Answer to Question #86963 in Mechanics | Relativity for Maria

Question #86963
A curve of radius 75 m is banked for a design speed of 75 km/h.
If the coefficient of static friction is 0.37 (wet pavement), at what range of speeds can a car safely make the curve? [Hint: Consider the direction of the friction force when the car goes too slow or too fast.]
Express your answers using two significant figures separated by a comma.
1
Expert's answer
2019-03-25T09:01:50-0400

A banked curve means a car should do the turn at the design speed without participation of acceleration. This information allows us to find out the angle of incline of the road:

"\\alpha = \\arctan\\frac{v_0^2}{gR}; \\\\"

For smaller speeds (e.g. car sliding downwards):

"F_n\\sin\\alpha - F_f\\cos\\alpha = m\\frac{v^2}{R}; \\\\\nF_n\\cos\\alpha + F_f\\sin\\alpha - mg = 0; \\\\"

Remembering that "F_f = {\\mu}F_n; \\\\" and dividing the equations we get

"v_{min} = \\sqrt\\frac{gR(\\tan{(\\alpha)}-\\mu)}{1+{\\mu}\\tan\\alpha} = \\sqrt\\frac{gR(v_0^2-{\\mu}gR)}{gR+{\\mu}v_0^2} \\approx 41\\frac{km}{h}. \\\\"

Similarly, for greater speeds (car sliding upwards):

"F_n\\sin\\alpha + F_f\\cos\\alpha = m\\frac{v^2}{R}; \\\\\nF_n\\cos\\alpha - F_f\\sin\\alpha - mg = 0; \\\\"

"v_{max} = \\sqrt\\frac{gR(\\tan{(\\alpha)}+\\mu)}{1-{\\mu}\\tan\\alpha} = \\sqrt\\frac{gR(v_0^2+{\\mu}gR)}{gR-{\\mu}v_0^2} \\approx 108\\frac{km}{h}. \\\\"

When doing the calculation please note that km/h should be converted to m/s and back likewise.

Answer: 41, 108.

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Comments

o
07.01.21, 14:25

nice

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