Final answer:
The frictional force required for a 1700 kg car to round a banked curve can be calculated using the centripetal force formula, adjusted for the banking angle of the curve.
Step-by-step explanation:
Calculate the Frictional Force on a Car in a Banked Curve
A 1700 kg car rounding a curve with a radius of 65 m and banked at 15 degrees travels at 82 km/h requires a specific amount of frictional force to prevent it from sliding. To calculate this force, we first need to convert the speed in km/h to m/s by using the conversion factor, 1 km/h = 0.277778 m/s. Thus, 82 km/h translates to 82 x 0.277778 m/s = 22.77756 m/s.
Next, we employ the centripetal force formula, which is $F_c = mv^2/r$, where $F_c$ is the centripetal force, $m$ is the mass of the car, $v$ is the speed, and $r$ is the radius of the curve. With the car's mass $m$ as 1700 kg, speed $v$ as 22.77756 m/s, and radius $r$ as 65 m, we get $F_c = 1700 \times (22.77756)^2 /65$.
However, since the road is banked, a part of the required centripetal force is provided by the normal force component acting towards the center due to the banking of the road. If the banking angle θ equals 15 degrees, the normal force component contributing to centripetal force can be calculated using $F_c = mg\sin(θ)$, leaving the remainder of the required centripetal force to be provided by the frictional force.
To find the friction force required, we deduct the normal force component from the total centripetal force needed and solve for the frictional force, which equals $f_f = F_c - mg\sin(θ)$. With the final calculation, we determine the exact magnitude of the friction force necessary to keep the car from slipping as it rounds the curve at the given speed.