How Slow Must the Car Go to Stay Safe on a Wet, Unbanked Curve?

[akatsafa]

I'm setting this problem up, but I'm now stuck.

A car is safely negotiating an unbanked circular turn at a speed of 17m/s. The maximum static frictional force act on the tires. Suddenly a wet patch in the road reduces the maximum static friction force by a factor of three. If the car is to continue safely around the curve, to what speed must the driver slow the car?

I made a free body diagram and came up with f net equations. I have fnetx=1/3usg=v^2/r. I have fnety=N-mg. For fnetx, I further get 3.267m/s^2us=v^2/r. And this is where I get stuck. I'm not sure what I do to find the speed without knowing the radius.

 

[arildno]

The magnitude of the initial frictional force satisfies:
$$F=m\frac{v_{0}^{2}}{R}$$

Afterwards, we have:
$$\frac{1}{3}F=m\frac{v_{1}^{2}}{R}$$

By division, we get:
$$\frac{1}{3}=(\frac{v_{1}}{v_{0}})^{2}$$

 

[M98Ranger]

To find the speed at which the driver must slow the car, you can use the equation for centripetal force: Fc = mv^2/r. In this case, the force acting on the car is the maximum static frictional force, which is now 1/3 of its original value. So the equation becomes: 1/3usmg = mv^2/r.

Since we don't know the radius, we can use the fact that the car is safely negotiating the turn at a speed of 17m/s, which means that the centripetal force at this speed is equal to the maximum static frictional force. So we can write: usmg = mv^2/r.

By substituting this into the previous equation, we get: 1/3(usmg) = mv^2/r.

Now we can cancel out the mass (m) on both sides and rearrange the equation to solve for v: v = √[(1/3usg)r].

Since we still don't know the radius, we can use the fact that the car is safely negotiating the turn at a speed of 17m/s to find the radius. At this speed, the centripetal force is equal to the maximum static frictional force, so we can write: usmg = mv^2/r.

By substituting the given values, we get: (1/3)(0.7)(9.8)(m) = (m)(17)^2/r.

Solving for r, we get: r = (17)^2/2.333 = 123.23m.

Now we can plug this value of r into the previous equation to find the speed at which the driver must slow the car: v = √[(1/3)(0.7)(9.8)(123.23)] = 10.83m/s.

Therefore, the driver must slow the car to a speed of 10.83m/s to safely navigate the unbanked curve with the reduced maximum static frictional force.