Solving Car Turning Round Bend: Speed Range

[Clari]

A car is turing round a bend of radius 100m and banking angle 15 degrees. If the coefficient of static friction between the tyres and the road is 0.1, determine the range of speeds within which the car can turn safely round the bend.

Here is what I have done:

$$R sin \theta + f >= \frac{mv^2}{r}$$
$$R sin \theta + \mu R >= \frac{mv^2}{r}$$
$$R (sin \theta + \mu) >= \frac{mv^2}{r}$$
$$mg cos \theta ( sin \theta + \mu) >= \frac{mv^2}{r}$$
$$v<= \sqrt{rgcos\thata (sin\theta +\mu)}$$
$$v<= 18.6 ms^{-1}$$

I don't know how to find the other one range, and the answer is 12.7 < v< 19.6...I am confused...please help..

 

[Doc Al]

To solve this problem, do this:

First identify all the forces acting on the car. I see three forces: weight, normal force (perhaps that's what you call R), and friction.

Now apply Newton's 2nd law: The net horizontal force on the car will produce the centripetal acceleration, while the net vertical force will be zero.

The range of speeds is obtained by realizing that at one extreme the friction points down the incline, while at the other it points up the incline.

 

[wais]

To find the other range, we can use the same equation but with the coefficient of static friction being negative instead of positive. This is because the car can also safely turn at a lower speed if the friction force is pointing in the opposite direction, preventing the car from sliding off the road.

So, our equation becomes:

R sin \theta - \mu R >= \frac{mv^2}{r}

Plugging in the values, we get:

100 sin 15 - 0.1 * 100 >= \frac{m*v^2}{100}

15 - 10 >= \frac{m*v^2}{100}

5 >= \frac{m*v^2}{100}

v^2 <= 500

v <= 22.4 ms^-1

Therefore, the range of speeds within which the car can safely turn round the bend is 12.7 < v < 19.6 ms^-1.

It is important to note that this is the theoretical range and other factors such as road conditions, driver skill, and vehicle condition may affect the actual safe speed for turning round a bend. It is always important to drive at a safe and controlled speed, especially when making turns.