Fig. 6-48 shows a conical pendulum, in which the bob (the small object at the lo
ID: 2132961 • Letter: F
Question
Fig. 6-48 shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass m, the string has length L and negligible mass, and the bob follows a circular path of circumference C. What are (a) the tension in the string and (b) the period of the motion? State your answers in terms of the given variables, using g and ? when appropriate.
Explanation / Answer
Let:
m be the mass of the bob,
v be its the linear velocity,
w be its angular velocity,
r be the radius of the horizontal circle it describes,
a be the semi-vertical angle of the cone,
L be the length of the string,
T be the tension,
c be the centripetal force,
P be the period,
g be the acceleration due to gravity.
Resolving vertically and horizontally, and using a for theta:
T cos(a) = mg ...(1)
T sin(a) = mv^2 / r = mv^2 / (L sin(a)) ...(2)
(a)
From (1) the tension in the cord is:
T = mg / cos(a)
(b)
The centripetal force is:
c = T sin(a)
= mg tan(a).
Dividing (2) by (1):
tan(a) = v^2 / gL sin(a)
v^2 = gL sin(a)tan(a)
The bob's linear velocity is:
v = sqrt(gL sin(a)tan(a)).
Its angular velocity is:
w = v / r
= v / (L sin(a))
= sqrt(gL sin(a)tan(a)) / (L sin(a))
= sqrt( gL sin(a)tan(a) / L^2 sin^2(a) )
= sqrt( g tan(a) / L sin(a) )
= sqrt( g / L cos(a) ).
The period is:
P = 2pi / w
= 2pi sqrt( L cos(a) / g )
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