Problem about an incomplete Loop-the-loop

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In summary, the conversation discusses a problem involving an automobile of mass M driving onto a loop-the-loop at a constant speed v, where v < v0. The coefficient of friction between the car and the track is μ, and the question is to find an equation for the angle θ where the car starts to slip. The equations for the forces acting on the car are discussed, and it is determined that the net force in the tangential direction must be zero for the car to maintain a constant speed. Substituting this into the equations, an equation for θ is obtained, but it cannot be solved. It is suggested to use a trig identity to relate sinθ and cosθ and solve for θ
  • #1
rivendell
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Homework Statement


An automobile of mass M drives onto a loop-the-loop, as shown. (click here for diagram) The minimum speed for going completely around the loop without falling off is v0. However, the automobile drives at constant speed v, where v < v0. The coefficient of friction between the auto and the track is μ. Find an equation for the angle θ where the auto starts to slip. There is no need to solve the equation.

Homework Equations

& 3. The Attempt at a Solution [/B]
FBD equations (where f represents F of friction)
Radial direction --> N-Wcosθ = M(v^2/R)
Tangential direction --> -f = MR(θ") = m(dv/dt)
so I get dv/dt = -f/M (1)
and N = Wcosθ + M(v^2/R) (2)
Then I substitute (2) into (1) to get dv/dt = -u(gcosθ + v^2/R).
But I can't solve this differential equation. I'd only be able to do that if I didn't consider gravity at all in this problem. If I could solve it, then I'd be able to get an equation for θ. Should I not consider gravity to make solving the diff eq easier? If these steps so far are right and I can continue this way, how can solve the diff eq I have above? Thank you!
 
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  • #2
If the car travels at constant speed, what is the value of dv/dt?

Is friction the only force acting in the tangential direction?

Does friction on the car act in the direction of increasing θ or decreasing θ?
 
  • #3
thanks for noting that! I'll correct my writeup:
-f - Wsinθ = MR(θ") = M(dv/dt)
--> dv/dt = -f/M - gsinθ
--> N = Wcosθ + M(v^2/R)
Subst: dv/dt = (-ugcosθ + v^2/R)/M - gsinθ = 0
I guess I can't solve this eq, which is ok, as long as it's right. friction acts in direction of decreasing θ, which I think my eqs show. Is this right?
 
  • #4
rivendell said:
--> dv/dt = -f/M - gsinθ
Still have a problem with the direction of f. Does f try to push the car down the slope or does it try to prevent the car from slipping down the slope?

--> N = Wcosθ + M(v^2/R)
Looks good.

Subst: dv/dt = (-ugcosθ + v^2/R)/M - gsinθ = 0
I don't believe you substituted correctly here. Check this.

I guess I can't solve this eq, which is ok, as long as it's right.

You could solve for θ using a trig identity to relate sinθ and cosθ. But you are not asked to do that.
 
  • #5
I thought f was just going against θ, so I had written that.
Should I have written dv/dt = f/M - gsinθ instead?
If I did that, then trying again subbing, dv/dt = u(Mgcos + Mv^2/R)/M - gsinθ = u(gcosθ + v^2/R) - gsinθ = 0. Does that work? and just if I wanted to solve for θ, would I use the sin double-angle identity (sin2θ = 2sinθcosθ)? thanks!
 
  • #6
rivendell said:
I thought f was just going against θ, so I had written that.
Should I have written dv/dt = f/M - gsinθ instead?
Think about a car going at constant speed up a slope. No acceleration implies that the net force along the slope is zero. If gravity has a component down the slope, then what direction does friction need to act in order to make the net force zero?
If I did that, then trying again subbing, dv/dt = u(Mgcos + Mv^2/R)/M - gsinθ = u(gcosθ + v^2/R) - gsinθ = 0. Does that work?
Yes. Looks right.

and just if I wanted to solve for θ, would I use the sin double-angle identity (sin2θ = 2sinθcosθ)?
I don't see how that identity would be directly helpful. Use sin2θ +cos2θ = 1 to write cosθ in terms of sinθ. Then you can get the equation in terms of one unknown: sinθ. You should be able to manipulate it into a quadratic equation for sinθ.
 
  • #7
got it. Thank you TSny!
 
  • #8
Why does net force in the tangential direction have to equal zero? I thought both the friction force and the force of the gravitational component would both oppose the direction of motion. Also, the question asks when the car starts slipping, does this mean when vT=0?
 
  • #9
Hello, Matt. Welcome to PF.

Friction does not always act opposite to the direction of motion. Sometimes, the friction force acting on an object is in the same direction as the motion of the object. A good example is a car starting from rest and accelerating along a horizontal surface. What force acting on the car causes the car to accelerate?

In the loop-the-loop problem, the car moves at constant speed along the circle. So, the car has zero acceleration in the tangential direction of the circle.

The point at which the car starts slipping is the point where the static force of friction between the tires and the loop is at maximum possible value.
 
  • #10
Oh ok, i was treating this as a block problem, this makes a lot more sence, thanx!
 

FAQ: Problem about an incomplete Loop-the-loop

1. What is a loop-the-loop?

A loop-the-loop is a type of roller coaster or amusement park ride where the track forms a complete loop, allowing the riders to experience a moment of weightlessness as they are inverted.

2. What is the problem with an incomplete loop-the-loop?

The problem with an incomplete loop-the-loop is that it can cause the riders to experience a sudden drop in acceleration, leading to a jarring and potentially dangerous experience. This is known as a "death drop" and can result in injuries or even fatalities.

3. What factors contribute to an incomplete loop-the-loop?

There are several factors that can contribute to an incomplete loop-the-loop, including the design and construction of the track, the speed of the ride, and the weight and position of the riders. Any of these factors can cause the train to lose momentum and fail to complete the loop.

4. How can scientists prevent incomplete loop-the-loops?

Scientists and engineers can prevent incomplete loop-the-loops by carefully designing and testing the track and train, taking into account factors such as speed, weight, and acceleration. They can also use computer simulations to identify potential problems and make adjustments before the ride is built.

5. What safety measures are in place for loop-the-loop rides?

Safety measures for loop-the-loop rides include regular inspections and maintenance of the track and trains, as well as strict regulations and standards set by governing bodies such as the International Association of Amusement Parks and Attractions. Additionally, riders are required to follow safety instructions and guidelines, such as keeping their hands and arms inside the train at all times.

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