Question

A conical pendulum has length 50 cm. Its bob of mass 100 g performs uniform circular motion in horizontal plane, so as to have radius of path 30 cm. 

Find:

(i) The angle made by the string with vertical

(ii) The tension in the supporting thread and

(iii) The speed of bob.

Answer 1

Given: l = 150 cm = 0.5 m, r = 30 cm = 0.3 m,

m = 100 g = 100 × 10^-3 kg = 0.1 kg

To find: (i) Angle made by the string with vertical (θ)

(ii) Tension in the supporting thread (T)

(iii) Speed of bob (y)

Formulae: (i) tan θ = – \(\frac{r}{h}\)

(ii) tan θ = \(\frac{v^2}{rg}\)

Calculation: By Pythagoras theorem, l² = r² + h²

h² = l² – r²

h² = 0.25 – 0.09 = 0.16

h = 0.4m

(i) From formula (1),

tan θ = \(\frac{0.3}{0.4}\) = 0.75

∴ θ = tan^-1 (0.75)

θ = 36°52’

(ii) The weight of bob is balanced by vertical component of tension T

∴ T cos θ = mg

cos θ = \(\frac{h}{l}=\frac{0.4}{0.5}\) = 0.8

∴ T = \(\frac{mg}{cos\,\theta}=\frac{0.1\times9.8}{0.8}\)

∴ T = 1.225 N

(iii) From formula (2),

v² = rg tan θ

∴ v² = 0.3 × 9.8 × 0.75 = 2.205

∴ v = 1.485 m/s

(i) Angle made by the string with vertical is 36°52′. ‘ 

(ii) Tension in the supporting thread is 1.225 N. 

(iii) Speed of the bob is 1.485 m/s