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Chapter 5: Problem 34

A circular curve of highway is designed for traffic moving at \(60 \mathrm{~km} / \mathrm{h}(=37 \mathrm{mi} / \mathrm{h}) .(a)\) If the radius of the curve is \(150 \mathrm{~m}\) \((=490 \mathrm{ft})\), what is the correct angle of banking of the road? ( \(b\) ) If the curve were not banked, what would be the minimum coefficient of friction between tires and road that would keep traffic from skidding at this speed?

Short Answer

Expert verified
The correct banking angle is approximately 8 degrees and the minimum coefficient of friction is approximately 0.28 to prevent skidding on an unbanked curve at a speed of 60 km/h.

Step by step solution

01

Determine the angle of banking

The angle of banking (θ) can be calculated for a perfect banked turn using the following physics formula: \(θ = tan^{-1} (v^2 / gr)\) where \(v\) is the speed of traffic, \(g\) is the acceleration due to gravity, and \(r\) is the radius of the curve. Given \(v = 60km/h = 16.67m/s, g = 9.8m/s^2, r = 150m\), plug these values into the formula to find the angle of banking.
02

Calculate the minimum coefficient of friction

The minimum coefficient of friction (μ) required to prevent skidding can be found using the equation: \(μ = (v^2 / gr) - tan(θ)\)We have already calculated the value of \(θ\) in the previous step and should be able to plug the values of \(v, g, r\) and \(θ\) into the formula and solve for \(μ\).
03

Coordination and checking of results

After calculating the angle of banking in degrees and the minimum coefficient of friction, check to see if the results make sense. The banking angle should not be excessively steep; the coefficient of friction should not exceed the typical maximum value of static friction coefficients (which is generally about 1).

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Most popular questions from this chapter

An elevator weighing \(6200 \mathrm{lb}\) is pulled upward by a cable with an acceleration of \(3.8 \mathrm{ft} / \mathrm{s}^{2}\). (a) What is the tension in the cable? (b) What is the tension when the elevator is accelerating downward at \(3.8 \mathrm{ft} / \mathrm{s}^{2}\) but is still moving upward?

A block slides down an inclined plane of slope angle \(\theta\) with constant velocity. It is then projected up the same plane with an initial speed \(v_{0} .(a)\) How far up the incline will it move before coming to rest? \((b)\) Will it slide down again?

An elevator and its load have a combined mass of \(1600 \mathrm{~kg}\). Find the tension in the supporting cable when the elevator, originally moving downward at \(12.0 \mathrm{~m} / \mathrm{s}\), is brought to rest with constant acceleration in a distance of \(42.0 \mathrm{~m}\).

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A baseball player (Fig. 5-31) with mass \(79 \mathrm{~kg}\), sliding into a base, is slowed by a force of friction of \(470 \mathrm{~N}\). What is the coefficient of kinetic friction between the player and the ground?
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