A satellite experiences a gravitational force of 228 N at an altitude of 4.0xE7 m above Earth. What is the mass of this satellite ?
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To calculate the mass of the satellite, we can use the formula for gravitational force:
\(F = \frac{{G \cdot m_1 \cdot m_2}}{{r^2}}\)
Where:
\(F\) is the gravitational force,
\(G\) is the gravitational constant (\(6.67430 \times 10^{-11}\) N m\(^2\)/kg\(^2\)),
\(m_1\) and \(m_2\) are the masses of the two objects,
and \(r\) is the distance between the two objects.
In this case, the satellite is experiencing a gravitational force of 228 N at a distance of 4.0 x 10\(^7\) m above Earth. We can assume that the mass of the Earth remains constant, so we can rearrange the equation to solve for the mass of the satellite:
\(m_2 = \frac{{F \cdot r^2}}{{G \cdot m_1}}\)
Substituting the given values into the equation, we have:
\(m_2 = \frac{{228 \, \text{N} \cdot (4.0 \times 10^7 \, \text{m})^2}}{{6.67430 \times 10^{-11}\, \text{N m}^2/\text{kg}^2 \cdot m_1}}\)
Now, we can solve for the mass of the satellite. However, we need the mass of the Earth (\(m_1\)) to complete the calculation.
To find the mass of the satellite, we can use Newton's law of universal gravitation:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.674 × 10^-11 N*m^2/kg^2)
m1 and m2 are the masses of the two objects (in this case, one of the masses is Earth)
r is the distance between the centers of the two objects
In this scenario, the satellite is the smaller object orbiting Earth. We know the gravitational force (228 N) and the distance (altitude of 4.0 x 10^7 m).
Let's rearrange the equation to solve for the mass of the satellite (m2):
m2 = (F * r^2) / (G * m1)
Since we're finding the mass of the satellite, we can assume the mass of Earth (m1) is known. For simplicity, let's assume it is 5.972 × 10^24 kg (the actual value).
Substituting the known values:
m2 = (228 N * (4.0 x 10^7 m)^2) / (6.674 × 10^-11 N*m^2/kg^2 * 5.972 × 10^24 kg)
Simplifying the equation gives:
m2 = (228 N * 1.6 x 10^15 m^2) / (3.993 x 10^14 N*m^2/kg)
m2 = 9.216 x 10^17 N*m^2 / (3.993 x 10^14 N*m^2/kg)
m2 = 2.308 x 10^3 kg
Therefore, the mass of the satellite is approximately 2.308 x 10^3 kg.
f=G*M*Me/(re+4E7)^2
solve for M