Circular motion: An aeroplane loops the loop

[Bad_Boy_Blue]

Question:

An aeroplane loops the loop in a vertical circle of radius 200m, with a speed of 40m/s at the top of thel oop. The pilot has a mass of 80kg.

What is the tension in the strap holding him into his seat when he is at the top of the loop?.

Answer:

The centrifugal force outwards due to the rotation is mv^2/r = 80*40^2/200 = 640N. Since its at the top of the loop, this points upwards, in the opposite direction to gravity.
The pilots mass is 80kg, and thus his weight is 80g = 800N (if you are supposed to take 9.8 or 9.81, then do so and change the numbers accordingly). This points downwards.
Total force in the direction of up/down (taking down as positive) is 800-640 = 160N
Thus his strap has to pull upwards with a force of 160N

But, I have been taught that centripetal (centrifugal) acceleration (and therefore force) is always directed towards the centre of the circle described by the body in circular motion. Therefore, wouldn't the centripetal force act downwards (in addition to the force on the pilot due to gravity)?

 

[NateTG]

But, I have been taught that centripetal (centrifugal) acceleration (and therefore force) is always directed towards the centre of the circle described by the body in circular motion. Therefore, wouldn't the centripetal force act downwards (in addition to the force on the pilot due to gravity)?

The centripetal acceleration needs to be provided by an outside force. It's not something that 'automatically' results from rotational motion. The tendency of things to 'fly out' of circular paths is due to the natural tendency of things to travel in straight lines.

The centripetal force is an 'external' force that provides the acceleration necessary to keep the object on it's circular path. In your example, the gravity is acting as the centripetal force at the top of the loop. Since gravity is providing more acceleration than necessary, there needs to be an upward force (the tension) to keep the pilot on his circular path.

The centrifugal force is a 'fictional' force that exists in rotating reference frames which is added to account for acceleration of the reference frame, and that acts outward from the center of rotation. In this example, the force of gravity is larger than the centrifugal force, so there is a net 'downward' force that must be compensated for by the strap.

 

[HallsofIvy]

But, I have been taught that centripetal (centrifugal) acceleration (and therefore force) is always directed towards the centre of the circle described by the body in circular motion.

I hope you haven't been taught that "centripetal" and "centrifugal" are the same thing! Centripetal force is directed toward the center. Because our brain wants to think of the circular motion as "natural", it interprets the force we have apply toward the center as necessary to counter the "ficticious" force (as NateTG said) that we call centrifugal force pulling outward.

 

[Bad_Boy_Blue]

OMG, this site is so cool! Now I can get help without having to wait for my teachers!

 

[LURCH]

That's the same way I felt when I found this site, Bad Boy!

To further illucidate; as has already been mentioned by Nate, centripedal force acts toward the center and, in this example, is being provided by gravity. This is true at the top of the loop because the plane's speed and angle of "climb" are not sufficient to generate one g of upward force. If the plane were going faster or turning more sharply, (or at any other point in the loop) centripedal force would be provided by the seat under the pilot, pushing him toward the center when his inertia tries to take him in a strait line.

 

[Cleonis]

But, I have been taught that centripetal (centrifugal) acceleration (and therefore force) is always directed towards the centre of the circle described by the body in circular motion.

As others have already pointed out, it depends on the circumstances whether required centripetal force is actually present.

If required centripetal force is present then it acts towards the center of of the circular motion. In 2009 the team of british television program 'fifth gear' set up a loop-the-loop with a car, looping inside a 40-foot arch that was built for the occasion.

That's a very dangerous stunt. The car needs to be very small, to fit into the curve. Go too fast and the suspension will bottom out, friction reduces speed and the car crashes. Go too slow and the car loses contact with the surface, possibly going out of control. (In fact in the 'fifth gear' stunt there was next to no margin of error. Minimum speed and maximum speed were within a few mph of each other.)

Another example is the shape of a loop-the-loop in a rollercoaster ride. Notice how the curvature is strongest at the top. That way the passengers do not experience too large a change in pulling G's.

When the rollercoaster cart is on a level track you are pulling 1 G, the Earth's gravity. Curving up the G-count probably goes up to 1.5 G or so. At the top the curvature is such that (I guestimate) the required acceleration is 1.5 times G. At the top of the loop gravity tends to pull the passengers out of their seats, so the seats only get to exert 0.5 G of centripetal force.

I think there are also rollercoaster rides where the passengers do experience negative G's. On that kind of rides passengers must be cushioned from all sides to prevent injury.

The usual setup, I think, is that the ride is such that the passengers always experience positive G's (say, between 0.5 and 2 G). That way the passengers remain pressed into the seats, and a bar across the hips is enough to secure them.

Some remars about flying a loop-the-loop.
If an unskilled pilot does a loop-the-loop without enough velocity then the airplane itself is like a car doing a loop-the-loop without enough velocity: the very airplane will fall like a brick. There's no reason for the pilot to fall out of his seat, as plane and pilot are falling together. The bad news is: the falling airplane is moving erratically, the unskilled pilot won't be able to make the it airborne again - he will crash.

 

[Danger]

I believe that you have just set a new record for necroposting.

 

[Cleonis]

I believe that you have just set a new record for necroposting.

Darn!

Sometimes I google terms that interest me. Google will say something like: "Last post 2 days ago".

When spam has been posted in a Physicsforums thread, and google visits that page before an administrator has deleted that spam, the thread comes up as a recent thread in Google results. So I always check the date, but with this %&#$* search result I forgot to check.

 

[Danger]

Not to worry, Mate. We've all done it at least once.