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Question

The force F1 that is necessary to move a body up an inclined plane is double the force F2 that is necessary to just prevent it from sliding down, then: ( Given that ϕ is angle of repose, θ is angle of inclined plane and w is weight of the body)

A
F2=w sin(θϕ)secϕ
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B
F1=w sin(θϕ)secϕ
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C
tanϕ=3tanθ
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D
tanθ=3tanϕ
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Solution

The correct options are
A F2=w sin(θϕ)secϕ
D tanθ=3tanϕ

F1=mg sin θ+μmg cos θ.
F2=mg sin θμmg cos θ.
But mg = w
μ=tan ϕF1=w (sin θ+sin ϕcos ϕcos θ)F1=w sin (θ+ϕ)sec ϕ
F2=w (sin θsin ϕcos ϕcos θ)F2=w sin (θϕ)sec ϕ
Now F1=2F2
mg sin θ+μmg cos θ=2(mg sin θμmg cos θ)
sin θ+μcos θ=2sin θ2μcos θ3μcos θ=sin θ
tan θ=3μ

tan θ=3tan ϕ.

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