Maximum distance the car accelerates in a circle without skidding

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1. Jul 20, 2017

Pushoam

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
At distance s, the speed of the car is v.
$$v^2 = 2wτs$$
$$\frac { mv^2} R ≤ kmg$$
Let's denote the maximum distance covered without sliding is smax.
$$\frac { m2wτsmax} R = kmg$$
$$smax = \frac {kgR} {2wτ}$$

Is this correct so far?

2. Jul 20, 2017

cnh1995

I think you have missed some syntax in your LaTex formatting.

3. Jul 20, 2017

ehild

Did you mean so?
It is correct.

Last edited: Jul 20, 2017
4. Jul 20, 2017

scottdave

But should you also consider the tangential acceleration and take the vector sum of that and the centripetal acceleration, when calculating the horizontal force on the car?

5. Jul 20, 2017

Hiero

I agree with @scottdave; if we assume that the tangential acceleration is due to the cars engine (and not, say, someone pushing the car) then the tangential acceleration must also be supplied by the frictional force, and so @Pushoam, your inequality is not quite right.

6. Jul 20, 2017

ehild

7. Jul 20, 2017

haruspex

There is a mistake in the question. It should provide the static friction coefficient, not that of "sliding" (kinetic) friction.

8. Jul 20, 2017

Pushoam

Yes, thanks for it.

Earlier, I had seen that in a uniform circular motion, when there is no force except friction acts on the body, the friction force provides the centripetal acceleration.

Here, you say that there is a component of friction in centripetal direction providing centripetal acceleration and another component in tangential direction.
But, why should one component of friction act in tangential direction?
It is not said in the question that the friction force provides tangential acceleration.

We are going to write the eqn. of motion when the body is moving, so,the question should provide kinetic friction. Isn't it so?

9. Jul 20, 2017

scottdave

Car tires work by maintaining static friction with the pavement for the small patch of tire in contact with the road. If there is no friction, or if you are sliding then the tires are skidding. So yes the friction is causing the car to acelerate.

10. Jul 20, 2017

Pushoam

The friction is causing the car to have centripetal acceleration. This I understood.
Do you mean that the friction is causing the car to have tangential acceleration, too?
And this tangential acceleration is wτ.

11. Jul 21, 2017

haruspex

The friction (static here) is providing the net acceleration, so both the centripetal and the tangential.
Static friction acts to oppose commencement of relative motion of the surfaces in contact; kinetic friction opposes actual relative motion.
Consequently, if the friction were kinetic it would be antiparallel to the relative velocity, so subsequent motion would be in a straight line. Kinetic friction will not get you round a bend.

12. Jul 21, 2017

Pushoam

How do we get to know this ?
How will I find out in which case friction provides only centripetal acceleration and in which case it provides both centripetal and tangential accelerations?

13. Jul 21, 2017

haruspex

Friction is the only horizontal force acting. $\Sigma\vec F=m\vec a$, where $\vec a$ is the acceleration of the mass, not just some component of it.

14. Jul 21, 2017

Pushoam

Car's engine couldn't provide centripetal acceleration, but, it could provide tangential acceleration. I am asking this because earlier I assumed that car's engine provides tangential acc. and friction provides centripetal acc.

So, if the question doesn't specify that it's car's engine which provides tan. acc. , then, we have to take friction providing tan. acc. Is it so?/

15. Jul 21, 2017

haruspex

The car's engine is not in contact with the road. The car only has acceleration in consequence of external forces acting on it, and those all come via the tyre/road contact, gravity and air resistance. Without friction between tyre and road, the engine won't get you anywhere.

16. Jul 21, 2017

Pushoam

If all of the accelerations come from the tyre/road contact, gravity and air resistance, then why do we need an engine?

17. Jul 21, 2017

haruspex

Think about what the engine does. It exerts a torque on the axle. Without friction on the road, that would cause the wheel to spin. The friction opposes the tyre sliding on the road, and that pushes the car along.