Physics Question

Question

A racetrack curve has radius 70.0 m and is banked at an angle of 21.0 ∘. The coefficient of static friction between the tires and the roadway is 0.400. A race car with mass 1200 kg rounds the curve with the maximum speed to avoid skidding. (A) As the car rounds the curve, what is the normal force exerted on it by the road? (B) What is the car's radial acceleration? (C) What is the car's speed?

Answer 1

Hi Kevin!

Lets start with our knowns

r = Curve Radius = 70m

∅ = curve bank angle = 21 deg

μ = coefficient of friction = .4

Lets define axes.

Let Y be the vertical axis (i.e up or down from the perspective of the car, where positive is up)

Let X be the horizontal axis (i.e. where positive is along the radius of the racetrack curve)

What forces are acting on the car?

The weight of the car (w = mg)

The normal force of the car (N)

The friction from the tires (Ff = μN)

I highly recommend drawing these on a freebody diagram to help you visualize what trig functions to use for each force component

Using these two axes lets sum the equations across the two axes defined:

ΣFx = -Ff* cos(ø) + N* sin(ø) = ma (we can't zero out this equation, because objects traveling in a circle have a center seeking acceleration)

ΣFy = N*cos(ø) - w + Ff*sin(ø)= ma = 0 (since the car is not accelerating vertically, a is zero!)

Plugging in the defined equations above, we can insert

ΣFx = - μN* cos(ø) + N* sin(ø) = ma

ΣFy = N*cos(ø) - mg + μN*sin(ø) = 0

For part a, all you need to do is solve one of those equations for N. (hint: solve the one with only one unknown variable)

For part b, since we now know the normal force, the equation for the radial acceleration (on the x axis) has only one unknown, so solve that equation for the acceleration!

For part c, we need to know the relationship between speed and center seeking accelerations. It is likely on your equation sheet. but here it is, a = v²/r. Solve for the unknown!