A O B is a swing suspended from vertical poles A A prime and B B prime as shown. If ropes O A prime and O B of length l1 and l2 respectively are massless, and are perpendicular to each other with a point mass m hanging from O, the time period of the swing for small oscillations perpendicular to the plane of paper is:

Question

A O B is a swing suspended from vertical poles A A prime and B B prime as shown. If ropes O A prime and O B of length l1 and l2 respectively are massless, and are perpendicular to each other with a point mass m hanging from O, the time period of the swing for small oscillations perpendicular to the plane of paper is: <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/physics/7aea393ed507000cefdee629807ec038-.png" /> (A) 2 π √(l1 l2/g √x2+y2) (B) 2 π √((x2+y2)3 / 2/l1 l2 g) (C) 2 π √(x2+y2/√l1 l2 g) (D) 2 π √(l1 l2/x ⋅ g)

Tardigrade Admin · Accepted Answer

The system acts as a pendulum of length

l

acting under effective gravity of

g cos θ

as shown in figure
Now,

l_{1} l_{2} = A B . l = x^{2} + y^{2} \cdot l

cos θ = \frac{x}{x ^{2} + y ^{2}}

\Rightarrow T = \frac{l _{1} l _{2} x ^{2} + y ^{2}}{x ^{2} + y ^{2} \cdot gx} = 2 π \frac{l _{1} l _{2}}{gx}

$2 π \frac{l _{1} l _{2}}{g x ^{2} + y ^{2}}$

$2 π \frac{( x ^{2} + y ^{2} ) ^{3/2}}{l _{1} l _{2} g}$

$2 π \frac{x ^{2} + y ^{2}}{l _{1} l _{2} g}$

$2 π \frac{l _{1} l _{2}}{x \cdot g}$

2πgx2+y2​l1​l2​​​

2πl1​l2​g(x2+y2)3/2​​

2πl1​l2​​gx2+y2​​

2πx⋅gl1​l2​​​

The system acts as a pendulum of length l acting under effective gravity of gcosθ as shown in figure Now, l1​l2​=AB.l=x2+y2​⋅l cosθ=x2+y2​x​ ⇒T=x2+y2​⋅gxl1​l2​x2+y2​​​=2πgxl1​l2​​​

$2 π \frac{l _{1} l _{2}}{g x ^{2} + y ^{2}}$

$2 π \frac{( x ^{2} + y ^{2} ) ^{3/2}}{l _{1} l _{2} g}$

$2 π \frac{x ^{2} + y ^{2}}{l _{1} l _{2} g}$

$2 π \frac{l _{1} l _{2}}{x \cdot g}$