An ideal spring with spring constant k is on a frictionless horizontal surface a

Question

An ideal spring with spring constant k is on a frictionless horizontal surface at the base of a frictionless inclined plane with height H as shown in the figure. A block with mass M is pressed against the spring, compressing it 2 cm from its equilibrium position. The block is then released and is not attached to the spring. If the block slides four-ninths the way up the inclined plane before coming to rest and then sliding back down, what minimum additional distance (in cm) must the spring be compressed so that the block will make it to the top of the incline plane?

The answer is 1 but how?

Explanation / Answer

Applying energy conservation

1/2Kx^2 = mgh

If x=2cm h =4/9 L

Taking ratio as K,mg = constant

(x1/x2)^2 = h1/h2

h2 =L for block to reac at top

(2/x2)^2 = (4/9L)/L =4/9

2/x = 2/3

x = 3cm

So additional distance needed to compress the spring =3-2 = 1 cm (ans)

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