Find the angle at which a curve of radius r should be banked in order that a car moving at a speed v will not need any frictional force to round it in the figure below. (Use the following as necessary: r, v, and g.)
θ =
To find the angle at which a curve of radius r should be banked, we need to consider the forces acting on the car.
The primary force that will keep the car moving in a circular path is the normal force (N) perpendicular to the surface of the curve. The weight force (W) acts vertically downwards and can be split into two components: one parallel to the surface of the curve (W_parallel) and one perpendicular (W_perpendicular).
We can break down the forces acting on the car as follows:
1. Normal force (N) acting perpendicularly to the surface of the curve.
2. Weight force (W) acting vertically downwards.
3. Centripetal force (Fc) acting towards the center of the circular path.
Since we want to find the angle at which the car will not require any frictional force to round the curve, the centripetal force (Fc) should be provided solely by the component of the weight force perpendicular to the surface of the curve (W_perpendicular).
Now, let's analyze the forces in more detail.
The weight force (W) can be calculated as the product of mass (m) and the acceleration due to gravity (g):
W = m * g
The perpendicular component of the weight force (W_perpendicular) can be calculated using trigonometry. It can be represented as:
W_perpendicular = W * cos(θ)
The centripetal force (Fc) acting towards the center of the circular path can be calculated using the following equation:
Fc = (m * v^2) / r
Now, equating the centripetal force (Fc) with the perpendicular component of the weight force (W_perpendicular), we have:
(m * v^2) / r = W * cos(θ)
Substituting W = m * g, we get:
(m * v^2) / r = (m * g) * cos(θ)
Simplifying the equation, we can cancel out common terms (m) on both sides:
v^2 / r = g * cos(θ)
Finally, to solve for the angle θ, we need to isolate it. Let's rearrange the equation:
cos(θ) = v^2 / (g * r)
To find the angle θ, take the inverse cosine (cos^-1) of both sides:
θ = cos^-1(v^2 / (g * r))
So, the formula for finding the angle θ at which a curve of radius r should be banked is:
θ = cos^-1(v^2 / (g * r))