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Physics Tutorial: Banked Curves

 

Banked curves were invented to prevent skidding. With banked curves, the road's normal force provides a

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component of force directing a vehicle towards the center of the curve. Banked curves reduce a vehicle's dependence on friction to safely navigate a curve.

 

Uniform circular motion is used to describe objects moving in a circular motion. With centripetal acceleration, the magnitude of the velocity remains unvarying; however the direction of this velocity is constantly changing. Acceleration is always directed to the center, thus force is always directed to the center.

 

This motion may be visible with vehicles rounding curves. On a flat surface, cars must rely on friction to prevent skidding. If conditions such as snow, ice, or rain reduce the coefficient of friction, then the turning force would be smaller. Without banked curves to help turn the vehicle the vehicle's inertia could carry it off the road.

 

PROBLEM: An exit ramp on a certain interstate has a radius of curvature of 75m. If the ramp is banked 20 degrees, what is the maximum speed that can safely be executed?

 

 


 

SOLUTION:

Horizontal component:

Use the formula:

 

Fnet = (mv2)/r

 

But the horizontal component of normal force, N, is equal to Fnet so we can substitute that

 

information into the equation.

 

Nsinø = (mv2)/r where ø is the slope of the surface above the horizontal (the bank).


Vertical component:

 

The vertical component of N is equal to weight, W.

 

N cos ø = mg

 

When we change the formula around, we get N.

 

N = (mg)/(cos ø)

 

The next step it to substitute the N in the horizontal component formula with the vertical component equation.

 

[(mg)/(cos ø)] sin ø = (mv2)/r

 

mg tan ø = (mv2)/r

 

Let's cancel out the m on each side.

 

g tan ø = v2/r

 

Since we are solving for speed, let's multiply each side by r to get v on its own.

 

rg tan ø = v2

 

Plug the values in.

 

v2 = (75m)(9.8m/s2)(tan 20º)

 

v2 = 267.5 m2/s2

 

Take the square root.

 

v = 16.4 m/s

 

The maximum speed is 16.4 m/s.

 

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Amanda Taphorn