Question

Космонавт вступил на недавно открытую планету за пределами Солнечной системы. Масса космонавта вместе с экипировкой составляет 154 кг, а его вес на планете равен 4065,6 Н. Радиус планеты равен 49300 км. Каково будет ускорение свободного падения на планете? Какова масса планеты?

Answer 1

Дано:
m=154 кг
P=4065,6 Н
R=49 300 км = 49,3*10⁶ м
______________
g - ?
M - ?
Ускорение:
g = P/m = 4065,6/154 ≈ 26,4 м/с²

Из формулы:
g=F/m = G*M*m/(m*R²) = G*M/R²
находим массу планеты:
M = g*R²/G = 26,4*(49,3*10⁶)² / 6,67*10⁻¹¹ ≈ 9,6*10²⁶ кг (почти в 160 раз больше массы Земли и сравнима с массой Сатурна)

Answer 2

Calculation of Acceleration Due to Gravity on the Planet

To calculate the acceleration due to gravity on the planet, we can use the formula:

g = G * (M / r^2)

where: - g is the acceleration due to gravity - G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2) - M is the mass of the planet - r is the radius of the planet

Let's calculate the acceleration due to gravity on the planet:

Given: - Mass of the astronaut with equipment (m) = 154 kg - Weight of the astronaut on the planet (W) = 4065.6 N - Radius of the planet (r) = 49300 km = 49300000 m

First, let's calculate the mass of the planet (M) using the weight of the astronaut:

Weight (W) = mass (m) * acceleration due to gravity (g)

From this equation, we can solve for the mass of the planet (M):

M = W / g

Now, let's calculate the acceleration due to gravity (g) on the planet:

g = G * (M / r^2)

Using the given values, we can calculate the acceleration due to gravity on the planet and the mass of the planet.

Calculation:

Given: - Mass of the astronaut with equipment (m) = 154 kg - Weight of the astronaut on the planet (W) = 4065.6 N - Radius of the planet (r) = 49300000 m

Using the equation M = W / g, we can calculate the mass of the planet (M):

M = 4065.6 N / g

Now, let's calculate the acceleration due to gravity (g) on the planet using the equation g = G * (M / r^2):

g = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (M / (49300000 m)^2)

Now, let's substitute the value of M in the equation:

g = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (4065.6 N / g) / (49300000 m)^2

Simplifying the equation, we get:

g^2 = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (4065.6 N) / (49300000 m)^2

Taking the square root of both sides, we get:

g = sqrt((6.67430 × 10^-11 m^3 kg^-1 s^-2) * (4065.6 N) / (49300000 m)^2)

Now, let's calculate the value of g:

g ≈ 1.709 m/s^2

Therefore, the acceleration due to gravity on the planet is approximately 1.709 m/s^2.

Now, let's calculate the mass of the planet (M) using the equation M = W / g:

M = 4065.6 N / 1.709 m/s^2

M ≈ 2381.7 kg

Therefore, the mass of the planet is approximately 2381.7 kg.

Please note that these calculations are based on the given values and assumptions.