University Physics AI

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Franklin Nash
7 years ago
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1 Uniesity Physics AI No. 4 The Gaitational Foce an the Gaitational Fiel Class Numbe Name I.Choose the Coect Answe. The manitue of the foce of aity between two ientical objects is ien by F 0. If the mass of each object is ouble but the istance between them is hale, then the new foce of aity between the objects will be ( A ) (A) 6 F 0 (B) 4 F 0 (C) F 0 (D) F 0 / Gm The manitue of the aitational foce is F a, accoin to the poblem, we et 4Gm Gm F a 6 6F 0 ( / ). A spheical symmetic nonotatin boy has a ensity that aies appeciably with the aial istance fom the cente. At the cente of the boy the acceleation of fee fall is ( C ) (A) Definitely lae than zeo. (B) Possibly lae than zeo. (C) Definitely equal to zeo. At the cente, the foce acte on the boy is zeo, so the acceleation of the boy is zeo. 3. The acceleation ue to aity in a hole u into a nonunifom spheically symmetic boy ( C ) (A) will incease as you o eepe, eachin a maimum at the cente. (B) will incease as you o eepe, but eentually each a maimum, an then ecease until you each the cente. (C) can incease o ecease as you o eepe. (D) must ecease as you o eepe. Fo a nonunifom spheically symmetic boy, the foce of aity epens on the istance fom the cente which is elate to how the ensity of the boy chane with espect to the istance fom the cente, fo instance: accoin to the Gauss s law of aity s 4πG i 4πG ρv s when ρ A, the acceleation will incease as you o eepe, when ρ A the acceleation will ecease as you o eepe.

2 II. Fillin the Blanks. Two masses m an m eet aitational foces of equal manitue on each othe. Show that if the total mass m +m is fie, the manitue of the mutual aitational foce on each of the two masses is a maimum when m m ( Fill < o o > ). Gm m Gm ( m ) The manitue of the aitational foce is F a, We et F m 0 G( m ) 0 m Then m m.. A mass m is insie a unifom spheical shell of mass an a mass is outsie the shell as shown in Fiue. The manitue of the total aitational foce on m is m F G. ( s + ) Ftotal F 'on m + F on m Q F ' on m 0 Ftotal F on m Gm Gm Ftotal ( s + ) s m Fi. 3. A cetain neuton sta has a aius of 0.0 km an a mass of k, about twice the mass of the Sun. The manitue of the acceleation of an 0.0 k stuent foolish enouh to be 00 km fom the cente of the neuton sta is m/s. The atio of the manitue of this acceleation an is If the stuent is in a cicula obit of aius 00 km about the neuton sta, the obital peio is s. (a) The manitue of the foce is Gm F ma So G a ( 0 ) m/s 5

3 (b) The atio of the manitue of this acceleation an is π 4π (c) Since the acceleation is a ω T T The obital peio is a T 3 5 4π 4π ( 0 ) ( s) 30 a G The manitue of the total aitational fiel at the point P in Fiue is m/s,the manitue of the acceleation epeience by a 4.00 k salt lick at point P is m/s, the manitue of the total aitational foce on the salt lick if it is place at P is 9.4 N m P m oon k ĵ î (a) The total aitational fiel at the point P is total + G ˆ G i + (cosθ + sinθ ˆ) j P P Fi. ath k (.6 0 ) (4.6 0 ) (0.3ˆ i ˆ) j 3 3 (.07 0 )ˆ i + (. 0 ) ˆj ( ˆ) j The manitue of the total aitational fiel at the point P is total m/s (b) The acceleation has no elation with anythin at some point. So the manitue of the acceleation epeience by a 4.00 k salt lick at point P is also equal to (a). a total m/s (c)the manitue of the total aitational foce on the salt lick if it is place at P is F ma N III. Gie the Solutions of the Followin Poblems

4 . Seeal planets (the as iants Jupite, Satun, Uanus, an Neptune) possess nealy cicula suounin ins, y pehaps compose of mateial that faile to fom a satellite. m In aition, many alaies contain in-like stuctues. θ Consie a homoeneous in of mass an aius. (a) Fin an epession fo the aitational foce eete by the in on a paticle of mass m locate a istance fom the cente of the in alon its ais. See Fi.3. (b) Suppose that the paticle falls fom est as a esult of the attaction Fi.3 of the in of matte. Fin an epession fo the spee with which it passes thouh the cente of the in. (a) Set up the cooinate system shown in fiue. Choose a pointlike iffeential masses shown as in fiue 3, the fiel pouce by is G + y G cosθ cosθ + G sinθ sinθ + Accoin to the symmety of the in-like stuctues, 0. So the aitational foce is i. ˆ Accoin to the aph, we et cosθ, so 3 / + G ( + ) Thus the aitational foce is G i ˆ 3 / ( + ) (b) Apply Newton s secon law of motion, we hae y G ( + ) 3 / Gm F m ( ) + 3 / Due to G, 3 / ( + ) t t We can et the spee 0 0 ( G + ) 3 / G ( + )

. A mass is in the shape of a thin unifom isk of aius. Let the z-ais epesent the symmety ais of the isk as inicate in Fiue 4. The ilky Way Galay is moele as such a mass isk to a fist appoimation.

5 . A mass is in the shape of a thin unifom isk of aius. Let the z-ais epesent the symmety ais of the isk as inicate in Fiue 4. The ilky Way Galay is moele as such a mass isk to a fist appoimation. (a) Fin the aitational fiel of the isk at a cooinate z alon the symmety ais of the isk? z θ 0 Fi.4 (b) What is the epession fo fo z >> in pat (a)? (c) Let σ be the suface mass ensity of the isk (the numbe of kiloams pe squae mete of the isk), so that σ. What is the aitational fiel in pat (a) as z 0 alon the positie π z-ais? (a) The mass is a unifom isk, so the suface mass ensity of the isk isσ. π Choose a cicula iffeential mass shown in fiue, π σ The aitational fiel establishe by the in at a cooinate z is Gz Gz σ π 3 / 3 / ( z + ) ( z + ) Thus the aitational fiel of the isk at a cooinate z alon the symmety ais of the isk is (b) Fo z G [ ] / Gzσ π ˆ ( z + ) k 3 / ( z + ) z G [ ] / ( z + ) 0 G z >>, the isk can be teate as a point mass, so z z > 0 z < 0 (c) In pat (a) G z, ( z ) + G When z 0, the aitational fiel is

6 The suface mass ensity of the isk σ π πgσ constant, so the aitational fiel is 3. The ath is not, in fact, a sphee of unifom ensity. A hih-ensity coe is suoune by a shell o mantle of lowe-ensity mateial. Suppose we moel a planet of aius as inicate in Fiue 5. A coe of ensity ρ an aius 3/4 is suoune by a mantle of ensity ρ an thickness /4. Let be the mass of the coe an m be the mass of the mantle. (This is not an accuate moel of the inteio stuctue of the ath but makes fo an inteestin an tactable poblem.) antle ensity ρ (a) Fin the mass of the coe. (b) Fin the mass of the mantle (c) What is the total mass of the planet (coe + mantle)? () Fin the manitue of the aitational fiel at the suface of the planet. (e) What is the manitue of the aitational fiel at the inteface between the coe an the mantle?. (a) The mass of the coe is ρvcoe ρ π ( ) π ρ 3 4 3/4 Coe ensity ρ Fi.5 /4 (b) The mass of the mantle is m ρ Vmantle ρ [ π π ( ) ] π ρ (c) The total mass of the planet is m π ρ + π ρ π 4 4 () The manitue of the aitational fiel at the suface of the planet is G( + m) 9 suface π Gρ. 96πGρ 4 (e) The manitue of the aitational fiel at the inteface between the coe an the mantle is G int eface G π ρ π Gρ 3 9 ( ) 4 3 ρ

7 4. What is the flu of the total aitational fiel of the ath an oon thouh a close suface that encloses the oon? If anothe suface encloses both the eath an oon, by how much is the flu of the aitational fiel chane? The sufaces ae shown in fiue. (a) The flu of the total aitational fiel of the ath an oon thouh a close suface S that encloses the oon is Φ s S S S 4πG (b) The flu of the total aitational fiel of the ath an oon thouh a close suface S encloses both the eath an oon is Φ s 4πG( + ) S The flu of the aitational fiel chane is Φ Φ Φ 4πG

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6 Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe