Question

A small sphere of mass m suspended by a thread is first taken aside so that the thread forms the right angle with the vertical and then released. Find: 

(a) the total acceleration of the sphere and the thread tension as a function of θ, the angle of deflection of the thread from the vertical; 

(b) the thread tension at the moment when the vertical component of the sphere's velocity is maximum; 

(c) the angle θ between the thread and the vertical at the moment when the total acceleration vector of the sphere is directed horizontally.

Answer 1

Let us depict the forces acting on the small sphere m, (at an arbitrary position when the thread makes an angle θ from the vertical) and write equation vector F=m vector w via projection on the unit vector u_t and u_m. From F_t=mw_t, we have