Using Newton's Law of Universal Gravitation, compare the weight of a 68.1 kg astronaut on the surface of the earth to the gravitational pull of the Earth on the same astronaut when she is on the International Space Station.
Constants: Average altitude of the ISS above the surface of the Earth = 387 km. You can find the other constants you will need on the Astronomy Formula Chart.
(In Moodle, enter large numbers with scientific E notation. Example: 100 = 1.00*102 in scientific notation, or 1.00E2 in scientific E notation. Don't forget measurement units! If there are two blanks, enter the numerical value in the first blank, and the measurement unit in the second blank. Example: 1.0 m + 1.0 m = .)
The weight of the astronaut near the surface of the earth:
Fg = mg = (68.1 kg)(strength of gravitational field, g m/s2) =
The pull of the Earth on the astronaut near the surface of the earth:
(G (Nm2)/kg2)(mass of earth, M♁ )(mass of astronaut, 68.1 kg) / (radius of earth, R♁ m)2 =
The pull of the Earth on the astronaut when she is on the ISS:
(G (Nm2)/kg2)(mass of earth, M♁ )(mass of astronaut, 68.1 kg) / (radius of earth, R♁ m + altitude of ISS: )2 =
So, why is the astronaut 'weightless' when she is on the space station?
To compare the weight of the astronaut on the surface of the Earth to the gravitational pull of the Earth on the same astronaut on the International Space Station (ISS), we can use Newton's Law of Universal Gravitation.
First, let's calculate the weight of the astronaut near the surface of the Earth. The weight (Fg) is given by the product of mass (m) and the strength of the gravitational field (g). In this case, the mass of the astronaut is 68.1 kg, and the strength of the Earth's gravitational field can be found on the Astronomy Formula Chart.
So, the weight of the astronaut near the surface of the Earth is given by:
Fg = mg = (68.1 kg)(strength of gravitational field, g in m/s^2)
Next, let's calculate the pull of the Earth on the astronaut near the surface of the Earth using Newton's Law of Universal Gravitation. The formula is:
Fg = (G * M♁ * m) / R♁^2
Where G is the gravitational constant (6.67 x 10^-11 Nm^2/kg^2), M♁ is the mass of the Earth (also found on the Astronomy Formula Chart), m is the mass of the astronaut (68.1 kg), and R♁ is the radius of the Earth (also found on the Astronomy Formula Chart).
Now, let's calculate the pull of the Earth on the astronaut when she is on the ISS. The formula remains the same, but we need to account for the altitude of the ISS (387 km) above the surface of the Earth.
Fg = (G * M♁ * m) / (R♁ + altitude of ISS)^2
where altitude of ISS is given in meters.
By comparing the weight of the astronaut near the surface of the Earth to the gravitational pull of the Earth on the astronaut when she is on the ISS, we can see that they are not the same. The weight of the astronaut near the surface of the Earth will be greater than the pull of the Earth on the astronaut when she is on the ISS.
Now, to address why the astronaut is 'weightless' when she is on the space station. The ISS is in a state of continuous free-fall around the Earth. It is constantly falling towards the Earth but moving fast enough to keep missing it. Due to this perpetual free-fall motion, the astronaut and other objects inside the space station experience a sensation of weightlessness. Though there is still gravitational pull acting on them, it is matched by the forward motion of the ISS, resulting in the feeling of weightlessness.