Why should a thread on a mathematical pendulum be necessarily loose? (For example, a thread of 1 km, why can't it stretch at least 1 cm during the oscillation)
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A simple pendulum consists of a ball (point-mass) m hanging from a (massless) string of length L and fixed at a pivot point P.
Explanation:
When displaced to an initial angle and released, the pendulum will swing back and forth with the periodic motion. By applying Newton's second law for rotational systems, the equation of motion for the pendulum may be obtained
τ = I α ⇒ − m g sin θ L = m L^2 d^ 2 θ /d t ^2
and rearranged as
d ^2 θ/ d t^ 2 + g/ L sin θ = 0
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HOPE THIS HELPS.
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