Question

show that for a pendulum to oscillate at the same frequency as an object on a spring, the pendulum's length must be L=mg/k

thank you

Answer 1

derivation for spring
F = -kx
m a = -kx
if x = A sin wt
a = -Aw^2 sin wt = -w^2 x
so
m (-w^2 )x = - k x
w^2 = k/m
w = 2 pi f = sqrt (k/m)
f = (1/2 pi) sqrt (k/m)
derivation for pendulum
Theta = A sin w t
so
angular velocity = A w cos w t
Potential energy at top = U = m g L (1-cos theta) = which is approximately theta^2/2 for small theta
= m g L (1/2) A^2 sin^2 w t
Kinetic energy at bottom = (1/2) m v^2 = (1/2) m L^2 A^2 w^2 cos^2 w t
max potential energy at top = max KE at bottom so
m g L = m L^2 w^2
w = sqrt (g/L
f = (1/2pi) sqrt (g/L)
if the two f values are the same than
sqrt (g/L) = sqrt(k/m)
g/L = k/m
L = g m/k