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A-horizontal-force-of-10N-just-prevents-a-mass-of-2kg-from-sliding-down-a-rough-plane-inclined-at-45-to-the-horizontal-Find-the-coefficient-of-friction-




Question Number 22892 by NECx last updated on 23/Oct/17
A horizontal force of 10N just  prevents a mass of 2kg from  sliding down a rough plane  inclined at 45° to the horizontal.  Find the coefficient of friction.
$${A}\:{horizontal}\:{force}\:{of}\:\mathrm{10}{N}\:{just} \\ $$$${prevents}\:{a}\:{mass}\:{of}\:\mathrm{2}{kg}\:{from} \\ $$$${sliding}\:{down}\:{a}\:{rough}\:{plane} \\ $$$${inclined}\:{at}\:\mathrm{45}°\:{to}\:{the}\:{horizontal}. \\ $$$${Find}\:{the}\:{coefficient}\:{of}\:{friction}. \\ $$
Commented by ajfour last updated on 23/Oct/17
Commented by ajfour last updated on 23/Oct/17
fsin θ+Ncos θ=mg        ...1(a)  Nsin θ=F+fcos θ           ...2(a)  and f=μN , so     (μ+1)N=mg(√2)            ...1(b)      (1−μ)N=F(√2)               ...2(b)  dividing we obtain       ((μ+1)/(1−μ)) =((mg)/F)  ⇒   ((μ+1)/2)=((mg)/(mg+F))  ⇒   μ=((mg−F)/(mg+F))  =((20−10)/(20+10)) =(1/3) .
$${f}\mathrm{sin}\:\theta+{N}\mathrm{cos}\:\theta={mg}\:\:\:\:\:\:\:\:…\mathrm{1}\left({a}\right) \\ $$$${N}\mathrm{sin}\:\theta={F}+{f}\mathrm{cos}\:\theta\:\:\:\:\:\:\:\:\:\:\:…\mathrm{2}\left({a}\right) \\ $$$${and}\:{f}=\mu{N}\:,\:{so} \\ $$$$\:\:\:\left(\mu+\mathrm{1}\right){N}={mg}\sqrt{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:…\mathrm{1}\left({b}\right) \\ $$$$\:\:\:\:\left(\mathrm{1}−\mu\right){N}={F}\sqrt{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\mathrm{2}\left({b}\right) \\ $$$${dividing}\:{we}\:{obtain} \\ $$$$\:\:\:\:\:\frac{\mu+\mathrm{1}}{\mathrm{1}−\mu}\:=\frac{{mg}}{{F}} \\ $$$$\Rightarrow\:\:\:\frac{\mu+\mathrm{1}}{\mathrm{2}}=\frac{{mg}}{{mg}+{F}} \\ $$$$\Rightarrow\:\:\:\mu=\frac{{mg}−{F}}{{mg}+{F}}\:\:=\frac{\mathrm{20}−\mathrm{10}}{\mathrm{20}+\mathrm{10}}\:=\frac{\mathrm{1}}{\mathrm{3}}\:. \\ $$
Commented by NECx last updated on 24/Oct/17
your answer is very accurate sir.  Thanks for the help
$${your}\:{answer}\:{is}\:{very}\:{accurate}\:{sir}. \\ $$$${Thanks}\:{for}\:{the}\:{help} \\ $$

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