Simple Harmonic Motion - Finding period of oscillation of block

[zorro]

Homework Statement

In the given figure, all the springs and pulley are ideal and surface is frictionless. The rod is massless. The time period of horizontal oscillations of the block is?

The Attempt at a Solution

I used two different approaches and both give different and wrong answers-

Consider a very small rotation of the rod by an angle θ. Let x be the displacement of the mass m.

1) Torque-

T_restoring = -(k/4*2*(lθ)*l + k(2lθ)*2l)

Simplfiying and writing L.H.S as 4ml²α

α = -9k/8m*θ

ω² = 9k/8m

2) Force-

F_restoring = -(k*2lθ + k/4*2*lθ) = -5klθ/2 = -5kx/4

a=-5kx/4m

ω² = 5k/4m

Can someone point out any mistake above and show me the correct method?

 

[tiny-tim]

Hi Abdul!

You need to treat x as independent of θ.

(and I think your T_restoring = -(k/4*2*(lθ)*l + k(2lθ)*2l) has too many 2s)

 

[zorro]

x and θ are dependent. How can they be independent of each other?

Which '2' do you think is redundant?

 

[tiny-tim]

x and θ are dependent. How can they be independent of each other?

The mass m is not fixed, so x and θ can vary independently.

 

[zorro]

ok...so the force due the lower spring is k(2lθ+/-x) ?
How do I decide whether it is + or - ?

 

[tiny-tim]

I don't really understand your question :confsued: …

you decide which way x goes, and which way θ goes.

 

[zorro]

I got the answer. Thank you!