Question
A science museum has asked you to design a simple pendulum that will make 25.0 complete swings in 85.0 {\rm s}.
How long should the pendulum be? in meters
How long should the pendulum be? in meters
Answers
Damon
No book handy so will derive pendulum
let theta be angle from vertical then
(1/2) m v^2 max = m g h = m g L (1- (1-theta^2/2)) maximum
(1/2) v^2 = g L(theta^2/2)
but v = L d theta/dt
L^2 d (theta/dt)^2 = g L theta^2
let Theta = A sin 2 pi t/T
max = A
then d theta/dt = A (2 pi/T) cos 2pit/T
max = A(2 pi /T)
so
L (2 pi/T)^2 = g
2 pi/T = sqrt(g/L)
T = 2 pi sqrt(L/g)
so now the problem
T =85/25 = 3.4 seconds per swing
2 pi/T = 1.85
T^2 = 3.42 = L/9.81
L = 33.5 m
let theta be angle from vertical then
(1/2) m v^2 max = m g h = m g L (1- (1-theta^2/2)) maximum
(1/2) v^2 = g L(theta^2/2)
but v = L d theta/dt
L^2 d (theta/dt)^2 = g L theta^2
let Theta = A sin 2 pi t/T
max = A
then d theta/dt = A (2 pi/T) cos 2pit/T
max = A(2 pi /T)
so
L (2 pi/T)^2 = g
2 pi/T = sqrt(g/L)
T = 2 pi sqrt(L/g)
so now the problem
T =85/25 = 3.4 seconds per swing
2 pi/T = 1.85
T^2 = 3.42 = L/9.81
L = 33.5 m
Damon
T =85/25 = 3.4 seconds per swing
T/2pi = .541 = sqrt(L/9.8)
L/9.8 = .293
L = 2.87 m
T/2pi = .541 = sqrt(L/9.8)
L/9.8 = .293
L = 2.87 m
Anonymous
A science museum has asked you to design a simple pendulum that will make 54 complete swings in 73 s. What length should you specify for this pendulum?