A 4
.
7 kg mass weighs 39
.
95 N on the surface
of a planet similar to Earth. The radius of
this planet is roughly 6
.
2
×
10
6
m.
Calculate the mass of of this planet. The
value of the universal gravitational constant
is 6
.
67259
×
10
−
11
N
·
m
2
/
kg
2
.
Answer in units of kg.
To calculate the mass of the planet, we can use the formula for gravitational force:
F = (G * m₁ * m₂) / r²
Where:
F is the gravitational force between two objects,
G is the universal gravitational constant,
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, we know the gravitational force (39.95 N), the mass of the smaller object (4.7 kg), and the radius of the planet (6.2 * 10⁶ m). We need to find the mass of the planet (m₂).
Rearranging the formula, we can solve for m₂:
m₂ = (F * r²) / (G * m₁)
Now we can substitute the given values into the formula:
m₂ = (39.95 N * (6.2 * 10⁶ m)²) / (6.67259 * 10⁻¹¹ N·m²/kg² * 4.7 kg)
First, calculate the expression in the numerator:
(39.95 N * (6.2 * 10⁶ m)²) = (39.95 N * 38.44 * 10¹² m²) = 1.537298 * 10¹⁵ N·m²
Next, substitute the calculated value into the formula:
m₂ = (1.537298 * 10¹⁵ N·m²) / (6.67259 * 10⁻¹¹ N·m²/kg² * 4.7 kg)
Now, divide the numerator by the denominator:
m₂ = 3.0058 * 10²⁵ kg
Therefore, the mass of the planet is approximately 3.0058 * 10²⁵ kg.