SHM equation

2010-07-15 4:56 am

Situation:

Two identicl pans ( each has mass M ) are connected by light string
which passes over a light pulley suspended from the ceiling.

There is a mass m on the left pan.

Under the left pan, there is a light spring with force constant k,
on which there is a light plate.

Assume that the pan moves together with the plate once they are in contact
and the string does not slack throughout.

Prove the SHM equation in detail
i.e. a = -w^2x
i.e. w = ?

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回答 (1)

天同

2010-07-17 7:59 am

✔ 最佳答案

At the equilibrium position when the spring is compressed a distance xo, k.xo = mg, where g is the acceleration due to gravityAssume the mass m is displaced a distance x from the equilibrium position, then the spring force = k(x+xo)

hence, mg - k(x+xo) = (m + 2M).a
where a is the acceleration of mass m
mg - kx - k(xo) = (m + 2M).a
mg - kx - mg = (m+2M).a
i.e. (m + 2M).a = -kx
a = -[k/(m+2M)].x

Therefore, the motion is simple harmonic, as acceleration is proportional to displacement x from the equilibrium position.

and w = sqaure-root[k/(m+2M)]

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