Gravitation 1

Example 1:a.) What is the acceleration due to gravity at

the surface of Venus?(MV = 4.88 x 1024 kg and RV = 6.07 x 106 m)

b.) What is the escape velocity for an objecton the surface of Venus?

c.) What is the total energy of a satellite witha mass of 15,000 kg orbiting in a circularorbit 200 km above the surface of Venus?

!

MV = 4.88"1024 kg and RV = 6.07"106 m

a.) gV = ?

!

Fg =GMV mRV

2

!

= mgV

!

gV =G MV

RV2

!

= 6.67"10#11 N $m2

kg2

%

& '

(

) *

4.88"1024 kg( )6.07"106 m( )2

!

gV = 8.83 ms2

b.) ve = ?

!

K = "U

!

so 12mve

2 =GMVmRV

!

ve = 2GMVRV

!

= 2 6.67"10#11 N $m2

kg2

%

& '

(

) *

4.88"1024 kg( )6.07"106 m( )

!

ve = 10,356 ms

Example 1:

!

MV = 4.88"1024 kg, RV = 6.07"106 m, m = 15,000 kg, and d = 200 km

c.) E = ?

!

Fg =GMV mr2

!

E = K +U

!

=12mv2 "GMVm

r

!

= m v2

r

!

= mac

!

so v = GMV r

!

E =12m GMV

r"

# $

%

& '

2

(GMVmr

!

=12GMVm

r"GMVm

r

!

E = "12

6.67#10"11 N $m2

kg2

%

& '

(

) *

4.88#1024 kg( ) 15,000 kg( )6.07#106 m + 2.0#105 m( )

!

E = "3.9#1011J

!

E = "12GMVm

r

!

RV

!

d

!

r = RV + d

!

= "12G MVmRV + d

Example 1:

Gravitation 4

Example 2:What is the weight of 85 kg person on the surfaceof Pluto?(MP = 1.2 x 1022 kg and RP = 1.15 x 106 m)

!

MP = 1.2"1022 kg, RP = 1.15"106 m, and m = 85 kg

Fg = ?

!

Fg =GMP mRP

2

!

Fg = 6.67"10#11 N $m2

kg2

%

& '

(

) *

1.2"1022 kg( ) 85 kg( )

1.15"106 m( )2

!

Fg = 51.4 N

!

gP =G MP

RP2

!

= 6.67"10#11 N $m2

kg2

%

& '

(

) *

1.2"1022 kg( )1.15"106 m( )2

!

gP = 0.61 ms2

!

Fg = mgP

!

= 85 kg( ) 0.61 ms2

" # $

% & '

!

Fg = 51.4 N

Example 2:

Gravitation 6

Gravitation 7

Example 3:Find the speed of a satellite that would orbit Mars200 km above its surface.(MM = 6.42 x 1023 kg and RM = 3.38 x 106 m)

!

MM = 6.42"1023 kg , RM = 3.38"106 m, and d = 200 km

v = ?

!

Fg =GMM mr2

!

= m v2

r

!

= mac

!

v = GMM r

!

RM

!

d

!

r = RM + d

!

v = G MM RM + d

!

v = 6.67"10#11 N $m2

kg2

%

& '

(

) *

6.42"1023 kg( )3.38"106 m + 2.0"105 m( )

!

v = 3459 ms

Example 3:

Gravitation 9

Example 4:How many minutes would it take a satellite toorbit Earth 150 km above its surface?(ME = 5.98 x 1024 kg and RE = 6.37 x 106 m)

!

ME = 5.98"1024 kg , RE = 6.37"106 m, and d = 150 km

T = ?

!

Fg =GME mr2

!

= m v2

r

!

= mac

!

v = GME r

!

RE

!

d

!

r = RE + d

!

T = 2"6.37#106 m +1.5#105 m( )3

6.67#10$11 N %m2

kg2

&

' (

)

* + 5.98#1024 kg( )

!

T = 5238 s

!

=2"rT

!

T = 2"r r GME

!

= 2" r 3 GME

!

T = 2" RE + d( )3 GME

!

= 87.3 min

Example 4:

The Law of Gravity 11

Example 5:A spaceship is fired from the Earth’s surfacewith an initial speed of 2.00 x 104 m/s. Whatwill be its speed when it is very far awayfrom the Earth? (Neglect friction)

(ME = 5.98 x 1024 kg and RE = 6.37 x 106 m)

12

!

M = ME = 5.98"1024 kg, v1 = 2.00"104 ms

, r1 = RE = 6.37"106 m, r2 =#, v2 = ?

Example 5:

!

K1 +U1 = K2 +U2

!

12mv1

2 "G Mmr1

=12mv2

2 "G Mmr2

!

v2 = 1.66"104 ms

!

r2 ="

!

v2 = v12 " 2G M

r1

!

v2 = 2.00"104 ms

# $ %

& ' (

2) 2 6.67"10)11 N *m2

kg2

#

$ %

&

' (

5.98"1024 kg( )6.37"106 m( )