L1. A particle with mass m is attached to one end of an inextensible string of length `. The other end of the string is

[answerhappygod]

L1. A particle with mass m is attached to one end of an
inextensible string of length `. The other end of the string is
attached at point O vertically above the particle. The particle is
then struck to move on a vertical circle of radius ` with initial
horizontal speed u.
(a) Show that when the string makes an angle θ with the downward
vertical, the tension in the string is T = m u 2 ` + mg(3 cos θ −
2). [7 marks]
(b) Show that the particle will describe at least a half-circle
provided u ≥ p 5g`. [3 marks]
(c) The particle is given an initial horizontal speed u = √ g`.
Compute (to the nearest degree) the angle at which the string
flexes. [3 marks]
(d) Also when the particle is given an initial horizontal speed
u = √ g`, compute (to the nearest degree) the angle at which it
comes to rest instantaneously while on the vertical circular
motion, and thus find which event occurs first. [3 marks]
(e) When the particle is oscillating back and forth with T >
0 at all times, what is the exact equation for θ(t) describing the
motion? What is the approximate equation for small oscillations? [4
marks]