A highway curve has a radius of \(825 \mathrm{m}\). At what angle should the road be banked so that a car traveling at $26.8 \mathrm{m} / \mathrm{s}(60 \mathrm{mph})\( has no tendency to skid sideways on the road? \)[$Hint: No tendency to skid means the frictional force is zero. \(]\)

The radius of the curve, \(r = 825 \mathrm{m}\). The velocity of the car, \(v = 26.8 \mathrm{m} / \mathrm{s}\).

The centripetal force is given by the formula \(F_c = \frac{mv^2}{r}\), where \(m\) is the mass of the car and \(v\) is its velocity.

To have no skidding, frictional force \(F_f\) must balance the centripetal force \(F_c\). Since the frictional force is zero, it turns into normal force \(N\) acting perpendicular to the banked angle. We can write: \(F_c = N \sin{\theta}\) With \(F_c = \frac{mv^2}{r}\), we get: \(\frac{mv^2}{r} = N \sin{\theta}\)

The normal force is also related to gravitational force, \(F_g = mg\). Since \(N\) acts perpendicular to the banked angle, we have: \(N = mg\cos{\theta}\)

Now, we have two equations with two unknowns, \(N\) and \(\theta\). We can eliminate \(N\) by substituting from the second equation into the first one: \(\frac{mv^2}{r} = mg\cos{\theta}\sin{\theta}\) We can cancel out \(m\) on both sides: \(\frac{v^2}{r} = g\cos{\theta}\sin{\theta}\) Now we need to solve for \(\theta\). To do this, divide both sides by \(g\cos{\theta}\) and use the identity \(\sin^2{\theta} + \cos^2{\theta} = 1\): \(\frac{v^2}{rg} = \sin{\theta}\cos{\theta}\) \(\Rightarrow \frac{2v^2}{rg} = 2 \sin{\theta}\cos{\theta}\) \(\Rightarrow \frac{2v^2}{rg} = \sin{2\theta}\) Now, we can solve for the angle \(\theta\) by taking the inverse sine: \(\theta = \frac{1}{2} \sin^{-1} \frac{2v^2}{rg}\)

Substitute the given values of \(v\), \(r\), and \(g = 9.81 \mathrm{m/s^2}\) into the equation: \(\theta = \frac{1}{2} \sin^{-1} \frac{2(26.8)^2}{(825)(9.81)}\) \(\theta \approx 0.152 \text{ rad}\) Now convert radians to degrees: \(\theta \approx (0.152 \times \frac{180}{\pi}) \approx 8.71^\circ\) So the road should be banked at an angle of approximately \(8.71^\circ\) for a car traveling at \(26.8 \mathrm{m} / \mathrm{s}\) to have no tendency to skid sideways on the road.