How do I answer these physics questions and what answer should I get?

Question

You want to design the automobile racing circuit in Los Angeles to attract tourists to boost

economy during Spring. First, you design the banked road for the racing cars to speed at 100mph

for a circular circuit of radius 1000 m. If you want to allow the speed of racing cars up to 200

mph, how much of the friction between the tire and the road should be added to banked road?

(a) Draw the free-body diagram with all the forces acting on a race car without friction.

(b) Set the equations of motion for the race car in your coordinate system.

(c) Find the banking angle for a speeding car at 100 mph.

(d) Include frictional force to your free-body diagram.

(e) Find the minimum coefficient of static friction to allow the speed of 200mph.

Answer 1

Let

m (unknown) be the mass of a car,

r = 1000 m be the radius of the track,

v = 100 mph (= 44.7 m/s) be car's speed without friction,

v' = 200 mph (= 89.4 m/s) be car's speed with friction,

g = 9.81 m/s² be gravitational acceleration,

α (unknown) be the banking angle

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(a)

Let me refer to your diagram, although it is a little misleading.

Given the picture of the incline, we are looking at the car from the

front or back, not side; so wheels should appear as rectangles, not circles.

Without friction there are three forces acting on the car in the car's coordinate system:

- The force of gravity, G, pointing down,

- Centrifugal force, C, pointing horizontally to the right,

- Normal force, N, pointing upward to the left, perpendicular to the surface.

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(b)

|G| = mg (1)

|C| = mv²/r (2)

N + G + C = 0 (3)

Some of the above equations refer to the norm, e.g. |G|, of a force, not to the force itself, such as G.

This is because the forces are vectors and some equations give only their sizes of the forces, not their direction.

Equation (1) for |G| is due to gravity.

Equation (2) for |C| is due the car's circular motion.

Equation (3) expresses the requirement that the car be balanced on the track.

That means the vector sum of all three forces must equal 0.

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(c)

Equation (3) is vector identity, and as such continues to be valid when the vectors are projected on any line.

To find the banking angle it is convenient to use the projection on the line parallel to the road surface.

The reason is that for the purpose of calculating α we do not care about N,

nor do we know its value, and the projection of N on the surface is conveniently 0.

Referring to the free-body diagram, equation (3) projects into

|G|sin(α) = |C|cos(α) (4)

Equation (4) basically expresses the balance between gravity and the car's speed

preventing it from sliding even in the absence of friction.

Substituting (1) and (2) into (4) and simplifying:

mgsin(α) = mv²cos(α)/r

tan(α) = v²/gr

Substituting actual numbers:

tan(α) = 44.7²/9.81×1000 = 0.2

α = 11.5°

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(d)

The force of friction always acts parallel to the surface.

Its direction depends on whether the car is below or above 100 mph --

the critical speed for which the track was designed to work even without friction.

Below 100 mph friction prevents the car for sliding down, therefore the direction of friction would be upward (to the right).

Above 100 mph friction prevents the car for sliding up, therefore the direction of friction would be downward (to the left).

The latter is of interest here because we are to consider v' = 200 mph.

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(e)

With the new velocity v' we have new equations

|C'| = mv'²/r (20)

N' + G + C' + F' = 0 (30)

The equations (20), (30) are simple rewrites of (2), (3) with the addition of the new force of friction F'.

The coefficient of friction (to be calculated) is defined as

f = |F'|/|N'|

(I) First calculate |F'|:

Subtract (3) from (30)

N'-N + C'-C + F' = 0 (31)

Again we take the projection of (31) on the line parallel to the surface for

similar reasons as above -- to make N'-N disappear, and to calculate F', which is our goal:

|F'| = (|C' - C|)cos(α) (40)

Substituting (2) and (20) into (40):

|F'| = (mv'²/r - mv²/r)cos(α) (41)

(II) To calculate |N'| take the projection of (30) on the line perpendicular to the surface.

The reason is that N' is now our goal.

|N'| = |G|cos(α) + |C'|sin(α) (50)

Now we can use (41) and (50) to calculate the coefficient of friction

f = |F'|/|N'| = (|C' - C|)cos(α) / (|G|cos(α) + |C'|sin(α))

Substituting from (1), (2), (20)

f = (mv'²/r - mv²/r)cos(α) / (mgcos(α) + mv'²sin(α)/r)

= (v'² - v²)cos(α) / (grcos(α) + v'²sin(α))

Substituting actual numbers

f = (89.4² - 44.7²)×cos(11.5°)/(9.81×1000×cos(11.5°) + 89.4²×sin(11.5°)) = 0.52