Question

A pendulum bob suspended on a flat car moves with velocity $\upsilon_{0}$ . The flat car is made to come to rest by a bumper
(i) What is the angle through which the pendulum swings.
(ii) If the swing angle is $\theta = 60^{o}$ and $l = 10 \, m ,$ what was the initial speed of the flat car?
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• A. $2 \left(\text{sin}\right)^{- 1} \left(\frac{\left(v ⁡\right)_{0}}{3 \sqrt{\text{g} l ⁡}}\right) \text{,} \text{5 m/s}$
• B. $2 \left(\text{sin}\right)^{- 1} \left(\frac{\left( v \right)_{0}}{2 \sqrt{\text{g} l ⁡}}\right) \text{,} \text{10 m/s}$
• C. $3 \left(\text{sin}\right)^{- 1} \left(\frac{\left(v ⁡\right)_{0}}{4 \sqrt{\text{g} l ⁡}}\right) \text{,} \text{15 m/s}$
• D. $4 \left(\text{sin}\right)^{- 1} \left(\frac{\left(v ⁡\right)_{0}}{2 \sqrt{\text{g} l ⁡}}\right) \text{,} \text{20 m/s}$

Answer 1

Answer: (B)

When the flat car collides with the bumper, due to inertia of motion the bob swings forward. No work is done by tension of string on the bob, therefore energy is conserved.
KE_A + PE_A = KE_B + PE_B
$\left(v ⁡\right)_{0}^{2} = \text{2 g} l ⁡ \left(1 - \text{cos} \theta \right)$
$v ⁡_{0}^{2} = \text{4 g} l ⁡ sin^{2} \theta / 2$
$∴ \, \, \, \theta = 2 \left(sin\right)^{- 1} \left(\frac{\left(υ\right)_{0}}{2 \sqrt{\text{g} l ⁡}}\right)$
For,
$\theta = 6 0^{∘} \text{, } l ⁡ = 1 0 m \text{, } \text{g} = 1 0 m/s^{2}$
$υ_{0}^{2} = 4 \times 1 0 \times 1 0 \left(sin\right)^{2} \left(3 0\right) = 1 0 0$
$υ_{0} = 1 0 m/s$