A curve of radius 60m is banked so that a car traveling with uniform speed 70km/hr can round the curve without relying on friction to keep it from slipping to its left or right. The acceleration of gravity is 9.8m/s^2. What the the Angle of the curve?
im using a=v^2/r, but when I get the acceleration, I don't know what to do no more.. Can anyone help me on this one? Thanks a bunch!
This is an "angle of lean" problem. You're right, first get the acceleration as you mentioned. You then have to set up your right triangle. The opposite side of the angle in question is the vertical component of force (the weight of the car), and the adjacent side of the triangle is the horizontal component of the force (in this case the mass x the centripetal acceleration...which you already calculated. Thus, the tangent of the angle of lean = opposite/adjacent
thus tangent angle = mg/ma, or just g/a.
To find the angle, we can use the tangent of the angle of lean, which is equal to the ratio of the weight of the car to the centripetal force. The weight of the car can be calculated using the formula weight = mass x acceleration due to gravity (mg).
From the given information, we can calculate the weight of the car as follows:
weight = mass x acceleration due to gravity
weight = (mass of the car) x (9.8 m/s^2)
Next, we can calculate the centripetal force using the formula centripetal force = mass x centripetal acceleration. In this case, the centripetal acceleration is the acceleration we calculated using the equation a = v^2/r.
Substituting the values given, we get:
centripetal acceleration = (70 km/hr)^2 / (60 m)
centripetal acceleration = (70,000 m^2/hr^2) / (60 m)
centripetal acceleration = (70,000 m^2/hr^2) / (60 m) * (1 hr/3600 s)^2
centripetal acceleration = (70,000 m^2/3600^2 hr^2) / 60 m
centripetal acceleration ≈ 5.58 m/s^2
Now, we can substitute the weight and centripetal acceleration into the tangent formula:
tangent angle = weight / centripetal acceleration
tangent angle = (mass x acceleration due to gravity) / (mass x centripetal acceleration)
tangent angle = acceleration due to gravity / centripetal acceleration
tangent angle = 9.8 m/s^2 / 5.58 m/s^2
Finally, to find the angle, we need to take the inverse tangent (arctan) of this ratio:
angle = arctan(tangent angle)
angle ≈ arctan(9.8 m/s^2 / 5.58 m/s^2)
Using a calculator, we find that the angle ≈ 61.04 degrees.
So, the angle of the curve is approximately 61.04 degrees.