A 2.5 kg mass weighs 23.75 N on the surface of a planet similar to Earth. The radius of this planet is roughly 7.7 × 10
6 m.Calculate the mass of of this planet. The value of the universal gravitational constant is 6.67259 × 10−11N · m2/kg2
To calculate the mass of the planet, we can use the formula for the gravitational force:
F = (G * m1 * m2) / r^2
where:
F is the gravitational force (23.75 N),
G is the universal gravitational constant (6.67259 × 10^-11 N · m^2/kg^2),
m1 is the mass of the planet (unknown),
m2 is the mass of the 2.5 kg object,
r is the radius of the planet (7.7 × 10^6 m).
Rearranging the formula, we get:
m1 = (F * r^2) / (G * m2)
Substituting in the given values:
m1 = (23.75 N * (7.7 × 10^6 m)^2) / (6.67259 × 10^-11 N · m^2/kg^2 * 2.5 kg)
Calculating the expression:
m1 = (23.75 * (7.7 × 10^6)^2) / (6.67259 × 10^-11 * 2.5)
m1 = (23.75 * 5.929 * 10^13) / (1.6681475 × 10^-10)
m1 = 140.811 * 10^13 / 1.6681475 × 10^-10
m1 = (140.811 / 1.6681475) * 10^13+10
m1 ≈ (84.444 * 10^13) kg
Therefore, the mass of the planet is approximately 8.4444 × 10^14 kg.
To calculate the mass of the planet, we can use the formula for calculating the weight of an object given its mass:
Weight = mass * acceleration due to gravity
In this case, we're given the weight of the object (23.75 N) and its mass (2.5 kg). Since the acceleration due to gravity is the same for all objects on the surface of the planet, we can rearrange the formula to solve for the mass of the planet:
Mass of planet = Weight of object / acceleration due to gravity
First, we need to calculate the acceleration due to gravity on the surface of the planet. We can use the formula:
acceleration due to gravity = (Universal gravitational constant * mass of planet) / (radius of planet) ^ 2
As mentioned in the question, the radius of the planet is 7.7 × 10^6 m, and the value of the universal gravitational constant is 6.67259 × 10^−11 N · m^2/kg^2.
Substituting these values into the formula, we have:
acceleration due to gravity = (6.67259 × 10^−11 N · m^2/kg^2 * mass of planet) / (7.7 × 10^6 m)^2
Now, we can substitute the known values for weight (23.75 N) and mass (2.5 kg) of the object back into the first formula to solve for the mass of the planet:
Mass of planet = Weight of object / acceleration due to gravity
Mass of planet = 23.75 N / acceleration due to gravity
By combining these equations, we can solve for the mass of the planet.