Work and Energy Problems

Transcription

1 09//00 Multiple hoice orce o strength 0N acts on an object o ass 3kg as it oes a distance o 4. I is perpendicular to the 4 displaceent, the work done is equal to: Work and Energy Probles a) 0J b) 60J c) 80J d) 600J e) 400J Since the orce is perpendicular to the displaceent, there is no work done by this orce. Multiple hoice Multiple hoice Under the inluence o a orce, an object o ass 4kg accelerates ro 3 /s to 6 /s in 8s. ow uch work was done on the object during this tie? a) 7J b) 54J c) 7J d) 96J e) an not be deterined ro the inoration gien This is a job or the work energy theore. bo o ass slides down a rictionless inclined plane o length L and ertical height h. What is the change in graitational potential energy? a) -gl b) -gh c) -gl/h d) -gh/l e) -ghl The length o the rap is irreleant, it is only the change in height W K i 4 kg 6 3 s s 54J

2 09//00 Multiple hoice Multiple hoice n object o ass is traelling at constant speed in a circular path o radius r. how uch work is done by the centripetal orce during one-hal o a reolution? a) π J b) π J c) 0J d) π r J e) π /r J Since centripetal orce always points along a radius towards the centre o the circle, and the displaceent is always along the path o the circle, no work is done. While a person lits a book o ass kg ro the loor to a tabletop,.5 aboe the loor, how uch work does the graitational orce do on the block? a) -30J b) -5J c) 0J d) 5J e) 30J The graitational orce points downward, while the displaceent i upward W gh kg9.8.5 s 30J Multiple hoice Multiple hoice block o ass 3.5kg slides down a rictionless inclined plane o length 6 that akes an angle o 30 o with the horizontal. I the block is released ro rest at the top o the incline, what is the speed at the botto? a) 4.9 /s b) 5. /s c) 6.4 /s d) 7.7 /s e) 9. /s The work done by graity is equal to the change in Kinetic Energy. W K gh gh gh i 9.8 sin 306 s 7.7 s block o ass 3.5kg slides down an inclined plane o length 6 that akes an angle o 60 o with the horizontal. The coeicient o kinetic riction between the block and the incline is 0.3. I the block is released ro rest at the top o the incline, what is the speed at the botto? a) 4.9 /s b) 5. /s Ki Ui W K U c) 6.4 /s d) 7.7 /s 0 gh L 0 e) 9. /s g Lsin g cos L This is a conseration o Mechanical Energy, including the negatie work done by riction. glsin cos 9.8 6sin cos60 s 9. s

3 09//00 Multiple hoice Multiple hoice n astronaut drops a rock ro the top o a crater on the Moon. When the rock is halway down to the botto o the crater, its speed is what raction o its inal ipact speed? a) /4 b) /4 c) / d) / e) / Total energy is consered. Since the rock has hal o its potential energy, its kinetic energy at the halway point is hal o its kinetic energy at ipact. K orce o 00N is required to keep an object at a constant speed o /s across a rough loor. ow uch power is being epended to aintain this otion? a) 50W b) 00W c) 00W d) 400W e) annot be deterined with gien inoration Use the power equation P 00N s 400W Understanding Understanding opare the work done on an object o ass.0 kg a) In liting an object 0.0 b) Pushing it up a rap inclined at 30 0 to the sae inal height opare the work done on an object o ass.0 kg a) In liting an object liting pushing liting W d gh s.0kg J 00J 3

4 09//00 Understanding 0 opare the work done on an object o ass.0 kg a) In liting an object 0.0 b) Pushing it up a rap inclined at 30 0 to the sae inal height bo slides down an inclined plane (incline angle = 40 0 ). The ass o the block,, is 35 kg, the coeicient o kinetic riction between the bo and the rap, u k, is 0.3, and the length o the rap, d, is pushing 0.0 a) ow uch work is done by graity? b) ow uch work is done by the noral orce? c) ow uch work is done by riction? d) What is the total work done? The distance traelled up the rap 0.0 d 0 sin30 pplied orce applied g sin Work W d applied W g sin d W.0kg 9.8 sin 30 0 s W 00J Solution Solution bo slides down an inclined plane (incline angle = 40 0 ). The ass o the block,, is 35 kg, the coeicient o kinetic riction between the bo and the rap, u k, is 0.3, and the length o the rap, d, is 8. a) ow uch work is done by graity? bo slides down an inclined plane (incline angle = 40 0 ). The ass o the block,, is 35 kg, the coeicient o kinetic riction between the bo and the rap, u k, is 0.3, and the length o the rap, d, is 8. b) ow uch work is done by the noral orce? Recall Work = orce distance with the orce being parallel to the distance, that is, the coponent down the rap Wgraity d g sin 760J d s 35kg 9.8 sin 408 Since the noral orce is perpendicular to the displaceent, NO WORK IS DONE. g 4

5 09//00 Solution Solution bo slides down an inclined plane (incline angle = 40 0 ). The ass o the block,, is 35 kg, the coeicient o kinetic riction between the bo and the rap, u k, is 0.3, and the length o the rap, d, is 8. c) ow uch work is done by riction? bo slides down an inclined plane (incline angle = 40 0 ). The ass o the block,, is 35 kg, the coeicient o kinetic riction between the bo and the rap, u k, is 0.3, and the length o the rap, d, is 8. d) What is the total work done? Recall Work = orce distance. Total Work = su o all works W d riction u g cos 0.335kg 9.8 cos J k d s W W W W total graity Noral riction 760J 0J 630J 30J Question truck oes with elocity 0 = 0 /s on a slick road when the drier applies the brakes. The wheels slide and it takes the car 6 seconds to stop with a constant deceleration. a) ow ar does the truck trael beore stopping? b) Deterine the kinetic riction between the truck and the road. truck oes with elocity 0 = 0 /s on a slick road when the drier applies the brakes. The wheels slide and it takes the car 6 seconds to stop with a constant deceleration. a) ow ar does the truck trael beore stopping? b) Deterine the kinetic riction between the truck and the road. d 0 t s s s 30 5

6 09//00 truck oes with elocity 0 = 0 /s on a slick road when the drier applies the brakes. The wheels slide and it takes the car 6 seconds to stop with a constant deceleration. a) ow ar does the truck trael beore stopping? b) Deterine the kinetic riction between the truck and the road. Question You and your bicycle hae a cobined ass o 80.0kg. When you reach the base o an bridge, you are traeling along the road at 5.00 /s. at the top o the bridge, you hae clibed a ertical distance o 5.0 and hae slowed to.50 /s. Ignoring work done by any riction: a) ow uch work hae you done with the orce you apply to the pedals? irst we need the acceleration the car eperiences. a t 0 0 s s 6s.6 s Since the only acceleration orce eperienced by the car is riction a a u n a ug a ug a u g You and your bicycle hae a cobined ass o 80.0kg. When you reach the base o an bridge, you are traeling along the road at 5.00 /s. at the top o the bridge, you hae clibed a ertical distance o 5.0 and hae slowed to.50 /s. Ignoring work done by any riction: a) ow uch work hae you done with the orce you apply to the pedals? onseration o energy states that initial kinetic energy equals inal kinetic energy plus work done by graity less work done by you W E T K U K Ui i W K K U U p i i i gh kg kg s s s 366.8J J Question The igure below shows a ballistic pendulu, the syste or easuring the speed o a bullet. bullet o =.0 g is ired into a block o wood with ass o M=3.0kg, suspended like a pendulu, and akes a copletely inelastic collision with it. ter the ipact o the bullet, the block swings up to a aiu height.00 c. Deterine the initial elocity o the bullet? Stage (conseration o Moentu) p p i i Mi V MV 0.00kg kg V 300V Stage (conseration o Energy) K U i M V M gh V s V 0.66 s Thereore: s 879 s 6

7 09//00 Question We hae preiously used the ollowing epression or the aiu height h o a projectile launched with initial speed 0 at initial angle θ: h sin 0 g Derie this epression using energy considerations We hae preiously used the ollowing epression or the aiu height h o a projectile launched with initial speed 0 at initial angle θ: Derie this epression using energy considerations This initially appears to be an easy proble: the potential energy at point is U =gh, so it ay see that all we need to do is sole the energyconseration equation K +U =K +U or U. oweer, while we know the initial kinetic energy and potential energies (K =½ =½ 0 and U =0), we don t know the speed or kinetic energy at point. We will need to break down our known alues into horizontal and ertical coponents and apply our knowledge o kineatics to help. h sin 0 g We hae preiously used the ollowing epression or the aiu height h o a projectile launched with initial speed 0 at initial angle θ: Derie this epression using energy considerations We can epress the kinetic energy at each point in the ters o the coponents using: y K K y y Question ousin Vinney skateboards down a playground rap. e (5.0 kg) oes through a quarter-circle with radius R=3.00. a) Deterine his speed at the botto o the rap b) Deterine the noral orce that acts on hi at the botto o the cure. c) Suppose the rap is not rictionless and that Vinney s speed at the botto is only 6.00 /s. what work was done by the riction orce acting on hi? onseration o energy then gies: K U Kgh U y 0 sin gh y 0 y 0 sin gh h g yh gy h gh y ut, we recall that urtherore, ut, the -coponents since projectile y is just the y- coponent o elocity does o initial not has change, zero ertical elocity: elocity at highest y = 0 sin(θ) points, y =0 h sin 0 g 7

8 09//00 ousin Vinney skateboards down a playground rap. e (5.0 kg) oes through a quartercircle with radius R=3.00. a) Deterine his speed at the botto o the rap b) Deterine the noral orce that acts on hi at the botto o the cure. c) Suppose the rap is not rictionless and that Vinney s speed at the botto is only 6.00 /s. what work was done by the riction orce acting on hi? ousin Vinney skateboards down a playground rap. e (5.0 kg) oes through a quartercircle with radius R=3.00. a) Deterine his speed at the botto o the rap b) Deterine the noral orce that acts on hi at the botto o the cure. c) Suppose the rap is not rictionless and that Vinney s speed at the botto is only 6.00 /s. what work was done by the riction orce acting on hi? ro onseration o Energy The ree body diagra o the noral is through-out his journey is: K U K U 0 gr 0 gr s 7.67 s a y N G R gr N g R g g 3g c gr ousin Vinney skateboards down a playground rap. e (5.0 kg) oes through a quartercircle with radius R=3.00. a) Deterine his speed at the botto o the rap b) Deterine the noral orce that acts on hi at the botto o the cure. c) Suppose the rap is not rictionless and that Vinney s speed at the botto is only 6.00 /s. what work was done by the riction orce acting on hi? n object o ass.0 kg traelling at 5.0 /s enters a region o ice where the coeicient o kinetic riction is 0.0. Use the Work Energy Theore to deterine the distance the object traels beore coing to a halt. The noral orce does no work, but the riction orce does do work. Thereore the non-graitational work done on Vinney is just the work done by riction. The ree body diagra o the noral is through-out his journey is: K U W K U W K U K U 0 0 gh 5.0 kg kg s s 85J.0 kg 8

9 09//00 Solution n object o ass.0 kg traelling at 5.0 /s enters a region o ice where the coeicient o kinetic riction is 0.0. Use the Work Energy Theore to deterine the distance the object traels beore coing to a halt. N orces We can see that the objects weight is balanced by the noral orce eerted by the ice. Thereore the only work done is due to the riction acting on the object. Let s deterine the riction orce. u k N u g k 0.0.0kg 9.8 s 0.98N Now apply the work Energy Theore and sole or d W KE d i 3 g 0.98N d.0kg 0.0kg 5.0 s s.5j d 0.98N bo o ass is released ro rest at point, the top o a long rictionless slide. Point is at height aboe the leel points and. lthough the slide is rictionless, the horizontal surace ro point to is not. The coeicient o kinetic riction between the bo and this surace is u k, and the horizontal distance between points and is. a) ind the speed o the bo when its height aboe the ground is / b) ind the speed o the bo when it reaches. c) Deterine the alue o u k, so that it coes to a rest at d) Deterine the alue o u k i was at a height o h aboe the ground. e) I the slide was not rictionless, deterine the work done by riction as the bo oed ro to i the speed at was ½ o the speed calculated in b) u k bo o ass is released ro rest at point, the top o a long rictionless slide. Point is at height aboe the leel points and. lthough the slide is rictionless, the horizontal surace ro point to is not. The coeicient o kinetic riction between the bo and this surace is u k, and the horizontal distance between points and is. a) ind the speed o the bo when its height aboe the ground is / bo o ass is released ro rest at point, the top o a long rictionless slide. Point is at height aboe the leel points and. lthough the slide is rictionless, the horizontal surace ro point to is not. The coeicient o kinetic riction between the bo and this surace is u k, and the horizontal distance between points and is. b) ind the speed o the bo when it reaches. E U K g 0 g total G U K E G total g g g g E U K g 0 g total G U K E G total 0 g g g u k u k 9

10 09//00 bo o ass is released ro rest at point, the top o a long rictionless slide. Point is at height aboe the leel points and. lthough the slide is rictionless, the horizontal surace ro point to is not. The coeicient o kinetic riction between the bo and this surace is u k, and the horizontal distance between points and is. c) Deterine the alue o u k, so that it coes to a rest at W K W i i d i guk i u k i uk g g bo o ass is released ro rest at point, the top o a long rictionless slide. Point is at height aboe the leel points and. lthough the slide is rictionless, the horizontal surace ro point to is not. The coeicient o kinetic riction between the bo and this surace is u k, and the horizontal distance between points and is. d) Deterine the alue o u k i was at a height o h aboe the ground. K U W K U g 0 L 0 gh g u g cos L gh k u cos L h k L u k h uk Lcos h bo o ass is released ro rest at point, the top o a long rictionless slide. Point is at height aboe the leel points and. lthough the slide is rictionless, the horizontal surace ro point to is not. The coeicient o kinetic riction between the bo and this surace is u k, and the horizontal distance between points and is. e) I the slide was not rictionless, deterine the work done by riction as the bo oed ro to i the speed at was ½ o the speed calculated in b) u k U K W U K U K W g 0W g g g W 4 g 3 W g g 4 4 n acrobat swings ro the horizontal. When the acrobat was swung an angle o 30 0, what is his elocity at that point, i the length o the rope is L? ecause graity is a conseratie orce (the work done by the rope is tangent to the otion o trael), Graitational Potential Energy is conerted to Kinetic Energy. U K U i i gl glsin 30 gl gl gl 30 L It s too diicult to use entripetal cceleration hlsin 30 0

11 09//00 The diagra below shows a roller-coaster ride which contains a circular loop o radius r. car (ass ) begins ro rest ro Point and oes down a rictionless rap to Point where it enters a ertical loop (rictionless), traelling once around the circle ( to to D to E back to ) it then traels along a lat portion ro to (which is not rictionless). a) ind the centripetal acceleration o the car when it is at Point b) Deterine the speed o the car when its position relatie to is speciied by the angle θ shown in the diagra. c) What is the iniu cut-o speed c that the car ust hae at D to ake it around the loop? d) What is the iniu height necessary to ensure that the car akes it around the loop? e) I =6r and the coeicient o riction between the car and the lat portion o the track ro to is 0.5, how ar along the lat portion o the track will the car trael beore coing to rest at? E D θ The diagra below shows a roller-coaster ride which contains a circular loop o radius r. car (ass ) begins ro rest ro Point and oes down a rictionless rap to Point where it enters a ertical loop (rictionless), traelling once around the circle ( to to D to E back to ) it then traels along a lat portion ro to (which is not rictionless). a) ind the centripetal acceleration o the car when it is at Point The centripetal acceleration is /r E D θ K U K U c 0 g gr g r g r r r g r a r The diagra below shows a roller-coaster ride which contains a circular loop o radius r. car (ass ) begins ro rest ro Point and oes down a rictionless rap to Point where it enters a ertical loop (rictionless), traelling once around the circle ( to to D to E back to ) it then traels along a lat portion ro to (which is not rictionless). b) Deterine the speed o the car when its position relatie to is speciied by the angle θ shown in the diagra. KU K U D 0 g g r rcos 80 E θ 0 g g r r cos The car s height aboe the botto o the track is gien by g r cos h=r+rcos(80 0 -θ). g r cos The diagra below shows a roller-coaster ride which contains a circular loop o radius r. car (ass ) begins ro rest ro Point and oes down a rictionless rap to Point where it enters a ertical loop (rictionless), traelling once around the circle ( to to D to E back to ) it then traels along a lat portion ro to (which is not rictionless). c) What is the iniu cut-o speed c that the car ust hae at D to ake it around the loop? E D θ When the car reaches D, the orces acting on the car are its weight, g, and the downward Noral orce. These orce just atch the entripetal orce with the Noral equalling zero at the cut-o elocity. G N cut o g r 0 rg gr

12 09//00 The diagra below shows a roller-coaster ride which contains a circular loop o radius r. car (ass ) begins ro rest ro Point and oes down a rictionless rap to Point where it enters a ertical loop (rictionless), traelling once around the circle ( to to D to E back to ) it then traels along a lat portion ro to (which is not rictionless). d) What is the iniu height necessary to ensure that the car akes it around the loop? K U K U D D E D pply the cut-o speed ro c) to the conseration o Mechanical Energy. g g r 0 g gr g r 5 gr 5 r The diagra below shows a roller-coaster ride which contains a circular loop o radius r. car (ass ) begins ro rest ro Point and oes down a rictionless rap to Point where it enters a ertical loop (rictionless), traelling once around the circle ( to to D to E back to ) it then traels along a lat portion ro to (which is not rictionless). e) I =6r and the coeicient o riction between the car and the lat portion o the track ro to is 0.5, how ar along the lat portion o the track will the car trael beore coing to rest at? K U K U D 0 g(6 r) 0 E alculate the Kinetic Energy at, then deterine how uch work riction ust do to reoe this energy. 6gr g k 6 gr 6r 6r r 0.5 k