Find the angle of banking for a highway curve of 300 ft radius design to accommodate cars traveling at 100 mph, if the coefficient friction between the tires and the road is 0.6. What is the rated speed of the curve?
show solution step by step
show FBD (if possible)
thanks
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ANCORA IMPARO
To find the angle of banking for the highway curve, we can use the concept of centripetal force. The centripetal force is the force required to keep an object moving in a curved path.
Step 1: Convert the speed from miles per hour (mph) to feet per second (fps). Since 1 mile = 5280 feet and 1 hour = 3600 seconds, we have:
Speed in fps = 100 mph × (5280 ft/mi) / (3600 s/hr) = 146.67 fps
Step 2: Determine the centripetal force required to keep the car moving along the curve. The centripetal force is given by:
Centripetal force = (mass × (velocity^2)) / radius
The mass of the car cancels out in this equation, so we only need to consider the velocity and radius of the curve.
Centripetal force = (velocity^2) / radius
Substituting the values, we have:
Centripetal force = (146.67 fps)^2 / 300 ft = 67.73 lb
Step 3: Calculate the frictional force. The frictional force is equal to the coefficient of friction (µ) multiplied by the normal force.
Frictional force = µ × normal force
In this case, the normal force acts perpendicular to the surface of the road, which is equal to the weight of the car. So:
Frictional force = µ × weight
Step 4: Substitute the value of frictional force obtained from Step 3 and solve for the weight (normal force). Rearranging the equation, we have:
Weight = Frictional force / µ
Weight = 67.73 lb / 0.6 = 112.88 lb
Step 5: Calculate the rated speed of the curve. The rated speed is the speed at which the car starts to slide.
Frictional force = Centripetal force
µ × Weight = (velocity^2) / radius
Substituting the known values, we can solve for velocity:
0.6 × 112.88 lb = (velocity^2) / 300 ft
Now, rearrange and solve for velocity:
velocity = √((0.6 × 112.88 lb × 300 ft) / 1 ft^2)
velocity = 32.85 fps
This is the rated speed of the curve, which means cars can safely navigate the curve at a speed of 32.85 fps or approximately 22.37 mph.
Regarding the free-body diagram (FBD) for the car on the banked curve, since this problem only involves the calculation of forces and angles, a FBD is not necessary. However, it is always good practice to draw a diagram to better understand the problem.
If you have any further questions, feel free to ask!