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Loop de loop answer part 1

Figuring out the minimum speed at the top of the loop de loop to stay on the track. Created by Sal Khan.

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• At , why is that we take acceleration at the topmost point to be equal to gravity? Why is the normal reaction taken as zero?
• To keep the car in a circular path, the centripetal acceleration (v^2/r) has to be greater than or equal to the acceleration due to gravity (g = 9.81 m/s^2). We set a = 9.81 because this gives us the minimum speed the car must have to stay in a circular path. As soon as the car goes slower than this, g will be greater than the centripetal acceleration, so the car will fall off the track.

At the top of the loop:
Fnet = ma
Fgravity + Fnormal = ma, and because a = centripetal acceleration = v^2/r, then
Fgravity + Fnormal = mv^2/r
So at the minimum speed for the car to stay in a circular path, Fnormal = 0. As the car's speed increases, however, so does Fnormal (because Fgravity stays constant).
• According to the video on fifth-gear website a Professor at Cambridge University calculated that he had to be going 36 miles per hour. Was he just building in a safety factor?
• I think the 36 mph is how fast you'd have to be going when you hit the ramp, assuming you weren't able to keep the engine running with enough traction on the wheels to keep the speed constant as you climbed up. So if you hit the ramp at 36 mph, the car will slow down but still be going faster than the minimum required (16 mph) when it gets to the top.
• It might sound stupid to ask this question, but I've got this question in mind that if the direction of the centripedal force is toward the center of the loop de loop, when the car was at the top, the centripedal force and the gravity force have the same direction, so the car will be pulled down with the two forces! What is the force that exert the car out of the center?
• For the car to not fall, it needs to stay in a circular path. Every object that moves in a circular path, with a constant speed, experiences a centripetal acceleration. So if something is moving along a circle it's experiencing centripetal acceleration, or the other way around, if something is experiencing centripetal acceleration, then it has to move in a circular path. So if we want to make sure that the car does not fall, we have to maintain it's circular motion, which means that all the forces on the car have to sum up to a centripetal force. For a minimal speed, the only force that the car is experiencing (at the top) is the force of gravity, so by simply saying that the force of gravity is a centripetal force, the car has to stay in a circular motion.

The centripetal force always points towards the center, but that does not mean that the car will go towards the center. Remember that the car also has forward velocity, so the centripetal force affects the direction of the forward velocity in such a way that it makes the car go in a circle. That's literally the definition of centripetal acceleration. If the car did not go in a circular path, then it would not experience centripetal acceleration.
• I'm having trouble with an initial concept Sal makes around - Why does AC = 9.81m/s^2? The 9.81 is from the force to due gravity, not centripetal acceleration. The 9.81 is in effect on the car at all points around the circle, not just at the top.
• I've had to revisit this topic because I realised that I didn't really understand it all!

I understand that at the top of the loop de loop, centripetal acceleration is attributed by gravity and normal force, but what EXACTLY IS PREVENTING the car from falling off? Gravity and normal force are pointed towards the centre of the loop de loop, and if centripetal acceleration needs to be equal to or greater than these combined forces, shouldn't the car just fall off the track because there is an overall net force towards the centre?

The only "logical deductions" that I've come up with are:

1) The vehicle experiences inertia in that it would otherwise travel on the tangential if it wasn't for the inward force attributed by centripetal acceleration

2) As per Newton's third law, the vehicle pushes on the track with a force that is equal in magnitude, but opposite in direction to the which the track pushes on the vehicle

I still, however, cannot grasp how the vehicle stays on the track in spite of the above mentioned factors. Any help would be greatly appreciated! :)
• So does the weight of the car not matter? I mean even if the car would weight 5 tons it would still make and loop de loop?
• Yes because it would have the kinetic energy associated with a 5 ton weight having been lifted to the launch height
• My text book derives the formula for critical speed as root(5gr) ..... If i calculate it by that formula i get a different answer.. And yes the derivation is very convincing :p
• sqrt(5gr) is the speed required at the BASE of the loop de loop.
So the kinetic energy at the base is 1/2 mv^2 = 5/2 mgr
Using the conservation of mechanical energy,
we get 5/2 mgr -2mgr = 1/2 mgr at the TOP of the loop. (this is kinetic energy)
and again using Ek = 1/2 mv^2,
The speed at the TOP is sqrt(2gr), as Sal has mentioned.
• How much harder do these kinds of problems become if the track has the more general shape of an ellipse?
In other words, how would you calculate this exact same problem if the width and height were different, and not perfectly equal as they are for a circle?

(Of course, a circle is a type of ellipse)
• I found your question very interesting!
Let's work with an ellipse of semi major axis a, semi minor axis b and b is parallel to the gravity.
First, think about what type of movement is naturally elliptical. The best example, I believe, is the planetary motion. So, from that we can withdraw the following results:
At a given point, the planet feels a central force directed towards the focus of the type
(GM).m/r² , where G is the gravitational constant and M is the star's mass;
The planet's speed v = sqrt[GM.(2/r - 1/a)] ------ This is a planet's speed equation as a function of the distance from the star;
So, writing out the constant GM, we can obtain
F = mv²/a
That's what happens for a planet in elliptical motion
We also need the following fact:
For any body running on a specified path, the perpendicular force acting upon it equals mv²/R , where R is the curvature radius of the path at that point.
So, what is the ellipse's curvature radius at the top? Here we use the planetary motion's result. The perpendicular force is F.(b/a) , since (b/a) is the cosine of the angle formed between the force directed towards the focus and the line perpendicular to the ellipse's surface at the top (which is vertical). Thus, mv²b/a² = mv²/R , so R = a²/b at the top of the ellipse.
Knowing the curvature radius R, we can easily find P + N, because of the fact mentioned above: "For any body running on a specified path, the perpendicular force acting upon it equals mv²/R , where R is the curvature radius of the path at that point" . Since the perpendicular force equals P + N, then P + N = mv²/(a²/b) , for minimum required speed N = 0, so v = a.sqrt(g/b)
Notice that this result coincides for what Sal found if a = b (a.k.a. circle)

EDIT : I changed my last comment because I believe the explanation with analogy is unsufficient, one must also include the curvature radius, making the result much more general (the car can have any value for its acceleration parallel to the surface) and mathematically accurate.