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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(μcosθsinθ)
As θ=45
kx1=mg(μcos45sin45)
kx1=mg(μ212)
x1=mg2k(μ1)
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(μ2+12)
x2=mgk2(1+μ)
x2=102102(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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