Car negotiating a bend

1. Dec 12, 2007

kudoushinichi88

I am asking this question on behalf of a forummer from Gendou.com's Physics Board... Sorry if a similar question has been posted before...

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Here is a mechanics question which totally stumped me:

A car together with its passengers has a combined mass of 2000kg. Its width, wheel-to-wheel is 2m and its center of gravity is 1m from the ground and midway between the wheels. It enters a roundabout which has a diameter of 10m.
Calculate the speed below which it must travel to ensure that it will not flip over about the outer wheels. Is the car were to travel at half this max speed, what is the combined frictional forces on the wheels?

Here are a couple of things that I'm quite confused about if you don't want to bother with the calculations:
a) what causes the car to flip in the first place?
b) in this case, if centripetal force is the resultant force of circular motion, what causes it in the first place?
c) does the difference of normal force on both sides of the car cause the car to overturn?
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Now, I understand that when the car accelerates around the roundabout, the center of gravity of the car moves away from the center of the roundabout due to it's tendency to move in a straight line. And when the center of gravity goes over the outer wheels, a torque is produced causing the car to flip.

But I don't know how to describe this mathematically. What puzzles me the most is, how do you explain mathematically what exactly causes the center of mass to move?

2. Dec 12, 2007

cryptoguy

a) On the roundabout the centripetal force is provided by the friction force alone. At the threshold speed, (mv^2)/r = $$\mu$$Fn. As soon as the centripetal force is greater then the max Friction (provided by the tires), the car will skid, or, flip.

Now I'm not completely sure about this but here's my try... The tires on the outside of the bend have a larger Fc (because v^2 is larger) so they do not immediately lose traction. The tires on the inside of the curve, however do lose traction which effetively means they are partially in air. This shifts the center of gravity toward the outer part of the car.

Not sure if this is a correct/adequate explanation, maybe someone will correct me.

3. Dec 12, 2007

kudoushinichi88

Yes, I initially thought that the centripetal force is provided purely by the frictional force. Does that mean in this question, we are missing the coefficient of friction between the tires and the road?

4. Dec 12, 2007

rl.bhat

When a vehicle is moving along unbanked road with forces of friction f1( inner tire) and f2( outer tire) and normal reaction R1 and R2, then taking moments about the center of gravity, we get (f1+f2)*h = (R1-R2)*b/2, where h is the height of cg from ground and b is the distance between the wheels. f1 +f2 = mv^2/r and R1 + R2 = mg. Solving the two equations we get R1 = 1/2*[ mg - 2mv^2h/rb] As velocity increases R1 goes on decreasesing and a stage is reached when R1 = 0 and inner wheel leaves the ground.

5. Dec 13, 2007

Shooting Star

A small correction: in the above, (f1+f2)*h = (R2-R1)*b/2. The final result is correct.

6. Dec 13, 2007

Shooting Star

You thought correctly. It's only the frictional force, preventing the car from slipping sideways, which provide the centripetal force.

Conceptually, perhaps going to the rotating frame may make things easier to understand. In that frame, there is a centrifugal force mv^2/r acting outward at the CM, weight acting downward at the CM, forces of friction acting inward at the points of contact, and normal reactions at the points of contact of the tires. The car does not topple over because the torque about the point of contact of the outer tire due to the three forces, viz., the weight, the normal reaction on the inner tire, and the centrifugal force, all balance out, as long as the speed is less than a certain value.

7. Dec 13, 2007

rl.bhat

A small correction: in the above, (f1+f2)*h = (R2-R1)*b/2. The final result is correct.
Moment due to R1 is clockwise and moment due to R2 is counterclockwise.

8. Dec 13, 2007

Shooting Star

Exactly the reason I have mentioned the correction. f1 and f2 both have CW moments in your picture, You are equating the magnitude of the CW moment (LHS) with the magnitude of the CCW moment (RHS), as a condition for the car not to rotate, right?

9. Dec 13, 2007

rl.bhat

Yes.

10. Dec 13, 2007

kudoushinichi88

(f1+f2)*h = (R1-R2)*b/2, where h is the height of cg from ground and b is the distance between the wheels. f1 +f2 = mv^2/r and R1 + R2 = mg. Solving the two equations we get R1 = 1/2*[ mg - 2mv^2h/rb] As velocity increases R1 goes on decreasesing and a stage is reached when R1 = 0

So, let me do the calculations step by step, so I can familiarise myself with LaTeX...

$$(F_1 + F_2)h = (R_2 - R_1) \cdot \frac {b}{2}$$

$$F_1 + F_2 = \frac {mv^2}{r}$$

$$R_2 - R_1 = mg$$

Solving for R1,

$$R_1 = \frac {1}{2} \left (mg - \frac {2mv^2h}{rb}\right)$$

Plugging in the values,

m = 2000kg,
r = 5 m,
b = 2 m,
and R1 = 0 N,

speed limit for car not to flip $$v = \sqrt {5g} \ \mbox{ms^{-1}$$

Combined frictional force on tires,

$$F = \frac {mv^2}{r} \\ = \frac {2000 \times \left (\frac {\sqrt {5g}}{2} \right)^2 }{5} \\ = 500g \ \mbox {N}$$

Is this correct?

Last edited: Dec 13, 2007
11. Dec 13, 2007

Shooting Star

Correct, if the value of g is in MKS.

12. Dec 13, 2007

kudoushinichi88

Wow, thanks a lot!

13. Jan 8, 2008

Volcano

I am very happy to find this topic. I was asking an almost the same question soon. But worrying about explaining the problem because of my percect English :) By the way, big thanks for helpers too.

May i add one more question? As Shooting star explain,

1. In rotating frame: We use the torque about the "point of contact of the outer tire"

2. In earth frame: We use the torque about the point of "center of mass"

I can not understand the reason of choice of this two different points. As i know, if a body is balanced, the point of moment which we may prefer is arbitrary. Even the point is anywhere in space but surely we pick an easy point for math operations.

14. Jan 8, 2008

Shooting Star

You can use either point – both will give the same result. I had intuitively mentioned moment about the point of contact of the outer tire because it would be the point which will remain fixed (in the rotating frame) once the car does topple. That’s all.

15. Jan 10, 2008

Volcano

Sorry, i am a bit late. Shooting star, as you said, once the car does topple, the forces rest on car is; mg, f2, R2. By the way, certainly if i choose earth reference frame, mg and R2 are balanced and f2 is the unbalanced force.

If i choose the torque about the point of contact of the outer tire, the torks of f2 and R2 are zero. So "mg" is the single force. If i choose this point how can i explain the moment(car) equilibrium as the choose of CW? Of course, we can not say, the car is in moment or force equalibrium, there is a net force(centripetal force) on car but for CW the tork is zero like the moment for rotating frame. This is really confusing me.

I am sorry, maybe i couldn't explain myself. If so please ask me.

16. Jan 10, 2008

Shooting Star

Where is mv^2/r? Or maybe I didn't understand the question?

17. Jan 10, 2008

Volcano

Thank you. Do you mean centripetal or centrifugal force? My problem is the point of moment for inertia reference system. I mean centripetal force. So CP(centripetal) force must be the f2 force because this is the net force as i know,

F=ma -> f2 = ma -> f2 = m(v^2/r)

thus, f2 is the centripetal force(on the ground). If i was prefer rotating frame, there must be another force(inertia force=centrifugal force) on car which effect about center of weight. At this point, both centripetal and centrifugal forces completely different forces and also effects from different points(ground and center of mass). What am i missing?

18. Jan 13, 2008

Volcano

Sorry for reply myself but is this threat forgotton? I can not solve this myself, any idea?

19. Jan 13, 2008

Shooting Star

Yes, it was temporarily forgotten. Sorry. But I'd have eventually answered, I have a huge list of "to be answered", which, unfortunately, grows everyday.