5m, it has a speed of 0 m/s. hat was the initial speed of the baseball? 5 m v v E = + U = (/) m v + mg E = + U = (/) m v + mg E = E Solve for v and plug in numbers (m cancels): v = 30 m/s

8 Example (continued) hat is the maximum height the ball will reach? E 3 = 3 + U 3 = 0 + mg max E = + U = (/)m(30 m/s) + mg E 3 = E mg max = (/)m(900 m /s ) + mg mg( max - ) = (/)m(900 m /s ) ( max - ) = (450 m /s )/ (0 m/s ) ( max - ) = h = 45 m

9 i-clicker Two blocks are placed on a frictionless ramp and held against a spring that is compressed one-half meter. The blocks are then released from rest, shooting the blocks up the ramp. Is the height that the block travels up the ramp greater in Case A, greater in Case B, or the same in both cases? A) The block in Case B travels further. B) Both blocks travel the same distance. C) The block in Case A travels further. D) It is impossible to predict without knowing the angle of the ramp. other E E mg ( h h)

10 i-clicker A 00-N box is initiall moving upward at 4 m/s. A student is appling a vertical force of 80 N to the box as shown. hich of the following is correct for the earth-box sstem as the box moves upward a distance of m? A. The kinetic energ will increase. B. The potential energ will increase. C. The kinetic energ will not change. D. The potential energ will not change. E. The total mechanical energ will not change.

11 HAT ABOUT MOTION ALONG A CURVED PATH? grav F grav ds d dx ds dxˆi dˆj ( mgˆ) j ( dxˆi dˆ) j ( mg ) d mg ( ) U g Ug CHANGE of gravitational potential energ onl depends on change of elevation other E E U mgh E

12 i-clicker 0 U g U g 0 U g U g v mg mg g

13 EXAMPLE: PROJECTILE MOTION M snow blower ejects snow with a velocit v v iˆ 0 ox v o ˆj v o v ox v o v h R x hat is the maximum height the snow flies? Initial state: o 0 0x v0 m( v ) U o 0 Final state (at maximum height): v v iˆ 0x 0 ˆj 0x U mgh o + U o = + U m( v 0x v 0 ) v0 gh 0x mgh h v 0 g

14 MOTION ALONG A CIRCULAR ARC Skate boarder drops into a (frictionless) quarter-pipe: hat is her speed when she exits? = 0 E E other mg n hat is normal? = 0 E = + U = (/) + 0 E = + U = 0 + mgr E = E (/) = mgr v = gr v gr If friction were present? f E E

15 i-clicker A skateboarder is launched b a giant spring initiall compressed a distance x as shown at right. His speed on the horizontal portion of the ramp is v. He then conducts a second launch with the spring initiall compressed a distance x. For the second launch, what can ou sa about the speed of the skateboarder on the horizontal portion of the ramp? A. The speed is v. B. The speed is v. C. The speed is 4v. D. None of the above. E. Cannot be determined. k kx ( x) m(v )

16 i-clicker A skateboarder is launched b a giant spring initiall compressed a distance x as shown at right. He rises to a height H after he leaves the ramp. He then conducts a second launch with the spring initiall compressed a distance x. For the second launch, how high will he rise? A. The height is H. B. The height is H. C. The height is 4H. D. None of the above. E. Cannot be determined. other kx k(x) E mgh E E mg (4H )

17 Phsics of

18 EXAMPLE: Roller Coaster Potential energ is function of x because is (x) : U(x) = mg(x) Plot U(x) on energ vs. x diagram At an point x, x a x b U(x) E m v ( x) E U( x) E mg( x) hat if energ = E? Then: x a < x < x b

19 Start INSIDE THE LOOP Top Free bod diagram h mg n At top of loop: min R v min i gr F i ma r hat is the minimum height h? mg R mg E top = top + U top = (/) min + mgr = (/)mgr + mgr = (5/)mgR E start = start + U start = 0 + mgh = E top n The smallest allowable speed at top of loop corresponds to n 0 mgh = (5/)mgR; h =.5 R

20 h OUTSIDE THE LOOP n mg Free bod diagram At top of n mg hill : R i v v max F ma i when mg n max 0 gr n mg R R So: E TOPmax = (/) max + mgr = (5/)mgR h max = (5/)R BUT: Need enough energ to get over the hill: x E TOPmin = U TOP = mgr = mgh min h min = R R < h <.5R

21 i-clicker A ball starts at rest at the top of the ramp and is allowed to roll down the frictionless track through the loop-the-loop as shown. At the instant it is at the top of the loop, the ball s total mechanical energ: A. Is greater than when it started B. Is less than when it started. C. Is equal to when it started. D. Is greater than it is at the bottom of the loop. E. Cannot tell without knowing the mass of the ball. No other forces, so : E E 0