Question

1. why are racetracks often banked at the corners? what would happen if they are not? give a written explanation, draw a diagram, and/or provide equations to back up your answers.

Explanation:

When a race - track is banked at the corners, the normal force component provides a centripetal force. Let the angle of banking be \(\theta\), the mass of the vehicle be \(m\), the velocity be \(v\), and the radius of the turn be \(r\). The normal force \(N\) can be resolved into vertical (\(N\cos\theta\)) and horizontal (\(N\sin\theta\)) components. In the vertical direction, \(N\cos\theta=mg\) (since there is no vertical acceleration). In the horizontal direction, the centripetal force \(F_c = N\sin\theta\). From \(N\cos\theta = mg\), we get \(N=\frac{mg}{\cos\theta}\). Substituting into the horizontal equation, \(F_c=\frac{mg\sin\theta}{\cos\theta}=mg\tan\theta\). Also, the centripetal force \(F_c=\frac{mv^{2}}{r}\). So, \(mg\tan\theta=\frac{mv^{2}}{r}\), and \(v = \sqrt{gr\tan\theta}\). This shows that the banked track allows for a certain speed \(v\) without relying solely on friction.

If the track is not banked, the centripetal force must come entirely from the frictional force between the tires and the track. The maximum frictional force \(f_{max}=\mu_sN\), where \(N = mg\) (vertical equilibrium). So, \(f_{max}=\mu_smg\). Setting \(f_{max}=\frac{mv^{2}}{r}\), we get \(v_{max}=\sqrt{\mu_sgr}\). Since \(\mu_s\) (coefficient of static friction) is usually less than 1, the maximum speed without banking (\(v_{max}=\sqrt{\mu_sgr}\)) is much less than the speed with banking (\(v = \sqrt{gr\tan\theta}\), when \(\tan\theta\) can be relatively large for well - designed banked tracks). So, without banking, the cars would have to slow down more significantly at the corners, and there would be a higher risk of skidding off the track as the speed limit for safe turning would be much lower.

Answer:

Racetracks are banked at the corners to provide a centripetal force component from the normal force, allowing higher safe speeds. If not banked, the centripetal force must come from friction, resulting in lower safe speeds and a higher risk of skidding.