The lift force on an airplane of mass 897 kg  with speed v is given by c v2 (N), where c is some constant depending on the air density and wing geometry.  The lift force points perpendicular to the wings of the plane, so that if the plane banks by, say 10 degrees, then the lift force turns 10 degrees from the vertical.  Now suppose that the pilot of this plane wishes to complete a flat turn of radius 931 m with speed 104 m/s.  How much should they bank the plane, in degrees?

A roller-coaster car has a mass of 513 kg when fully loaded with passengers. The path of the coaster from its initial point shown in the figure to point B involves only up-and- down motion (as seen by the riders), with no motion to the left or right. Assume the roller-coaster tracks at points and are parts of vertical circles of radius ₁ = 10.0 m 15.0 m, respectively. and r2 = B (a) If the vehicle has a speed of 19.9 m/s at point, what is the force exerted by the track on the car at this point? N (b) What is the maximum speed the vehicle can have at B and still remain on the track? m/s

A puck is sliding around a circular track that is banked at an angle ϕ with respect to the horizontal such that no friction between the puck and the track is required to keep the puck from slipping up or down the track. If the puck is moving at a speed 34.1 m/s and the radius of the track is 401 m, determine ϕ in degrees.

A ball of mass 4.7 kg and a diameter of 68.2 cm is attached to a string of length 1.1 m. It moves in a circular path at constant speed, 3.0 m above the ground. The string makes a constant angle of 17 ° with respect to vertical. L m (a) What is the speed of the ball? (b) What is the tension in the string?

A point mass m slides without friction from O = (0,0) to P = (a, b) on a curve C under the action of constant gravity (see Figure) with vanishing initial velocity. The time it takes for m to reach P is given by P 1 -ds, Jo T = where ds = V(dr)² + (dy)² and v is the speed. The goal is P=(a,b) y to find the curve C that minimises T. (a) Write T as T = Sº F[y(x), y'(x)]dx and determine the function F. (b) Making use of F – y = const (see Problem 1), derive the relation y' y ƏF dy' V# - 1, where d is a constant. (c) Use the parametric representation y(@) = d sin² = $(1 – cos ø) and determine r(6). %3D The extremal curve is a cycloid.