a curve of radius 50 m is to be banked, so that the automobile may make a turn at a speed of 15 m/s without depending on friction. at what angle should the road be banked?
Formula: tanØ=v^2/gr
Given:
radius=50m
Velocity=15m/s
Required:Ø
Solution:
tanØ= (15m/s)^2/ (9.8m/s)(50m)
Ø= arctan(15m/s)^2/(9.8m/s)(50m)
Ø=arctan(45/98)
Ø=24°40"
To determine the angle at which the road should be banked, we can use the concept of centripetal force. When a vehicle moves in a curved path, a centripetal force is required to keep it in that path. In this case, the centripetal force is provided by the horizontal component of the normal force acting on the vehicle.
The normal force is perpendicular to the surface of the road, and we can resolve it into two components - the vertical component (mg) and the horizontal component (Ncosθ), where N is the magnitude of the normal force and θ is the angle of bank.
To find the angle of bank, we need to set up an equation involving forces. The centripetal force (Fc) is given by the formula:
Fc = (mv^2) / r
where m is the mass of the automobile, v is the velocity, and r is the radius of the curve. In this case, we are given that the velocity is 15 m/s and the radius is 50 m.
So, Fc = (m * (15^2)) / 50
Now, since the centripetal force is provided by the horizontal component of the normal force, we can equate it to the horizontal component - Ncosθ:
(m * (15^2)) / 50 = Ncosθ
Next, we can consider the vertical component of the normal force, which is balanced by the weight of the automobile (mg):
mg = Nsinθ
Since we are considering a scenario where the automobile is not dependent on friction, the net force in the vertical direction is zero, which means:
mg = Nsinθ
Now, we can substitute the value of N from the second equation into the first equation:
(m * (15^2)) / 50 = (mg) * cosθ / sinθ
Simplifying the equation, we get:
15^2 / 50 = cosθ / sinθ
225 / 50 = cosθ / sinθ
4.5 = tanθ
Finally, taking the inverse tangent of both sides, we can find the angle θ:
θ = tan^(-1)(4.5)
Using a calculator, the approximate value of θ is approximately 78.69 degrees.
Therefore, the road should be banked at an angle of approximately 78.69 degrees.