3. The force F required to move a box by pulling on an attached rope that forms an angle w with thehorizontal is given byfmgF(w) =cOSw + f sin wwhere f, m, g > 0 are all constant.(a) Show that the angle wo that minimizes the required force to set the box in motion is given byarctan(f)(b) Show that this minimal force F(wo) is proportional to (a constant multiple of)

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Asked Nov 20, 2019
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3. The force F required to move a box by pulling on an attached rope that forms an angle w with the
horizontal is given by
fmg
F(w) =
cOSw + f sin w
where f, m, g > 0 are all constant.
(a) Show that the angle wo that minimizes the required force to set the box in motion is given by
arctan(f)
(b) Show that this minimal force F(wo) is proportional to (a constant multiple of)
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3. The force F required to move a box by pulling on an attached rope that forms an angle w with the horizontal is given by fmg F(w) = cOSw + f sin w where f, m, g > 0 are all constant. (a) Show that the angle wo that minimizes the required force to set the box in motion is given by arctan(f) (b) Show that this minimal force F(wo) is proportional to (a constant multiple of)

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Expert Answer

Step 1

a)

Force required to move the box of mass m is,

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fing F(0) cos fsin a

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Step 2

The quantities f, m and g are constant. ω is variable.

Differentiate the above equation with respect to ω by applying chain rule

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d fmg -F() = do da cosa+ f sin o d =fimg- da (cos o+ fsin @)| d(cos af sin a)d(cos f sin ) fing d(cos af sin a = da .d(sin) --1 dcos o f da - fimg(-1)(cos + f sin ) do -fmg (-sin fcos a (cos+f sin a)

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Step 3

Calculate ω for mini...

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-fing (-sin a,fcos a,) = 0 (cos a+f sin a -sin fcosa, = 0 sin a fcos an o = f aarctanf

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