Question 6(15 marks) Use a SEPARATE writing booklet.
(a) A particle of mass m is suspended by a string of length lfrom a point directly above the vertex of a smooth cone, which has a vertical axis. The particle remains in contact with the cone and rotates as a conical pendulum with angular velocity ω. The angle of the cone at its vertex is 2α, where α> , and the string makes an angle of α with the horizontal as shown in the diagram. The forces acting on the particle are the tension in the string T, the normal reaction to the cone Nand the gravitational force mg.
(i) Show, with the aid of a diagram, that the vertical component of N is Nsinα.
(ii) Show that , and find an expression for T−N in terms of m, landω.
(iii) The angular velocity is increased until N=0, that is, when the particle is about to lose contact with the cone. Find an expression for this value of ωin terms ofα,land g.
Question 6 continues on page 10
2 T N mg 3
+ = sinα
1 α α
N
mg T l α
NOT TO SCALE
π 4
– 9 –
Marks
HSC 2002 - Mathematics Extension 2
Question 6 (continued) (b) For n=0, 1, 2, … let
.
(i) Show that .
(ii) Show that, for n≥2,
.
(iii) For n≥2, explain why In< In−2 , and deduce that .
(iv) By using the recurrence relation of part (ii), find I5and deduce that .
End of Question 6 2
3 2 3
<ln < 4
2 1
2 1
1
2 1
n I
n n
(
+)
< <(
−)
3 I I
n + n = n
−2 − 1
1
3 I1 1 1
2 2
= ln
In = ⌠ n d
⌡ tan θ θ
π 0 4
– 10 –
Marks
HSC 2002 - Mathematics Extension 2
