A car is safely negotiating an unbanked circular turn at a speed of 29 m/s. The road is dry, and the maximum static frictional force acts on the tires. Suddenly a long wet patch in the road decreases the maximum static frictional force to one-ninth of its dry-road value. If the car is to continue safely around the curve, to what speed must the driver slow the car?
To find the speed at which the driver must slow the car, we need to use the concept of centripetal force and the frictional force acting on the car.
1. First, let's consider the car's initial state when negotiating the unbanked circular turn at 29 m/s. In this situation, the maximum static frictional force between the tires and the road provides the centripetal force needed for the car to move in a circular path. Let's denote this maximum static frictional force as Fs and the mass of the car as m.
2. The centripetal force required for circular motion is given by the equation: Fc = (m * v^2) / r, where v is the velocity of the car and r is the radius of the circular turn.
3. In this case, the maximum static frictional force (Fs) is providing the centripetal force (Fc) needed for the car to safely negotiate the turn. So, we can equate Fs to Fc: Fs = Fc = (m * v^2) / r.
4. However, when the car encounters the wet patch, the maximum static frictional force decreases to one-ninth (1/9) of its dry-road value. Let's denote this reduced maximum static frictional force as Fs_wet.
5. Since the reduced maximum static frictional force (Fs_wet) is one-ninth (1/9) of the original maximum static frictional force (Fs), we can say that: Fs_wet = Fs / 9.
6. To find the speed at which the driver must slow the car, we need to determine the new maximum static frictional force (Fs_wet) that can provide the necessary centripetal force. Let's solve for this new speed.
7. We can rewrite the equation Fs_wet = (m * v^2) / r by substituting Fs_wet with Fs / 9: (Fs / 9) = (m * v^2) / r.
8. Rearranging the equation, we can solve for v: v^2 = (Fs * r) / (9m).
9. Finally, to determine the speed at which the driver must slow the car, take the square root of both sides of the equation: v = √((Fs * r) / (9m)).
10. Substitute the known values: Fs = maximum static frictional force, r = radius of the circular turn, and m = mass of the car, into the equation to get the final answer.