the space shuttle typically orbits 400 km above the earths surface. The earth has a mass of 5.98 * 10^24 kg and a radius of 6380km.
A) How much would a 2000 kg part for the space station weigh when it has been lifted to that orbit in the shuttles cargo bay?
B)Use your result from Part A to determine the acceleration due to gravity at that altitude.
C) Use your knowledge of circular motion to determine the orbital speed of the shuttles cargo.
a. if the shuttle were in orbit, the 2000kg will weigh zero.
b. Maybe the part a was not interested in weight at all, but the gravitational attracation
W=GMe*2000/(6.380+.4)^2
That is the graviataional attraction,perhaps that is what the teacher wanted in part a.
Then part b would be
g'*2000=W so g'=W/2000
c. g'=V^2/(6.38+.4)^2 solve for V
I erred on units.
W=GMe*2000/(6.380E6+.4E6)^2
g'=V^2/(6.38E6+.4E6)^2 solve for V
A) To determine the weight of a 2000 kg part in the shuttle's cargo bay when lifted to the orbit, we can use the equation for weight:
Weight = mass * acceleration due to gravity
In this case, the mass of the part is 2000 kg. The acceleration due to gravity can be calculated using Newton's law of universal gravitation:
acceleration due to gravity = (gravitational constant * mass of Earth) / (radius of Earth + altitude)^2
The gravitational constant is approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2.
Plugging in the given values:
acceleration due to gravity = (6.67430 x 10^-11 m^3 kg^-1 s^-2 * 5.98 x 10^24 kg) / (6.38 x 10^6 m + 400,000 m)^2
acceleration due to gravity = 9.815 m/s^2
Now, we can calculate the weight:
Weight = mass * acceleration due to gravity
Weight = 2000 kg * 9.815 m/s^2
Weight = 19,630 N
Therefore, the 2000 kg part would weigh approximately 19,630 Newtons when lifted to that orbit in the shuttle's cargo bay.
B) The acceleration due to gravity at that altitude can be obtained from the previous calculation, which is approximately 9.815 m/s^2.
C) The orbital speed of the shuttle's cargo can be determined using the concept of circular motion, which relates the gravitational force and the centripetal force. The gravitational force provides the necessary centripetal force to keep the cargo in orbit.
The centripetal force is given by the equation:
centripetal force = (mass of the cargo) * (orbital speed)^2 / (radius of Earth + altitude)
Setting the centripetal force equal to the gravitational force:
(mass of the cargo) * (orbital speed)^2 / (radius of Earth + altitude) = (mass of the cargo) * (acceleration due to gravity)
Simplifying and solving for orbital speed:
(orbital speed)^2 = (acceleration due to gravity) * (radius of Earth + altitude)
orbital speed = sqrt((acceleration due to gravity) * (radius of Earth + altitude))
Plugging in the given values:
orbital speed = sqrt((9.815 m/s^2) * (6.38 x 10^6 m + 400,000 m))
orbital speed ≈ 7,664 m/s
Therefore, the orbital speed of the shuttle's cargo is approximately 7,664 m/s.
A) To find the weight of the 2000 kg part when lifted to the orbit, we need to calculate the gravitational force acting on it.
The weight of an object can be found using the formula: Weight = mass * acceleration due to gravity.
The mass of the part is given as 2000 kg, and the acceleration due to gravity can be calculated using the formula:
acceleration due to gravity = gravitational constant * (mass of the Earth / distance from the center of the Earth squared).
The gravitational constant denoted by "G" is approximately equal to 6.674 × 10^-11 N(m/kg)^2.
First, convert the distance to meters: 400 km = 400,000 meters.
Then, calculate the acceleration due to gravity:
acceleration due to gravity = (6.674 × 10^-11 N(m/kg)^2) * ((5.98 × 10^24 kg) / (400,000 + 6,380,000)^2)
Calculate the expression inside the brackets: (400,000 + 6,380,000)^2 = 6,780,000^2.
Calculate the value of the expression: 6,780,000^2 ≈ 4.595 × 10^13.
Substitute the values into the equation:
acceleration due to gravity = (6.674 × 10^-11 N(m/kg)^2) * ((5.98 × 10^24 kg) / (4.595 × 10^13))
Calculate the value: (5.98 × 10^24 kg) / (4.595 × 10^13) ≈ 1.303 m/s^2.
Now, we can find the weight:
Weight = mass * acceleration due to gravity = 2000 kg * 1.303 m/s^2.
Multiply the values together:
Weight ≈ 2606 N.
Therefore, the weight of a 2000 kg part in the shuttle's cargo bay when lifted to that orbit would be approximately 2606 Newtons.
B) To determine the acceleration due to gravity at that altitude using the result from Part A, we can directly use the formula for gravitational acceleration:
acceleration due to gravity = (gravitational constant * mass of the Earth) / (distance from the center of the Earth + radius of the Earth)^2.
Substitute the values into the equation:
acceleration due to gravity = (6.674 × 10^-11 N(m/kg)^2 * (5.98 × 10^24 kg)) / (400,000 + 6,380,000)^2.
Calculate the value of the expression in the brackets: (400,000 + 6,380,000)^2 = 6,780,000^2.
Calculate the value of the expression: 6,780,000^2 ≈ 4.595 × 10^13.
Substitute the values into the equation:
acceleration due to gravity ≈ (6.674 × 10^-11 N(m/kg)^2 * (5.98 × 10^24 kg)) / (4.595 × 10^13).
Calculate the value: (6.674 × 10^-11 N(m/kg)^2 * (5.98 × 10^24 kg)) / (4.595 × 10^13) ≈ 0.906 m/s^2.
Therefore, the acceleration due to gravity at that altitude is approximately 0.906 m/s^2.
C) To determine the orbital speed of the shuttle's cargo, we can use the formula for circular motion:
orbital speed = √(acceleration due to gravity * radius of the orbit).
In this case, the radius of the orbit is given as 400 km = 400,000 meters.
Using the given acceleration due to gravity from Part B (approximately 0.906 m/s^2) and the radius of the orbit (400,000 meters), we can substitute the values into the equation:
orbital speed = √(0.906 m/s^2 * 400,000 meters).
Calculate the value: √(0.906 m/s^2 * 400,000 meters) ≈ 6004 m/s.
Therefore, the orbital speed of the shuttle's cargo is approximately 6004 m/s.