A block of mass 1 kg falls freely on a spring form a height of 20 cm as shown in figure . Find the compression in the spring if its force constant is `10^(3)` N/m . ( Take ` g = 10.0 m//s^(2)`)

Explanation

Let xm be the compression in the spring . Block is considered to fall through theheight `(x+0.2)` m . The potential energy of the block at height `(x+0.2)` m is used up in compressing the spring xm and converted in the gravitational energy =`mg(h+x)`
`= 1 xx10 (0.2 +x)`
Work done by compressing spring =` 1/2 kx^(2)`
which is the gravitational energy .
`1/2 kx^(2) =10(0.2 +x)`
`:. 1/2 xx 600 x^(2)=2+10x`
` :. 300 x^(2) - 10x -2=0`
` :. 150x^(2)-5x -1 = 0 `
` :. x = -b pmsqrt((b^(2)-4ac))/(2a)`
` :. x - 5 pm sqrt(25 - 4 xx150 xx(-1))/(2xx150)`
` :. x = (5pm sqrt(625))/300`
` :. x = (5+25)/300 or x = (5-25)/300 `
` :. x = 0.1 m or x = - 0.0833 ` m which is non practical .
` :. ` The compression in the spring will be 10 cm .