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A box with weight ww is pulled at constant speed along a level floor by a force F\overrightarrow{\boldsymbol{F}} that is at an angle θ\theta above the horizontal. The coefficient of kinetic friction between the floor and box is μk\mu_{\mathrm{k}}. (a)(a) In terms of θ,μk\theta, \mu_{\mathrm{k}}, and ww, calculate FF. (b)(b) For w=400 Nw=400 \mathrm{~N} and μk=0.25\mu_{\mathrm{k}}=0.25, calculate FF for θ\theta ranging from 00^{\circ} to 9090^{\circ} in increments of 1010^{\circ}. Graph FF versus θ\theta. (c)(c) From the general expression in part (a)(a), calculate the value of θ\theta for which the value of FF, required to maintain constant speed, is a minimum. (Hint: At a point where a function is minimum, what are the first and second derivatives of the function? Here FF is a function of θ\theta.) For the special case of w=400 Nw=400 \mathrm{~N} and μk=0.25\mu_{\mathrm{k}}=0.25, evaluate this optimal θ\theta and compare your result to the graph you constructed in part (b)(b).

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Given :\textbf{\textcolor{#c34632}{Given :}}

  • μk=0.25\mu_{_{k}}=0.25
  • w=400  Nw=400\;\mathrm{N}

Required \textbf{\textcolor{#c34632}{Required }}

a)\textbf{\textcolor{#c34632}{a)}} Find FF in terms of [μk  ,  θ,  w][\mu_{_{k}}\; , \;\theta , \; w] .

b)\textbf{\textcolor{#c34632}{b)}} Find FF for in range [0°\text{\textdegree} : 90°\text{\textdegree}] , and graph F  F\;versus   θ\;\theta

c)\textbf{\textcolor{#c34632}{c)}} Find the angle θmin\theta_{_{min}} required to make FF minimum .

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