Two uniform rods AB and CD , each of length ℓ and mass m are fixed to two gears (Which are disc's of radius 2r and r pivoted to fixed points A and C respectively at same horizontal level) as show

Question

Two uniform rods AB and CD , each of length $ℓ$ and mass m are fixed to two gears (Which are disc's of radius 2r and r pivoted to fixed points A and C respectively at same horizontal level) as shown in figure. The system is in vertical plane. When rod AB is slightly rotated clockwise by a small angle, rod CD rotates in the opposite sense (anticlockwise) by another small angle such that there is no slipping and no dissipation of energy at the point of contact of the gears due to their teeth in contact. Neglecting the mass of gears, find the time period of small oscillations of the system. [g is acceleration due to gravity] [Consider $cos θ = 1 - \frac{θ ^{2}}{2 !} \dots$ . neglecting higher terms]

Options:

Answer:

B

Solution:

When rod AB turns through $θ$ , the gear ratio turns rod CD through $2 θ$ ; writing the total kinetic plus potential energy and setting $\frac{d E}{d t} = 0$ gives the angular SHM equation $\frac{10}{3} ℓ α + 3 g θ = 0$ .

Hence $ω^{2} = \frac{9 g}{10 ℓ}$ and $T = 2 π \frac{10 ℓ}{9 g}$ - option (B).

TestHub · Accepted Answer

When rod AB turns through θ , the gear ratio turns rod CD through 2 θ ; writing the total kinetic plus potential energy and setting d t d E ​ = 0 gives the angular SHM equation 3 10 ​ ℓ α + 3 g θ = 0 . Hence ω 2 = 10 ℓ 9 g ​ and T = 2 π 9 g 10 ℓ ​ ​ - option (B).

Physics - SHM Question with Solution | TestHub

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