Mathematical pendulums

Question

the periods of oscillations of n different pendulums (mathematical) are T1, T2, ..., Tn. The strings of these pendulums are joint into one without length loss and I got a new pendulum. Find its period of oscillation.

Answer 1

T₁ = 2π√L₁/g, So L₁ = gT₁²/(4π²). So L = g∑₁ⁿT_i²/(4π²).

Now T = 2π√ g∑₁ⁿT_i²/(4π²)/g = sqrt(∑_i=1ⁿT_i²).

Answer 2

the periods of oscillations of n different pendulums (mathematical) are T1, T2, ..., Tn. The strings of these pendulums are joint into one without length loss and I got a new pendulum. Find its period of oscillation.

Hello Armad, Time period of a pendulum is proportional to square root of its length

So T = K√ l Where l = length

Thus, l = T²/K²

l₁ = T1²/K²

l₂ = T ₂ /K² ........ and so on

Total length of new pendulum = l₁ +l₂ + l₃ +..... = L

So new time period = T_n = K √L = K √ (l₁ +l₂ + l₃ +...)= K √.( T1²/K² +T ₂ /K² + ..... ) = √ ( T₁² +T₂² +....