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**1. The problem statement, all variables and given/known data**

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.

**2. Relevant equations & 3. The attempt at a solution**

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!