Theoretical Model for Mathematical Pendulum · IZhO 2021 #1

Physics · mechanics

Problem

Consider a mathematical pendulum, which is a small massive ball suspended on an inextensible thread of length ll. The pendulum is subject to the gravity field with the free fall acceleration gg. In the following neglect the air resistance completely. Assume that at the initial moment of time t=0t = 0 the angle of the thread deflection from the vertical is ϕ0\phi_0, and the initial velocity of the ball is equal to zero. The ball moves along an arc of a circle, therefore, its position is determined by the angle of the thread deflection from the vertical ϕ\phi, and the rate of change of this angle in time is determined by the angular velocity ω=dϕdt\omega = \frac{d\phi}{dt}. 1.1 Write down a formula for the period TT of small oscillations of a mathematical pendulum. 1.2 Obtain an exact formula for the dependence of the angular velocity of the pendulum on the deflection angle ω(ϕ)\omega(\phi) for a given angular amplitude ϕ0\phi_0 and known values of ll, gg. 1.3 Write down an exact expression for calculating the time t1t_1 from the known dependence of the angular velocity on the deflection angle ω(ϕ)\omega(\phi). 1.4 Express a period of oscillations TT in terms of time t1t_1. 1.5 Write down a formula relating the angular velocity in dimensionless units ω~=dϕdτ\tilde{\omega} = \frac{d\phi}{d\tau} to the previously defined angular velocity ω\omega, where the dimensionless time is τ=tgl\tau = t\sqrt{\frac{g}{l}}. 1.6 Determine a period T~\tilde{T} of small oscillations of the mathematical pendulum in the dimensionless units of time. 1.7 Determine a dependence of the angular velocity ω~\tilde{\omega} on the deflection angle ϕ\phi: ω~(ϕ)\tilde{\omega}(\phi).

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