Definition of Weight: Weight is the force generated by the gravitational attraction on an object, such as a rocket. It is a force that can be measured, making it more familiar to individuals than other forces acting on a rocket.
Comparison with Other Forces:
Mechanical Forces: These include aerodynamic forces like lift and drag, as well as thrust. Mechanical forces require physical contact with gases that generate them.
Gravitational Force: Unlike mechanical forces, gravitational force is a field force, meaning physical contact with the source of this force is not necessary.
Nature of Gravity
Historical Context: The understanding of gravitational force has progressed significantly, with current research still ongoing among theoretical physicists. For objects the size of rockets, the gravitational framework provided by Sir Isaac Newton over 300 years ago remains applicable.
Newton's Law of Gravitation:
Published theories while he was only 23, combining gravity with his laws of motion.
Gravitational force depends on two factors:
The mass of the two objects involved.
The inverse square of the distance between them.
Gravitational Force Equation:
F = G rac{m_1 m_2}{d^2}
Where:
$F$: gravitational force between two particles.
$G$: universal gravitational constant.
$m_1$ and $m_2$: masses of the two particles.
$d$: distance between the two particles.
Calculation of Gravitational Acceleration Near Earth
Contribution of Particles Near Earth: For objects near the Earth, to calculate gravitational force, the mass of all nearby particles is equivalent to the mass of the Earth, with distance measured from the center of the Earth.
Distance from Earth's Center: Approximately 4000 miles on Earth's surface.
Earth's Gravitational Acceleration:
Given by:
g_e = G rac{m_{Earth}}{r_{Earth}^2}
Where:
$g_e$: gravitational acceleration.
$m_{Earth}$: mass of the Earth.
$r_{Earth}$: radius of the Earth.
Standard values:
g_e = 9.8 ext{ m/sec}^2
Approx. 32.2 ext{ ft/sec}^2
Weight Formula:
W = m g_e
Where:
$W$: weight of an object.
$m$: mass of the object.
$g_e$: gravitational acceleration.
Weight Variation with Location
Weight vs. Mass: An object's mass does not change depending on location, but weight does change as it is dependent on gravitational acceleration, which varies with the square of the distance from the Earth's center.
Example Calculation: Weight of a Space Shuttle in Low Earth Orbit
Weight on ground: Approx. 250,000 pounds.
Distance in orbit: 200 miles above Earth's surface → distance from Earth's center: 4200 miles.
Then solve for W_{orbit} = 250,000 imes ext{(computed ratio)} leading to approximately 226,757 pounds.
Astronaut Experience: Astronauts feel "weightlessness" not due to the absence of gravity but because they are in free-fall when in orbit.
Effects of Mass and Distance in Space Exploration
Weight Variation Across Celestial Bodies: The weight of an object, influenced by the mass of both the object and the attracting celestial body and inversely by the distance squared, varies significantly across different planets:
Moon: Gravitational acceleration given by:
g_m = G rac{m_{Moon}}{d_{Moon}^2} = 1.61 ext{ m/sec}^2
Consistency of Mass: The mass of a rocket remains unchanged whether it's on Earth, the Moon, or Mars.
Weight Differences: Weight on the Moon is about 1/6 that on Earth; weight on Mars is about 1/3. Therefore, less thrust is required to launch from these celestial bodies.
Determining Weight:
Since forces are vector quantities, weight acts towards the center of the Earth and is dependent on the total mass of the rocket, including fuel and payload.
Weight distribution can be simplified through a point known as the center of gravity, around which the rocket rotates during flight, always directed towards Earth's center.
Dynamic Weight Changes During Rocket Flight
Weight Variation During Launch: As fuel is burned, the rocket's weight decreases, necessitating consideration in motion equations for accurate modeling.
Percentage Change: In model rocketry, weight changes are small, but in full-scale rocketry, the changes are significant and must be integrated into calculations.
Staging: Engineers design rockets with staging systems - where smaller rockets are discarded during flight to enhance performance and maintain thrust efficiency as mass decreases.