i have been trying to solve this past exam problem, a simple pendulum of length(adsbygoogle = window.adsbygoogle || []).push({}); land bob with massmis attracted to a massless support moving horizontally with constant acceleration a. Determine the lagrange's equations of motion and the period of small oscillations.

here's what i solved for lagrange's equation:

Coordinates of mass

x = l cos θ

y = l sin θ + f(t)

Velocity of mass

x˙ = l(−sin θ)θ˙

y˙ = (cosθ)θ˙ + f˙

Kinetic energy

T =1/2m( ˙ x2 + ˙y2)

=1/2m[l2θ˙2 + (2lf˙ cos θ)θ˙ + f˙2]

Potential energy

U = −mgx

= −mgl cos θ

∂T/∂θ=1/2m · 2lf˙θ˙(−sin θ) = −mlf˙θ˙ sin θ

∂T/∂θ˙=1/2m[2l2θ˙ + 2lf˙ cos θ] = ml2θ˙ + mlf˙ cos θ

∂U/∂θ= −mgl(−sin θ) = mgl sin θ

Lagrangean

L = T − U =1/2m[l2 ˙ θ2 + (2l ˙ f cos θ)˙θ + f˙2] + mgl cos θ

Lagrange’s eqs.

∂L/∂θ −d/dt(∂L/∂θ)= 0

∂T/∂θ −∂U/∂θ −d/dt∂T/∂θ˙= 0

−mlf˙θ˙ sin θ − mgl sin θ −d/dt[ml2˙θ + mlf˙ cos θ] = 0

mlf˙θ˙ sin θ + mgl sin θ + ml2θ¨+ mlf¨cos θ + mlf˙(−sin θ)θ˙ = 0

Finally,

ml2 ¨θ + mgl sin θ + ml ¨ f cos θ = 0

I am not sure if i have it done correctly and aslo still trying to figure out the next part... Thanks for your concern...

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# Pendulum motion lagrange's equation

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