A pendulum bob has a speed 4 m / s at its lowest position. The pendulum is 1 m long. When the length of the string makes an angle of 6 0 ∘ with the vertical, the speed of the bob at that position is (

Question

A pendulum bob has a speed 4 m / s at its lowest position. The pendulum is 1 m long. When the length of the string makes an angle of 6 0 ∘ with the vertical, the speed of the bob at that position is (acceleration due to gravity, g = 10 m / s 2 , cos 6 0 ∘ = 0.5 )

TestHub · Accepted Answer

By the principle of conservation of mechanical energy, the total energy at the lowest point equals the total energy at an angle $θ$ .

$\frac{1}{2} m v_{0}^{2} = \frac{1}{2} m v^{2} + m g h$

The height $h$ of the bob when the string makes an angle $θ$ with the vertical is given by $h = L (1 - cos θ)$ .

Substituting the given values $L = 1 m$ and $θ = 6 0^{\circ}$ :

$h = 1 (1 - cos 6 0^{\circ}) = 1 (1 - 0.5) = 0.5 m$

Using $v_{0} = 4 m/s$ and $g = 10 m/s^{2}$ :

$\frac{1}{2} v_{0}^{2} = \frac{1}{2} v^{2} + g h$

$v^{2} = v_{0}^{2} - 2 g h$

$v^{2} = 4^{2} - 2 (10) (0.5)$

$v^{2} = 16 - 10 = 6$

$v = 6 m/s$

Answer: $6 m / s$

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