Effects of Gravitation

The Gravitational Field

• Region surrounding a mass or body where another body experiences a force of attraction due to the first body

• There is a gravitational field around the earth, and it affects a body by giving it an acceleration due to gravity g, as determined by Newton’s Law of Gravitation.

• Using Newton’s Law of Gravitation, the force of attraction between the earth and an object is given by,

• F = G (mema / re2)

• The weight of a body, w, is the total gravitational force exerted on it by the earth equivalent to m x g. So the acceleration due to gravity of a planet of mass M and radius r is given by

g = G (M/r2)

Pendulum Motion

• Another way of deriving g is by using pendulums. A simple pendulum is just a mass suspended by a “mass less” or a string with negligible mass. Starting from a certain height an allowed to swing freely, it will move along a curved path, moving back and forth.

A Simple Pendulum

• This could be assumed as uniform circular motion with a period τ, so we can use the equation for the centripetal acceleration ac. The radius of curvature will just be the length L, so

ac = (4π2r/τ2) = (4π2L/τ2)

• At the lowest point, ac is just equal to the acceleration due to gravity g, so

g = (4π2L/τ2)

Satellite Motion

• Simplifying, we assume the satellite of mass m follows a uniform circular motion of radius r around a large body, like earth. We can then use this equation for centripetal acceleration,

ac = (v2/r)

directed towards the center of the larger body, like earth.

• The centripetal acceleration is also equal to g, the acceleration due to gravity, given by

g= G(me/r2)

Since F = mg and from Newton’s Second Law, F = mac, we can equate the two, so

F = mg = msac

• Substituting G(me/r2) for g and (v2/r) for ac ,

mG (mₑ/r2) = m(v2/r) G (mₑ/r2) = (v2/r)

Solving for v,

v=

r

Gme

r

Gme

• To solve for period τ,

τ= (2 r/v)π

τ = 2 πr

eGm

r

Human Beings in Space

• Satellites and spacecraft are said to be weightless when they are in orbit around the earth because of the influence of gravity.

• Aerodynamic forces on the lifting surfaces of an aircraft keep it up against the force of gravity, but a space vehicle cannot stay aloft in this way because of the absence of air in space.

• Spacecraft must orbit if it is to remain in space.

• Aircraft flying in the earth’s atmosphere can cause propellers and wings for propulsion and maneuvering, but spacecraft cannot do so because of the lack of air.

• When spacecraft fires a rocket blast in one direction, the reaction imparts momentum to the spacecraft in the opposite direction.

• Space – hostile environment for human beings• The vacuum of space can destroy an

unprotected human body in a few seconds by explosive decompression.

• Temperatures can become fatally high under direct solar radiation.

• Energetic solar and cosmic radiations – fatal to an unshielded person who is not protected by the Earth’s atmosphere.

• Hermitically sealed cabin or space suit, with a supply of pressurized air or oxygen to approximate conditions on Earth.

• Absorbing and reflecting surfaces – regulate the amount of heat radiation affecting the craft.

• Heavy shielding – protect against solar radiation storms

• Crews might be sheltered in a central position within the spacecraft with supplies and equipment to surround and shield them.

• Spacecraft might be shaped like a large wheel that spins slowly around its own axis, or like a dumbbell, rotating end over end.