Application of Calculus: Banked turn with friction
Banked turn with friction Ken & Simer
What is centripetal & centrifugal force?• Centripetal force – the force that acts towards the centre of the circular motion. It keeps the object in the circular motion.• Centrifugal force – the force acts against the centripetal force. It acts outwards of the circular motion. One of the forms of inertia.
Centripetal force/acc. = Centrifugal force = Both forces are balanced as the car does not go off track (“dragged out”) or slip to the centre (“pushed into”). Since both forces are equal,
What is gravitational component & normal force?• Gravitational component – the force of gravity acting on the object at an angle, .• Normal force, N – the force that opposes the effect of gravity on the object. It seems as though the surface the object is on, exerts this force. Following Newton’s Third Law: Every action has an equal but opposite reaction.
Normal force =Gravitational component = Both forces are balanced as the car does not float above the track and topple or crash into it. Since both forces are equal,
(1)……………………...(2)………………..By solving the above equation for mass and substitutingthis value into our previous equation we get: Solving for v we get:
Angle of banking, ( radians) Velocity, v (km/h) 90.1 90.2 90.3 90.5 90.6 90.7 90.8 91.0 1 91.1
Application• Depending on the value of the second order differentiation of the angle, it could be a minimum or maximum velocity.• However, in most cases, like this one, it would be the minimum or MOST SUITABLE velocity as the second order differentiation will always be positive.• Therefore, in order to find the maximum velocity which is used to set speed limits on certain terrain of roads, the value of the velocity is inserted into the equation first and then it is inversed to find the magnitude of the forces acting on the car.• If the forces are not balanced, the car will “fly off” the road.• Thus, the value of the velocity is increased or decreased until the forces acting on the car are relatively equal.