A Banked Curve

Banked Curve• Train in Philadelphia

was going to fast• Banking too little• Example 5.42 pg 157• For a car traveling at a

certain speed, it is possible to bank a curve at just the right angle so that no friction at all is needed to maintain the car’s turning radius. Then a car can safely round the curve even on wet ice.

• Forces on Car• Gravity• Road (normal)• No Friction

n = n sinβ i + n cosβ j

Horizontal & Vertical Components• The acceleration in the x-direction is the centripetal

acceleration: arad = v2/R

• ΣFx = marad = mv2/R = nx = n sinβ• There is no acceleration in the y-direction• ΣFy = 0 = −mg + n cosβ

• mv2/R = n sinβ• mg = n cosβ

Divide the equations:

v2/Rg = sinβ/cosβ

sin = opp/hypcos = adj/hyp sin / cos = opp/adj = tan

Final Result

• mv2/Rg = sinβ/cosβ = tan β

• β = tan-1(v2/Rg)