Q: A team of engineers is building a spaceship to go to Jupiter. Their design includes a gravity ring, a spinning section where (in the ring’s reference frame) astronauts are pushed outwards by the centrifugal force.
In their current design, the ring has to spin once every ten seconds in order to provide the same sensation of gravity as on Earth—exactly 1g of acceleration. However, this rate of spinning creates far too much stress on the bearings!
Can the engineers adjust their design to allow the ring to spin at a slower rate while keeping 1g of acceleration at the edge?
A: We are asked to conserve artificial earth gravity ” g ” = 9.8m/s^2 by slowing the rotation of a body revolving in space. To continue producing earth gravitational acceleration ( g ) in a situation in which angular velocity has been DECREASED, we must successfully conserve the angular momentum of the body by increasing its Moment of Inertia ” I “.
Two bodies of differing mass ” m ” can possess equal momentum ” p ” if they are moving at differing velocities. If a cannon fires, the cannonball will move outward with lots of force ( and momentum ). The cannon, which is much more massive, recoils at a much lower velocity; however, the momentum of the two objects is equal. Momentum, whether it be linear or angular in nature, is a measure of the ” quantity ” of motion a body possesses. In linear motion, momentum ” p ” is quantified as being a product of mass ” m ” times velocity ” v “. p = ( m )( v ). The question above, however, deals with the conservation of angular momentum. Therefore, we must examine angular momentum, and an important concept called the ” moment of inertia ” = ” I “.
A spinning ice skater increases ( or decreases ) rotational velocity by either extending their arms or pulling their arms inward. This effectively increases or decreases their ” moment of inertia “, denoted with the letter ” I “.
A circle can be drawn in which a ” slice of pie ” is isolated by two lines ( radii ) coming from a common center. These two lines define an angle. The outer arc of the circle contained between the two lines is defined as being some angular distance ” l “. It can be shown that for any angle ( theta ), ” l ” divided by ” r ” = theta. ( l / r ) = theta.
Going further, l = ( r )( theta ). If the distance ” l ” moves relative to some point in a certain amount of time, dividing ” l ” by time ” t ” will give us the angular velocity ( w ). So diving l = ( r )( theta ) by time will give us an expression for angular velocity: v = ( r )( w ), where ” w ” is ( theta / time ). Furthermore, centripetal acceleration ( center seeking acceleration ) is defined by ( v^2 / r ). We can derive ( v^2 / r ) from the above equation by first squaring both sides: v^2 = ( r^2 )( w^2 ). We can now divide both sides by ” r ” to get the desired quantity. a = ( r )( w^2 ).
Force equal mass times acceleration : F = ( m )( a ). Therefore, a centripetal ( center seeking ) force ( F ) = ( m )( r )( w^2 ). Finally, if such a force is extended outward from a center of mass by a distance of ” r “, we have forces of motion working perpendicular to ” r “. Work is defined by ( F )( r ). In the above derivation, this gives us ( m )( r^2 )( w^2 ). The average energy is defined as ( 1 /2 )( m )( r^2 )( w^2 ).
Momentum is a quantity that is conserved when no outside forces act on a body ( or system ), and the angular velocity of a body can be influenced if its effective radius is increased or decreased. Within the expression ( 1 / 2 )( m )( r^2 )( w^2 ), the ( m )( r^2 ) term is defined as the ” Moment of Inertia “, ( I ). If the moment of inertia ( I ) is increased in this example, the angular speed ( w ) will decrease, but the rotational energy and momentum ” p ” will be conserved. This will be accomplished by making the spacecraft WIDER.
RENAISSANCE PHYSICS 