PHYSICS 211 MIDTERM II 12 May 2004

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1 PHYSIS IDTER II ay 004 Exa i cloed boo, cloed note. Ue only your forula heet. Write all wor and anwer in exa boolet. The bac of page will not be graded unle you o requet on the front of the page. Show all your wor and explain your reaoning (except on #). Partial credit will be given (not on #). No credit will be given if no wor i hown (not on #). If you have a quetion, raie your hand or coe to the front.. (0 point) or each of thee ultiple choice quetion, indicate the correct repone (, B,, or D (where needed)) on the page for proble in your exa boolet. i) bloc i on a rap that i at an angle to the horizontal. the angle i increaed, the bloc reain tationary (due to the tatic frictional force on it fro the rap) until a axiu angle ax when the bloc begin to lide down the rap. If we double the a of the bloc, and repeat the experient, i the new axiu angle greater than, le than, or the ae a the original axiu angle? ) Greater than. B) Le than. ) The ae a. ii) ball i tied to the end of a tring and wung in a vertical circle on earth. The tenion in the tring i contant. I the peed of the ball larger at the top of the circle, the botto of the circle, or the ae at top and botto? ) Larger at top. B) Larger at botto. ) Sae at top and botto. iii) n apple fall fro a tree and hit Sir Iaac on the head. The apple fall due to the force of gravity that the earth exert on the apple. I the agnitude of the force of gravity that the apple exert on the earth greater than, le than, or the ae a the agnitude of the force of gravity that the earth exert on the apple? ) Greater than. B) Le than. ) The ae a. iv) Kepler' nd law tate that a line that connect a planet to the Sun weep out equal area in equal tie. Thi iplie that the planet ha it larget peed when it i in which relative poition in it it? ) loet to the Sun. B) arthet fro the Sun. ) Sae peed at all point in it. v) The plot at right how the value of a force that will act on a particle at the correponding value of x. The force i along the x- axi and the particle tart at x 0 with a poitive velocity. t which one of the four labeled point i the inetic energy of the particle the greatet? B D x

2 . (0 point) all ball of a hang by a cord of length L fro the ceiling. The ball wing in a horizontal circle uch that the cord ae an angle with the vertical, a hown at right. The tie for the ball to ae one revolution i τ. Expre anwer below in ter of, L, g, τ, and other contant a needed. a) ind an expreion for the angle. L b) ind an expreion for the tenion in the cord. c) If the ball revolve fater, doe the angle increae or decreae? 3. (0 point) box of a 5.0 g i thrown down an inclined plane with an initial peed v 0 3 /. fter traveling a ditance d 4 down the incline, the box hit a pring ( 08 N/) whoe other end i anchored to the incline. There i friction (µ 0.5) between the box and incline during the firt part of the otion, but not once the pring i contacted. The incline i at an angle above the horizontal, where in 3/5. ue g 0 /. Ignore the ize of the box. a) What i the peed of the box a it contact the pring? b) How far i the pring copreed before the box coe to ret? d µ 4. (0 point) Planet laire it around the tar ntare in a circular it at a ditance of r 3.0 x 0. Planet laire ha a radiu of R 8.0 x 0 6 and a a of 6.0 x 0 4 g and ntare ha a a of 6.0 x 0 30 g. ue G 6.0 x 0 - N /g. a) What i the period of Planet laire' it? b) What i the gravitational acceleration on Planet laire? c) Where i the net gravitational force due to Planet laire and ntare zero? 5. (0 point) bloc of a i in contact with a bloc of a a hown at right. There i no friction between bloc and the floor, but there i friction between bloc and bloc, with a tatic coefficient of µ. n external force i applied to the bac of bloc and the two bloc ove to the right, without bloc lipping down the front of bloc. µ a) What i the acceleration of the two-bloc yte? b) What iniu force in ut be applied to bloc to prevent bloc fro lipping with repect to bloc? c) What i the noral force of the floor on bloc?

3 PHYSIS IDTER II SOLUTIONS ay 004. i) The angle ax i the angle where the axiu poible frictional force ( f,ax µ N µ gco) equal the weight coponent ( gin) down the rap. Doubling the a will double both of thee together, leaving the equality ( ginµ gco µ tan) unchanged. Hence the angle i unchanged. ii) t the top of the circle T and g both point down, in the ae direction a the centripetal acceleration. t the botto of the circle T point up and g point down, while the centripetal acceleration point up. pplying a to each ituation give T + g v / r at the top and T g v / r at the botto. Since T i contant, the peed will be larger at the top of the circle. iii) The force are equal and oppoite according to Newton third law. iv) n elliptical it i hown at right. Kepler' nd law of equal area in equal tie ean that the area d wept out in a tie dt i a contant, or that d/dt i contant. or all tie, we can approxiate the area d a an iocele triangle with height r and bae v dt, where v i the coponent of velocity perpendicular to the poition vector. We thu find that rv i a contant during the ital otion. Thi iplie that the larget peed occur when the planet i cloet to the un. r d v) B The change in inetic energy i given by the integral of the force, or the area under the curve. If that area increae a x increae, then the inetic energy will increae. we integrate fro x 0, the area increae until we reach point B, after which it decreae, o the inetic energy will increae until point B and then decreae.

4 PHYSIS IDTER II SOLUTIONS ay 004. the ball travel in a circle it experience an acceleration toward the center of the circle. Hence there ut be a force toward the center of the circle acting on the ball. Since gravity point downward, only the tenion in the cord can provide thi centripetal force. Hence the ball ut hang away fro the center of the circle a hown below, with the horizontal coponent of the tenion providing the needed centripetal force. L T a r g a) ro the force diagra above, we can write down the two equation of otion for the vertical and horizontal direction. Note that the radiu of the circle i r L in. Tin a v r Tco g 0 Tco g Now divide thee two equation and write the velocity in ter of the period: in in π π π v r 4 r 4 L co rg rg τ co π 4 L arcco 4π L b) Ue the vertical equation of otion to find the tenion: T g T g τ co g 4π L L T 4π τ c) If the ball revolve fater, then the period τ decreae. The equation for tell u that if τ decreae, then co alo decreae. The function co between 0 and π/ decreae a increae. T hu a τ decreae, increae.

5 PHYSIS IDTER II SOLUTIONS ay onider the otion in tep: 0-> gravity and friction act, -> gravity and pring act. d 0 µ N f (0->) or (->) g a) The peed of the box at pring contact (point ) i found by conidering the wor done by gravity and friction: r r Wg gd π g d co( ) gd in r r Wf f d fd µ gd co K W + W K K v v gd in µ gd co gd in µ co v v + gd in µ v v 0 0 g f 0 ( co) v + gdin µ co b) During copreion of the pring the bloc travel fro x 0 to x. Gravity alo act during thi travel, giving: Wg gx in, W x x x K W + W K K 0 v gx in x x g x gx in v 0 gin± ( gin) + v gin v ± + g in 5g x 08N5 ± + 08N 5g x [ ] 3

6 PHYSIS IDTER II SOLUTIONS ay a) The force of gravity provide the centripetal force to eep the planet oving in a circle. obine Newton' econd law with hi law of gravity to find the circular ital peed. a G v c π r r r τ τ 4π G 3 r 3 4π G r 4π τ N g ( g) 7 τ yr b) The gravitational acceleration i found by equating gravity to g : g g G g R 4 G N g ( g) g R 6 ( ) g c) The net gravitational force will be zero oewhere between laire and ntare, where both pull on an object and the force cancel (ee diagra below): 0 G G G Rx Rx ( Rx) ( Rx) Rx Rx Rx Rx Rx r R x R x 4

7 PHYSIS IDTER II SOLUTIONS ay The free body diagra for each a are hown below. f N N g N g f a) Neither bloc accelerate vertically, and they both accelerate to the right with acceleration a, giving the equation of otion N a f g 0 N a N f g 0 Plugging the firt equation into the third allow u to find the acceleration of the yte: a a a+ a ( + ) a a + b) To find the iniu poible force in before bloc lip, ue the equation decribing the liitation of tatic friction: f µ N g µ a g µ + + g µ g in ( + ) µ c) The noral force i found fro the fourth equation of otion N f g 0 N g+ f g+ g N + g 5