Question

A concrete highway curve of radius 84 m is banked at a 15.3 degree angle. What is the maximum speed with which a 1500 kg rubber-tired car can take this curve without sliding?

First of all, what is the relevant coefficient of friction? (It's 1)

What is the magnitude of the normal force acting on the car?

What is the maximum speed the car can take this curve without sliding?

Does this maximum speed depend on the mass of the car?

Answer 1

We need to derive an equation for velocity on the curve. The forces acting horizontally (the centripetal forces) are the horizontal component of the normal force (nsinθ) and the horizontal component of friction (μncosθ). These are both acting toward the center of the curve, so both are added together and their sum set equal to the centripetal acceleration:

mv²/r = nsinθ + μncosθ

mv²/r = n(sinθ + μcosθ)------------------->(1)

The normal force is our only unknown (besides velocity), but we may write an expression for it from the vertical forces. These forces are the vertical component of the normal force (ncosθ), the vertical component of friction (-μnsinθ), and the weight of the car (-mg). Since there is no acceleration vertically, these forces net to zero:

ncosθ - μnsinθ - mg = 0

n(cosθ - μsinθ) = mg

n = mg / (cosθ - μsinθ)---------------------->(2)

So normal force:

n = (1500*9.8)/(cos15.3 – 1*sin15.3)

= 2.1*10^4 N (ans)

Now plug (2) into (1) and solve for v:

mv²/r = [mg / (cosθ - μsinθ)](sinθ + μcosθ)

v = √[gr(sinθ + μcosθ) / (cosθ - μsinθ)]

= √[(9.80m/s²)(84m)(sin15.3° + 1.00cos15.3°) / (cos15.3° - 1.00sin15.3°)]

= 38 m/s

Max speed does not depend on the mass of the car.