PHYS - Summe 007 - Pofesso Caillault Homewok Solutions Chapte 9
3. Pictue the Poblem The owne walks slowly towad the notheast while the cat uns eastwad and the dog uns nothwad. Stategy Sum the momenta of the dog and cat using the component method. Use the known components of the total momentum to find its magnitude and diection. Let noth be in the y diection, east in the x diection. Solution. Use the component method of vecto addition to find the owne s momentum. Divide the owne s momentum by his mass to get the components of the owne s velocity 3. Use the known components to find the diection and magnitude of the owne s velocity p total p d + p c m d vd + m c vc 0.0 kg p total 5.0 kg m/s p 0 m v0 0 p total p total 5.0 kg m/s v 0 m 0 (.50 m/s ŷ) + ( 5.00 kg) ( 3.00 m/s ˆx ) ˆx + ( 50.0 kg m/s) ŷ 0.4 m/s ˆx + ( 50.0 kg m/s) ŷ 70.0 kg ˆx + ( 0.74 m/s) ŷ θ tan 0.74 0.4 73.3 v 0 ( 0.43 m/s) + ( 0.743 m/s) 0.746 m/s Insight We bent the ules of significant figues a bit in step 3 in ode to avoid ounding eo. The owne is moving much slowe than eithe the cat o the dog because of his lage mass. 5. Pictue the Poblem The baseball dops staight down, gaining momentum due to the acceleation of gavity. Stategy Detemine the speed of the baseball befoe it hits the gound, then use equation - to find the height fom which it was dopped. Solution. Use equation 9- to find the speed of the ball when it lands. (b) Use equation - to solve fo y 0. Let y 0 and v 0 0 v p m 0.780 kg m/s 0.50 kg ( 5.0 m/s) v v 0 g y y 0 y 0 v g 9.8 m/s 5.0 m/s.38 m Insight Anothe way to find the initial height is to use consevation of enegy, setting mgy 0 mv and solving fo y 0. 3. Pictue the Poblem The ball ebounds fom the floo in the manne indicated by the figue at ight. Stategy The impulse is equal to the change in the y-component of the momentum of the ball. Use equation 9-6 in the vetical diection to find the impulse. Solution Apply equation 9-6 in the y diection I Δp y mδv y m v 0 cos65 ( v 0 cos65 ) 0.60 kg ( 5.4 m/s) ( cos65 ).7 kg m/s Insight Thee is no impulse in the x diection because the ball does not change its hoizontal speed o
momentum. 9. Pictue the Poblem The two pieces fly in opposite diections at diffeent speeds.. Stategy As long as thee is no fiction the total momentum of the two pieces must emain zeo, as it was befoe the explosion. Combine the consevation of momentum with the given kinetic enegy atio to detemine the atio of the masses. Let m epesent the piece with the smalle kinetic enegy. Solution. Set p 0 and solve fo p + p 0 m v + m v m m. Set K K 3. Combine the expessions fom steps and m m v v K K v v m v m v m m m m 4. The piece with the smalle kinetic enegy has the lage mass. m m v v v v m m m m Insight The smalle mass caies the lage kinetic enegy because kinetic enegy inceases with the squae of the velocity but is linea with mass. Its highe speed moe than compensates fo its smalle mass.. Pictue the Poblem The lumbejack moves to the ight while the log moves to the left. Stategy As long as thee is no fiction the total momentum of the lumbejack and the log emains zeo, as it was befoe the lumbejack stated totting. Combine vecto addition fo elative motion (equation 3-8) with the expession fom the consevation of momentum to find speed of lumbejack elative to the shoe. Let v L, log speed of lumbejack elative to the log, and v log, s speed of the log elative to the shoe. Solution. (a) Wite out the v L, log + v log, s equation fo elative motion. Let the log tavel in the negative v L, log v log, s diection v log, s v L, log. Wite out the consevation of momentum with espect to the shoe 3. Substitute the expession fom step into step and solve fo p 0 m L m log v log, s m L m log v log, s m log ( v L, log ) ( m L + m log ) m log v L, log m log v L, log m L + m log (.7 m/s) 380 kg 85 + 380 kg. m/s 4. (b) If the mass of the log had been geate, the lumbejack s speed elative to the shoe would have been geate than that found in pat (a), because the log would have moved slowe in the negative diection. 5. (c) Use the expession fom step 3 to find the new speed of the m log v L, log m L + m log (.7 m/s) 450 kg 85 + 450 kg.3 m/s
lumbejack Insight Taking the agument in (b) to its exteme, if the mass of the log equaled the mass of the Eath the lumbejack s speed would be exactly.7 m/s elative to the Eath (and the log). If the mass of the log wee the same as the mass of the lumbejack, the speed of each elative to the Eath would be half the lumbejack s walking speed.
5. Pictue the Poblem The initial and final momentum vectos fo this collision ae depicted at ight. Stategy Assuming thee is no fiction between the playes skates and the ice, we can use consevation of momentum togethe with the fact that the playes stick togethe afte the collision to find the final velocity. Let the motion of playe be in the positive x-diection and the motion of playe be at an angle of 5 measued counteclockwise fom the positive x-axis. Solution. Wite out the consevation of momentum and solve fo v f p i + p i p f ( mvˆx ) + ( mv cosθ ˆx + mv sinθŷ) mv f v ( + cosθ) ˆx + v ( sinθ ) ŷ v f ( 5.5 m/s) ( + cos5 ) ˆx + ( 5.5 m/s) ( sin5 ) ŷ v f (.5 m/s) ˆx + (.38 m/s) ŷ v f. Detemine the magnitude of v f v f v fx + v fy (.5 m/s) + (.38 m/s).8 m/s Insight The two playes slide away fom the collision at 57.4 above the positive x axis. Playe s initial momentum povides the y component of the final momentum, but his x momentum in the ˆx diection is smalle than playe s momentum in the ˆx diection, and so the two playes have a total momentum in the positive x diection.
33. Pictue the Poblem The apple and oange collide in the manne depicted in the figue at ight. The mass of the apple is 0.30 kg and the mass of the oange is 0.60 kg. Stategy Use consevation of momentum in the y diection to find the y component of the apple s speed, and consevation of momentum in the x diection to find the x component of the apple s speed. The known components can then be used to find the total speed and diection of the apple afte the collision. Solution. Conseve momentum in the y diection to find v f,y. Conseve momentum in the x diection to find v f,x p y 3. Find the apple s speed v f v f,x 0 m v f,y m v f,y ( 0.60 kg).03 m/s v f,y m v f,y m ( 0.30 kg) p x m v i,x + m v i,x m v f,x + 0 + v i,x + m m v i,x v f,x. m/s + v f,y 0.38 m/s.68 m/s 0.60 kg 0.30 kg. m/s (.68 m/s) + ( 0.38 m/s).3 m/s v 4. Find the apple s diection θ tan f,y tan.7 m/s 0.38 m/s 73 + 80 07 v f,x This angle is measued counteclockwise fom the positive x axis. Insight Since the collision is elastic you can also set K f m v f + m v f K i and use the K i 0.97 J given in the example to find v f.3 m/s, which is coect to within ounding eo. Note that we bent the ules of significant figues in steps and 3 to avoid such ounding eo. This appoach leaves an ambiguity in the x component of the apple s final velocity, howeve, and you still need consevation of momentum in the x diection to esolve it. 43. Pictue the Poblem The geomety of the sulfu dioxide molecule is shown at ight. Stategy The cente of mass of the molecule will lie somewhee along the y axis because it is symmetic in the x diection. Find Y cm using equation 9-5. Both oxygen atoms will be the same vetical distance y O fom the oigin. Let m epesent the mass of an oxygen atom, m s the sulfu atom. Solution. Use equation 9-5 to find Y cm Y cm my M 6 u my O + my O + m s y s my O + 0 m + m s m + m s sin 30 0.036 nm 0.43 nm ( 6 u) + 3 u
. Recalling that nm 0 9 m, we can wite ( X cm, Y cm ) ( 0, 3.6 0 m) Insight If the angle wee to decease fom 0 the cente of mass would move upwad. Fo instance, if the bond angle wee only 90, the cente of mass would be located at (0, 5. 0 m). 6. Pictue the Poblem The placement of the lead weight moved the cente of mass of the tie fom a cetain location to the geometic cente. Stategy Let the coodinate system be placed so that the oigin is at the geometic cente of the wheel. Teat the entie mass of the wheel as a point mass at some small distance d fom the oigin. Togethe with the lead weight the cente of mass of the system is exactly at zeo, so wite out equation 9-4 and solve fo d. Solution. Solve equation 9-4 fo d X cm 0 m lw x lw + m wh d m lw + m wh d m lw x m lw wh 0.050 kg 35.5 kg 5.0 cm 0 mm/cm 0.354 mm. Befoe the lead weight was added, the cente of mass was 0.354 mm fom the cente of the wheel. Insight Teating the entie wheel as if it wee a point mass located at its cente of mass is an impotant concept in mechanics. The attempt to pove this concept led Isaac Newton to develop the mathematical method we call calculus!