Artificial Gravity & Centripetal Acceleration

Background Resources

The Need for Artificial Gravity

Medical studies of astronauts exposed to long periods in orbit show serious declines in their bone strength, muscular tone and in their cardiovascular conditioning.

Transparency Master

There is also documented evidence for a decrease in the efficiency of the body's immune system.

The body generally recovers quickly when the force of gravity returns but bone decalcification takes along time to recover.

Also, in a free-fall environment (where there are no specific directions of "up" or "down") there many minor inconveniences that range from the inability to drink liquids from a cup to using the toilet or taking a shower.

For very long space missions these minor problems can become sources of major discomfort and annoyance adding to the psychological pressures of isolation and limited space.

To avoid some of the physiological and psychological problems created by long term exposure to the free-fall conditions associated with space travel, a simulated gravitational environment must be provided for humans traveling in space.

The only way to duplicate the effect of being at rest in a gravitational field is to provide an accelerated environment that simulates gravity and a uniformly rotating environment does this very nicely.


Long Rockets

In a nuclear rocket the reactor and the crew module should be as far apart as possible for safety reasons. The result is that nuclear powered spacecraft, which are designed to transport humans, are somewhat longer than spacecraft of a more conventional design.

This `extra' length has certain advantages. For example, it is possible to create an environment that simulates gravity by rotating the spacecraft. We can capitalize on the increased length of the spacecraft because by increasing the length of the spacecraft the rotation period for a given gravitational effect is reduced. This is a huge benefit since longer rotation periods are much more comfortable for the human passengers, and are less likely to cause motion sickness which is frequently caused by rotational motion.


Transparency Master
To fully understand the dynamics of rotating the spacecraft we need to investigate how the mass of the ship is distrubuted along its length, from the thrusters, at one end, to the crew module at the opposite end.

The nuclear reactor is very massive since nuclear fuel, radiation shielding, and other structural requirements result in a very massive and dense rocket engine.

This gives the spacecraft a centre of mass further aft of its geometric center than it is for a conventional, chemically powered, spacecraft

We can use centripetal acceleration to create an artificial gravitational field by the rotation of the nuclear powered spacecraft. Let's begin our investigation by considering the rotational behavior of a familiar object, a hammer.

Rotation and The Centre of Mass

An object that has a non-uniform structure, like a hammer, usually balances at a point that does not coincide with its geometric centre.

If you pick up a hammer you will notice that it has a balance point the is much closer to head of the hammer than it is to handle of the hammer.

The hammer can be balanced in three possible orientations, face down (as shown), on its side (the way you typically lie a hammer down) and on the end of its handle (a bit tricky).


Transparency Master
The point at which the hammer balances is sometimes called its "centre of gravity" or, more correctly, its " center of mass".

The centre of mass is actually a point inside the hammer where the lines of balance in each of three possible orientations intersect.

It is possible for the centre of mass of a complex shape such a metal crescent (a horseshoe shape) to lie outside of the structure itself.

Suppose we toss a hammer up in the air so that its spins. If we do, it will spin as shown in the figure below.

Can you see why it might be easier to catch the rotating hammer by the handle than by the head? If you have tried this experiment you might have also discovered that if you fail to catch the handle in your palm, it can give you painful crack on your knuckles instead.

This because the linear speed of the handle is much greater than the speed of the hammer's head.


Transparency Master
Dynamically the hammer will behave as if all of it's mass were concentrated at the centre of mass.

The centre of mass will rise and fall in a uniform vertical path, whereas the head and handle will rotate around the centre of mass (as shown in the illustration to the left).

The key things to note are

  1. the larger the distance from the rotational (centre of mass) the greater the speed of the object in motion.
  2. The rotational axis passes through the centre of mass.

Ballistic Trajectory and The Centre of Mass

If the hammer is thrown horizontally, so that plane of rotation is in the plane of the flight path, then the centre of mass will follow the ballistic trajectory (as if it were a small rock or ball),as shown.

The centre of mass will follow a trajectory that is in the shape of a parabola ..shown below in red.


Transparency Master
Notice again, (as pointed out above) that the hammer's handle has a large radius of rotation whereas the head of the hammer has a much smaller radius of rotation.

Imagine now that the hammer is a nuclear powered spacecraft.

The mass distribution of the spacecraft is similar to that of the hammer having much of the mass distributed towards one end.

The period of rotation of the head of the hammer and the handle of the hammer are both the same. But the radius of rotation of the handle is much larger than the radius of rotation for the head of the hammer, this means that the centripetal acceleration at the handle end is much larger than it is at the head of the hammer.

In the case of the spacecraft, the larger the distance of crew module is from the centre of rotation the slower the rotation speed (longer period) needed for a given centripetal force.

Rotational Orientation and The Centre of Mass

The direction of rotation for a spacecraft is somewhat arbitrary and has more to do with the direction of the astronaut's preferred line-of-sight than with the physics of the problem.

In the leftmost case below, the spacecraft rotates with a propeller-like motion. Provided that the spacecraft does not rotate along it's longitudinal axis, the same side of the spacecraft always faces in the direction spacecraft's motion through space.


Transparency Master
In the rightmost case, the rotation is in the plane of the spacecraft's motion through space.

As odd as it might seem at first, the astronauts inside the crew module cannot tell the difference between the left and right cases unless they make astronomical observations of the sun and stars outside the spacecraft.

As far as the effect of centripetal accelerations are concerned both cases are indistinguishable (from each other).

Apparent Weightlessness


Transparency Master
Inside a spacecraft, either in orbit, or in a planetary transfer ellipse, astronauts experience and effect which creates the illusion of weightlessness; that is, the force of gravity appears to vanish.

In the photo to the left, an astronaut "floats" in the apparent weightless environment of NASA's Skylab.

Gravity of course does not vanish, but the result of everything undergoing exactly the same acceleration disguises the effect of gravity, creating the illusion that the force of gravity has disappeared.

In a circular orbit we call this a "free-fall" environment, since the spacecraft is simply falling along a line directed at the planet's centre of mass while its tangential speed causes it to remain at a constant distance from the planet's surface. (Even spacecraft in elliptical orbits are in a state of free-fall)

The Spin-Up

A spacecraft in an Earth-Mars transfer orbit is in state of free-fall towards the Sun, and all gravitational effect seem to disappear.

The simulate the effect of gravity it will required that we place the rocket and the crew-module in a rotational mode so the centripetal acceleration directed towards the rocket's centre of rotation simulates the acceleration due to gravity.

The magnitude of the simulated gravitational acceleration can be selected by choosing the appropriate rotational speed.


Transparency Master

Transparency Master
In the above illustration the rotation of the spacecraft is counterclockwise (large red arrow). The smaller red arrows in the right hand figure represent the relative tangential velocity vectors of objects that are at various distances from the rotational axis of the spacecraft.

Notice that if an astronaut moves from a crouching position to a standing position the velocity of their head must decrease. In this example, the fact that one's head is moving (tangentially) too fast when they move from sitting (or crouching) to standing creates an interesting illusion called the Coriolis effect.

Coriolis Effect

The artificial gravity produced by rotation is far from ideal, especially when the rotational radius is small. When the radius of rotation of small an effect called the Coriolis Effect is easily noticed by those within the rotating environment.

The photo below is taken inside NASA's Skylab module. The radius of the Skylab module is about 6.6m. To produce 0.7g (70% of the Earth's gravitational acceleration) inside Skylab it would have to rotate at 9.7 rpm.


Transparency Master

Consider what happens when an astronaut is crouched down in a rotating spacecraft. Both feet and head are rotating with the same (i.e. constant) angular velocity, but the rotational circle of the feet is larger than the rotational circle of the astronaut's head.

This means that the linear speed of the astronaut's feet is greater (faster) than that of her head.

In other words, the closer an object is to the centre of rotation, the slower it must move (i.e. smaller linear velocity) in order to preserve a constant angular velocity.
Consider now what happens when she suddenly changes from crouching position to a standing position in the rotating spacecraft.

She cannot stand "straight up", the Coriolis effect causes her upper body to move in the direction of the rotation so that she tumbles (in the direction of the spacecraft's rotation).


Transparency Master
The Coriolis effect is an unavoidable result of artificial gravity produced by rotation. The effect is especially noticeable when the rotational of radius is small as in this Skylab simulation.

The Coriolis effect gives the illusion that mysterious forces are at work, the truth is that as one moves towards the centre of rotation the linear speed required to maintain the same rotational period ( i.e. the same rotational velocity) is less.

In simulating the effects of gravity in rotating environments, large rotational radii and long period (low frequency) rotations are best since the Coriolis effects are smaller.


Summary

In Skylab, where the radius of rotation is small, it is possible to throw an object upwards with a velocity that causes it to cross the lab in exactly one half a rotation, so that the object can be caught on the opposite side. The Coriolis effect makes the path of the object appear to be a loop!

In the animation below the leftmost image shows the motion of a tossed ball within a rotating cylinder, the rightmost animation illustrates how this motion is perceived by the astronaut within the rotating cylinder.

Outside (left above) ---- Inside (right above)

Transparency Master

When astronauts move around within the crew module the Coriolis effect creates the illusion of tangential forces.

These forces not only cause visual disorientation but the also act upon the balance sensitive organs within the inner ear and cause a loss of balance and create a sense of vertigo.

Large rotational radii are preferred. Luckily nuclear powered craft for human travel in space tend to be long in order to help isolate the astronauts from the nuclear reactor. Also the centre of mass is close to the reactor, making the rotation radius of the crew module larger still.
IMPORTANT: In our examples with Skylab two important things must be stated;
  1. Unlike our Skylab example, the trans-Martian spacecraft will rotate around an axis perpendicular its longitudinal axis, not around its longitudinal axis.

  2. As far as is known, Skylab was never deliberately put into a rotational mode to simulate gravity. Skylab was selected as an example because of its appealing geometry.


Student Activities

The Physics of Centripetal Acceleration

Background


Transparency Master

Exploring Centripetal Forces

The acceleration (m/s2) of an object moving in circular path of radius R metres and having a period T seconds is given by the expression

a =42 R/T2

Therefore the force (F, in newtons N) required to keep an object of mass m kilograms moving in circular path is given by

F =42 mR/T2

A derivation of the centripetal acceleration for objects moving in a circular path is given in the attached document called A Derivation of the Equations of Centripetal Acceleration

Student Activity 1

A simple experiment to investigate the relationship between the centripetal force, the rotational radius, and the rotation period can be performed using a short piece of plastic pipe (10cm), some study string (several metres) and some large (1-2cm bolts), a golf ball and a small sock.

Experiment

The experiment can be printed from the following document Activities Which Investigate Centripetal Acceleration

Student Activity 2


Transparency Master

Safe and Critical Rotation Speeds

The local centripetal acceleration that the astronauts experience in their spacecraft depends upon only two factors.
  1. The period of rotation
  2. The radius of rotation
The graph to the above shows the effects of increasing rotational period and increasing rotational radius on the centripetal acceleration.

Assignment

Referring to the graph determine each of the following.
  1. The period of rotation for 1.0g, if the radius of rotation is 50m.

  2. The period of rotation for 0.5g if the radius of rotation of 50m.

  3. The maximum rotational period for a structure whose rotational radius is 100m.

  4. The change in rotational period to increase from 0.5g to 1.0g if the rotational radius is 150m.

  5. The decrease in the period of rotation from 1g to structural failure for spacecraft having a 60m rotational radius.

Curriculum Outcomes:

Knowledge Outcomes GR 11-12

WORK-11-325.05: use vectors to represent force, velocity, and acceleration

WORK-11-325.06: analyze quantitatively the horizontal and vertical motion of a projectile

WORK-11-325.07: identify the frame of reference for a given motion

WORK-11-325.08: apply Newton"s laws of motion to explain inertia, the relationship between force, mass, and acceleration, and the interaction of forces between two objects

WORK-11-325.09: analyze quantitatively the relationship among force, distance, and work

WORK-11-325.10: analyze quantitatively the relationship among work, time, and power

WORK-11-325.11: analyze quantitatively two-dimensional motion in a horizontal plane and a vertical plane Resources WORK-11-325.12: describe uniform circular motion, using algebraic and vector analysis

WORK-11-325.13: explain quantitatively circular motion using Newton"s laws