A pendulum of mass m is suspended from a trolley which runs on a horizontal rail. The trolley and pendulum are initially at rest with θ = 0. If the trolley is given a constant acceleration a = g, the tension in the cord at θ = 45° is

Engineering

Physics

Relative Motion

Circular Motion in Vertical Plane

Free Body Diagram

Question

A pendulum of mass m is suspended from a trolley which runs on a horizontal rail. The trolley and pendulum are initially at rest with θ = 0. If the trolley is given a constant acceleration a = g, the tension in the cord at θ = 45° is

$(5 \sqrt{2} - 2) mg$

$(3 \sqrt{2} - 2) mg$

$\sqrt{2} mg$

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T – mg cos θ – ma sin θ = $\frac{{mv}^{2}}{ℓ}$ .... (1)
By work energy theorem
$\frac{{mv}^{2}}{2}$ = ma ℓ sin θ – mgℓ (1 – cos θ) ... (2)
on solving eq. (1) & (2) for a = g
t = mg $(3 \sqrt{2} - 2)$ = 2.24 mg