The Moon has a mass of 7.35 � 1022 kg and its
equatorial radius is 1.74 � 106 m. Earth’s mass
is 5.97 � 1024 kg and its equatorial radius is
6.38 � 106 m.
(a) Calculate the magnitude of the gravitational
force exerted by
i) the Moon on a fully suited 100-kg
astronaut standing on the Moon’s
surface, and
ii) Earth on a fully suited 100-kg astronaut
standing on Earth’s surface.
the gravitational constant
G =6.67•10⁻¹¹ N•m²/kg²,
Earth’s mass is M₁ = 5.97•10²⁴kg,
Earth’s radius is R₁ = 6.38•10⁶ m.
Moon’s mass is M₂= 7.35•10² kg
Moon’s radius is R₂=1.74•10⁶ m
i) F(Moon)= mg₂= G•m•M₂/R₂²
ii)F(Earth)= mg₁= G•m• M₁ /R₁²
To calculate the magnitude of the gravitational force exerted by the Moon and the Earth on a 100-kg astronaut, we can use the formula for gravitational force:
F = G * (m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant (6.67430 × 10^-11 N m^2 /kg^2), m1 and m2 are the masses of the objects involved, and r is the distance between the centers of the objects.
(a) i) The Moon on a 100-kg astronaut standing on the Moon's surface:
m1 = astronaut's mass = 100 kg
m2 = Moon's mass = 7.35 × 10^22 kg
r = Moon's radius = 1.74 × 10^6 m
Using the formula:
F = (6.67430 × 10^-11 N m^2 /kg^2) * (100 kg * 7.35 × 10^22 kg) / (1.74 × 10^6 m)^2
Calculating this, we get:
F = 1.91 × 10^3 N (rounded to two significant figures)
Therefore, the magnitude of the gravitational force exerted by the Moon on a fully suited 100-kg astronaut standing on the Moon's surface is approximately 1.91 × 10^3 N.
ii) Earth on a 100-kg astronaut standing on Earth's surface:
m1 = astronaut's mass = 100 kg
m2 = Earth's mass = 5.97 × 10^24 kg
r = Earth's radius = 6.38 × 10^6 m
Using the formula:
F = (6.67430 × 10^-11 N m^2 /kg^2) * (100 kg * 5.97 × 10^24 kg) / (6.38 × 10^6 m)^2
Calculating this, we get:
F = 981 N (rounded to three significant figures)
Therefore, the magnitude of the gravitational force exerted by the Earth on a fully suited 100-kg astronaut standing on Earth's surface is approximately 981 N.
To calculate the magnitude of the gravitational force exerted by the Moon on a fully suited 100-kg astronaut standing on the Moon's surface, we can use Newton's law of universal gravitation formula:
F = ( G * m1 * m2 ) / r^2
Where:
F represents the magnitude of the gravitational force,
G is the gravitational constant (6.67 x 10^-11 N*m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, the mass of the Moon (m1) is 7.35 x 10^22 kg, the mass of the astronaut (m2) is 100 kg, and the radius of the Moon (r) is 1.74 x 10^6 m.
Plugging in these values, we can calculate the force:
F = (6.67 x 10^-11 N*m^2/kg^2) * (7.35 x 10^22 kg) * (100 kg) / (1.74 x 10^6 m)^2
Calculating this expression will give us the magnitude of the gravitational force exerted by the Moon on the astronaut.
Similarly, to calculate the magnitude of the gravitational force exerted by Earth on a fully suited 100-kg astronaut standing on Earth's surface, we use the same formula:
F = ( G * m1 * m2 ) / r^2
The mass of the Earth (m1) is 5.97 x 10^24 kg, and the radius of the Earth (r) is 6.38 x 10^6 m. The mass of the astronaut (m2) remains the same (100 kg).
Now, let's calculate the force:
F = (6.67 x 10^-11 N*m^2/kg^2) * (5.97 x 10^24 kg) * (100 kg) / (6.38 x 10^6 m)^2
Calculating this expression will give us the magnitude of the gravitational force exerted by Earth on the astronaut.