the net force ,F networks , needed to push a block along a horizontal surface is directly proportional to the mass, m kg , of the block. when m =5, f = 49. (1) find an equation connecting F and m . (2) find the value of F and m =14. (3) calculate the value of m when F =215.6. (4) draw the graph of F against M.
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Answer:
Let us first calculate the force F required to bring m into motion in terms of angle θ. Equation using laws of motion are as follows:
N=mg−Fsinθ
and Fcosθ=μ
s
N
Fcosθ=μ(mg−Fsinθ)
⇒F=
cosθ+μ
s
sinθ
μmg
We have to find the angle θ for which this force F is minimum.
Substituting μ
s
=tanϕ ( for simplification), we get
F=
cosθ+tanϕsinθ
mgtanϕ
=
cos(θ−ϕ)
mgsinϕ
F is minimum if cos(θ−ϕ) is maximum. Hence, F is minimum for θ=ϕ=tan
−1
μ
s
and F
min
=mgsinϕ.
To bring m into motion with least effort, force should be applied at an angle tan
−1
μ
s
and should have a magnitude equal to
F
min
=mgsinϕ=
1+μ
s
2
μ
s
mg
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