3. A car is driving at a constant speed y around a banked curve of radius R and

Question

3. A car is driving at a constant speed y around a banked curve of radius R and banking angle theta. The coefficient of static friction between the car tires and the roadway is mu s. a. (5 pts.) Find the optimum banking angle for this curve in terms of y, R, and known constants? The optimum banking angle is the angle for which friction is unnecessary. b. (5 pts.) What are the minimum and maximum speeds with which a car can traverse the curve without slipping? Assume the curve is optimally banked. Use the result from (a). c. (5 pts.)If v= 20*m/s, R=50 m/rad, and theta = 30 degree , what minimum coefficient of static friction (mu s) would be required for the car to stay on the road?

Explanation / Answer

part A)

for optimum angle ,

as there is no frictional force acting on the car ,

centripetal force = mg*sin(theta)

m*v^2/R = m*g*sin(theta)

sin(theta) = v^2/(g*R)

theta = arcsin(v^2/(g*R))

the optimum angle is arcsin(v^2/(g*R))

part B)

for maximum speed friction will act as the centipetal force

m*Vmax^2/R = m*g * sin(theta) + us*mg*cos(theta)

solving for Vmax

Vmax = sqrt(g*(sin(theta) + us*cos(theta))*R)

the maximum speed is sqrt(g*(sin(theta) + us*cos(theta))*R)

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for minimum speed , friction acts oposite to banking force ,

m*Vmin^2/R = m*g * sin(theta) - us*mg*cos(theta)

solving for Vmin

Vmin = sqrt(g*(sin(theta) - us*cos(theta))*R)

the minimum speed is sqrt(g*(sin(theta) - us*cos(theta))*R)

Part C)

for v = 20 m/s

R = 50 m

theta = 30 degree

Now, let the coefficient of friction is us

20^2/50 = 9.8 * sin(30) + us* 9.8 * cos(30)

solving for us

us = 0.365

the coefficient of friction needed is 0.365

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