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Question

# A block of mass m is pulled slowly by a minimum constant force on a horizontal surface through a distance x. The coefficient of kinetic friction is μ. Find the work done by force F. Diagram

A
2lV2v2
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B
2vlV2v2
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C
μmgx1+μ2
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D
2VlV2v2
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Solution

## The correct option is C μmgx1+μ2Draw FBD for the given system. Diagram Find minimum value of the force. From FBD, Fcosθ=f …(i) N+Fsinθ=mg …(ii) And force of friction, f=μN …(iii) Using equation (ii) and (iii) in equation (i), Fcosθ=μ(mg−Fsinθ) Fcosθ+μFsinθ=μmg F=μmgcosθ+μsinθ for F to be minimum cos θ+μsin θ must be maximum let dF(θ)d(θ)=0=−sinθ+μcosθ tanθ=μ For θ=tan−1μ, F is minimum. Find work done by force F. Formula used: W=Fscosθ Given, distance travelled by block, =x So, work done by force F, W=Fxcosθ W=μmgcosθ+μsinθ(xcosθ) W=μmgx1+μtanθ W=μmgx1+μ2

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