A banked circular highway curve is designed for traffic...

A banked circular highway curve is designed for traffic moving at 60 $\mathrm{km} / \mathrm{h}$ . The radius of the curve is 200 $\mathrm{m}$ . Traffic is moving along the highway at 40 $\mathrm{km} / \mathrm{h}$ on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to take the turn without sliding off the road? (Assume the cars do not have negative lift.)

A banked circular highway curve is designed for traffic moving at 60 $\mathrm{km} / \mathrm{h}$ . The radius of the curve is 200 $\mathrm{m}$ . Traffic is moving along the highway at 40 $\mathrm{km} / \mathrm{h}$ on a rainy day. What is the minimum coefficient of friction between tires and road that will
allow cars to take the turn without sliding off the road? (Assume the cars do not have negative lift.)

Key Concepts

Banked Curve Dynamics

This concept explains how a road is designed with a tilt (banking) so that a component of the normal force supplies the necessary centripetal force to keep a vehicle on a curved path without solely relying on friction. It involves understanding how the geometry of the road, intended vehicle speed, and gravitational forces interact to enable safe turning.

Circular Motion and Centripetal Force

Vehicles moving in a circular path require an inward force to continually change direction. This centripetal force is typically provided by a combination of the friction between the tires and the road and the horizontal component of the normal force on a banked curve. Understanding this balance is crucial to maintaining stable circular motion without skidding.

Coefficient of Friction

This is a measure of the frictional resistance between two surfaces, in this case, the tires and the road. It quantifies how much grip is available to counteract forces that would otherwise cause sliding. Knowing the minimum coefficient of friction is essential for ensuring that a vehicle can safely negotiate a turn even under less-than-ideal conditions like wet roads.

Effect of Speed Variation on Friction Requirements

The vehicle’s speed relative to the ideal design speed for a banked curve affects the force balance. Deviations from the design speed can lead to an imbalance where friction must compensate for either a deficit or excess of the inward force provided by the road's banking. This concept is important when determining the minimum coefficient of friction required for safety under various driving conditions.

Transcript

00:01 So essentially, most of the analysis is done in sample problem, a car in a banked circular turn.

00:21 And the analysis leads to equation 626, which gives us the theta equaling arc tan of v squared over the acceleration due to gravity times the radius of curvature.

00:40 Now, we know that here v is equaling 60 kilometers per hour.

00:45 We're going to multiply this by 1 ,000, rather we can say 1 meter per second for every 3 .6 kilometers per hour, giving us approximately 17 meters per second.

01:00 And we know that r is 200 meters.

01:04 So given this, we can then solve for theta.

01:08 And theta would be arc tan of 17 meters per second, quantity squared, divided by 9 .80.

01:20 Meters per second squared multiplied by r of 200 meters and this is going to give us 8 .1 degrees.

01:31 Now if we were to consider a vehicle taking this banked curve at 40 kilometers per hour, we can say that v prime would be equal to 40 kilometers per hour.

01:44 Let's convert one meter per second for every 3 .6 kilometers per hour.

01:52 This is going to equal 11 meters per second.

01:57 And we can then say that it's horizontal acceleration here would be considered a prime.

02:03 This would be again v prime squared divided by r and it has components parallel to the incline and perpendicular to the incline.

02:13 So we can say that a parallel would be equal to a prime cosine of theta where a perpendicular would be equal to a sign a prime sine of theta...

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