Further Examples

Horizontal Motion
When you accelerate in a car you feel shoved backward into the seat.  Really what's happening is the seat is shoving you in the back, but we can't tell the difference between the two descriptions.  What I mean by that is that there really is no difference between the sensation of gravity and the sensation of acceleration.  That's why accelerating in some direction will effectively change the way gravity seems to behave. [br][br]Suppose you have a cup of coffee on the dashboard of your car when a traffic light turns green and you accelerate(bad idea).  Naturally it'd fly toward the rear of the car.  Why?  [br][br]In light of prior conversations friction between the cup and dashboard was apparently not enough to allow it to keep pace accelerating with the vehicle.  In light of our present conversation, we can see it as a change in effective gravitation.  If you are accelerating in the x-direction at 5m/s[sup]2[/sup], and gravity is 10m/s[sup]2[/sup] downward in what I'll call here the -y direction, then the effective gravitational constant that the cup and you feel is [math]\vec{g}_{effective}=\vec{g}-\vec{a}=-10m/s^2\hat{y}-5m/s^2\hat{i}.[/math]  This means gravity now feels as if it is pointing downward and toward the rear of the vehicle.  The angle is [math]\tan\theta=5/10[/math] or roughly 27[sup]o[/sup] toward the rear of the car from downward.  It stands to reason that it would slide off the dashboard since effectively gravity is pulling it not downward, but 27 degrees rearward.  Assuming the friction between the cup and dashboard isn't very big, it slides.
Accelerating Train
Have you ever tried standing on a train without holding on while it accelerates away from a station? If you have, or if you could imagine it anyway, you need to lean forward in the direction of the acceleration a bit. That's because of the exact same reason. If the train is accelerating at the rate of the previous example (not likely), you'd need to lean forward 27 degrees to stay balanced. If you dropped something from your hand, it'd fall backward and downward at a 27 degree angle with respect to vertical. The rate of acceleration of your keys from your hand would be [math]|\vec{g}_{effective}|=\sqrt{(5m/s^2)^2+(10m/s^2)^2}=11.2m/s^2.[/math]
ISS Astronauts
Even without knowing any physics about orbits, there's one thing you can know for certain. Since earth's gravity pulls all things toward earth's center, and since at the location of the International Space Station (ISS) gravity is 89% as strong as it is here, and since they are clearly experiencing zero gravity, they must be accelerating directly toward the center of the earth at 89% of 9.8m/s[sup]2[/sup]=8.7m/s[sup]2[/sup]. This must be true since the effective gravitation is zero. This implies that the downward gravitational field must be cancelled by an equal downward acceleration.[br][br]How this is possible in a sustained fashion is that they are undergoing centripetal acceleration while in orbit.  While centripetal acceleration is directed toward the center of the circular path, recall that such motion nonetheless does not result in getting any closer to the center of the path.  So in a very real sense, to orbit means to continually fall toward earth while never getting any closer to earth!
Skating Down a Ramp
If you ever try skateboarding down an incline you'll notice that you have to stand perpendicular to the incline to keep your balance.  In other words you certainly do not remain vertical if you want to stay on the skateboard for long.  Why is this?  Look at the diagram below.  When the acceleration is down the incline at a rate [math]a=g\sin\theta[/math] the direction of [math]\vec{g}_{effective}[/math] is perpendicular to the incline.  That's the way gravity seems to be pulling you as you skate down a hill, and is also the direction your phone will fall out of your hand relative to you if you drop it while filming yourself skating down the hill.
In this centrifuge one places test tubes.  Each of the holders sits vertically when at rest, but as the centrifuge rotates, the individual buckets holding the tubes swing outward on pivots.  At very high rotation rates the tops of the test tubes will all be facing the center of the centrifuge because the outside edge will effectively be "down" to them.
Centrifugal (not Centripetal) Force
Many physics textbooks make a strong statement about the virtual (not real) nature of the centrifugal force, which is the one that you feel pushes you outward as you travel in circular motion - as when rounding a corner in a car.  They tell you that the centrifugal force doesn't actually exist.  [br][br]While it's true that in the coordinate system of the road the centrifugal force doesn't exist, in your accelerated reference frame it is very real and is indistinguishable from gravity.  It is real because you can measure it with accelerometers or force probes or any of the tools commonly used in physics labs.  [br][br]Why do you feel pushed toward the right side of a car when it turns left?  Based on our previous studies, in the non-accelerated refernce frame of the earth, your centripetal acceleration is directed toward the center of the circular path (left) and is given by [math]\vec{a}_c=-\frac{v^2}{r}\hat{r}.[/math]  The effective gravity that you feel while being inside the accelerating car is made up of the opposite of this vector plus earth's gravitational field, or [math]\vec{g}_{effective}=\vec{g}-\vec{a}=\vec{g}+\frac{v^2}{r}\hat{r}.[/math]   Inside the car where you feel this, things do in fact slide to the outside of the car because gravity acquires an additional[i] outward [/i]radial component.  [br][br]This principle is used to design [b]centrifuges[/b], or rotating devices used to separate fluids based on density.  In a cup resting on a table, the denser fluids eventually find their way to the bottom of a cup - as when you mix oil and water.  However, if the fluids don't flow very quickly or if there is only a subtle difference in density then we'd need stronger gravity to cause them to separate.  With the goal of making stronger gravity in mind, such fluids are placed in rapidly spinning vessels and can experience hundreds or thousands of times earth's gravity toward the outside of the spinning centrifuge.  The densest materials end up on the outside edge and the least dense closest to the center.
Lean Angle
When you ride a bicycle or a motorcycle, to stay balanced means to keep the contact patch where the rubber meets the road, directly beneath your center of mass.  However, when rounding a turn you must lean.  As the rider, there is a clear sensation of being pushed outward by the centrifugal force.  This is because to you the rider there is a component of effective gravity pointing outward that has a magnitude exactly equal to the centripetal force, but in the opposite direction.  [br][br]How then should we find the required lean angle for a given turn?  Simple!  Effective gravity tells us which way is 'down'.  The wheels need to be under you in that direction.  Look at the diagram below.  From the angle we can easily see the connection between gravity, the centripetal acceleration and the lean angle.
Lean Angle due to Effective Gravitation