Question

# A block of mass √2 kg is kept on an inclined plane having an angle of inclination 45∘ attached with a spring of force constant 10 N/m as shown in the figure. If the coefficient of friction between block and incline is 0.4, then which of the following statements is true.

A
If spring is elongated by x>2 m, then the block has no slipping.
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B
If spring is compressed by x<1.4 m, then the block has no slipping.
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C
If spring is compressed by x>1.4 m, then the block has no slipping.
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D
If spring is elongated by 0<x<2 m, then the block has no slipping.
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Solution

## The correct option is B If spring is compressed by x<1.4 m, then the block has no slipping.Case 1: When the spring is elongated For no slipping, at limiting condition ΣFy=0 ⇒ N=mgcosθ ⇒ f=μN=μmgcosθ and, ΣFx=0 ⇒ f=kx1+mgsinθ ⇒ kx1=mg(μcosθ−sinθ) As θ=45∘ ⇒ kx1=mg(μcos45∘−sin45∘) ⇒ kx1=mg(μ√2−1√2) ⇒ x1=mg√2k(μ−1) as μ=0.4<1, if string is elongated, there always will be slipping. Case 2: For compression For no slipping, at limiting condition ΣFy=0 ⇒ N=mgcosθ ⇒ f=μN=μmgcosθ and, ΣFx=0 ⇒ kx2=mgsinθ+f ⇒ kx2=mg(sinθ+μcosθ) As θ=45∘ ⇒ kx2=mg(μcos45∘+sin45∘) ⇒ kx2=mg(μ√2+1√2) ⇒ x2=mgk√2(1+μ) ⇒ x2=10√210√2(1+0.4)=1.4 m Hence, If spring is compressed by x<1.4 m, then the block has no slipping.

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