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Question-214248

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Question Number 214248 by mr W last updated on 03/Dec/24
Commented by mr W last updated on 02/Dec/24
a man with mass M is standing on the top of a twin step ladder. each ladder with length l has a mass m. both ladders form an angle of 45°. suddently the spreaders are brocken and the ladders begin to slip on the smooth surface of thefloor. after what time and with what speed will the man hit the floor?
$${a}\:{man}\:{with}\:{mass}\:{M}\:{is}\:{standing}\:{on} \\ $$$${the}\:{top}\:{of}\:{a}\:{twin}\:{step}\:{ladder}.\:{each} \\ $$$${ladder}\:{with}\:{length}\:{l}\:{has}\:{a}\:{mass}\:{m}. \\ $$$${both}\:{ladders}\:{form}\:{an}\:{angle}\:{of}\:\mathrm{45}°. \\ $$$${suddently}\:{the}\:{spreaders}\:{are}\:{brocken} \\ $$$${and}\:{the}\:{ladders}\:{begin}\:{to}\:{slip}\:{on}\:{the} \\ $$$${smooth}\:{surface}\:{of}\:{thefloor}. \\ $$$${after}\:{what}\:{time}\:{and}\:{with}\:{what} \\ $$$${speed}\:{will}\:{the}\:{man}\:{hit}\:{the}\:{floor}? \\ $$
Answered by a.lgnaoui last updated on 02/Dec/24
hauteur h=Lcos( ((45)/2)) h=L((√(2+(√2)))/2) a_y =g v=gt+v_(0 ) v_0 =0 v=gt h=(1/2)gt^2 ⇒ v^2 =2gh v=(√(2gh)) =(√(gL(√(2+(√2))))) t =((√(Lg(√(2+(√2))) )))/g)
$$\mathrm{hauteur}\:\:\mathrm{h}=\mathrm{Lcos}\left(\:\frac{\mathrm{45}}{\mathrm{2}}\right) \\ $$$$\:\:\mathrm{h}=\mathrm{L}\frac{\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$$$\:\mathrm{a}_{\mathrm{y}} =\mathrm{g}\:\:\:\mathrm{v}=\mathrm{gt}+\mathrm{v}_{\mathrm{0}\:\:\:} \\ $$$$\mathrm{v}_{\mathrm{0}} =\mathrm{0}\:\:\:\:\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{gt}}\:\: \\ $$$$\:\boldsymbol{\mathrm{h}}=\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{gt}}^{\mathrm{2}} \:\:\Rightarrow\:\boldsymbol{\mathrm{v}}^{\mathrm{2}} =\mathrm{2}\boldsymbol{\mathrm{gh}} \\ $$$$\:\:\boldsymbol{\mathrm{v}}=\sqrt{\mathrm{2}\boldsymbol{\mathrm{gh}}}\:=\sqrt{\boldsymbol{\mathrm{gL}}\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}} \\ $$$$\:\mathrm{t}\:=\frac{\sqrt{\left.\boldsymbol{\mathrm{Lg}}\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}\:\right)}}{\boldsymbol{\mathrm{g}}} \\ $$$$ \\ $$
Commented by a.lgnaoui last updated on 02/Dec/24
Commented by mr W last updated on 03/Dec/24
you seem not to understand what the question is about. the man falls with the ladder. he doesn′t fall from the ladder! besides the total mass is M+2m. the center of mass of each ladder is at its middle point, not on its top.
$${you}\:{seem}\:{not}\:{to}\:{understand}\:{what} \\ $$$${the}\:{question}\:{is}\:{about}. \\ $$$${the}\:{man}\:{falls}\:{with}\:{the}\:{ladder}.\:{he} \\ $$$${doesn}'{t}\:{fall}\:{from}\:{the}\:{ladder}! \\ $$$${besides}\:{the}\:{total}\:{mass}\:{is}\:{M}+\mathrm{2}{m}. \\ $$$${the}\:{center}\:{of}\:{mass}\:{of}\:{each}\:{ladder}\:{is} \\ $$$${at}\:{its}\:{middle}\:{point},\:{not}\:{on}\:{its}\:{top}. \\ $$
Commented by mr W last updated on 03/Dec/24
Answered by a.lgnaoui last updated on 03/Dec/24
mouvement de l′ escalief: force apliques au centre de chaque escalier mg+T=ma escslier gisse de x /axe x θ=90−(π/8) x=(L/2)cos (90−(𝛑/8)) x=(L/2)sin (π/8) d autre psrr h passe a y=(L/2)cos (𝛑/8) { ((a_x =0 v=cte x=vt =(L/2)sin (π/8))),((a_y =+g v=gt y=(1/2)gt^2 =(L/2)cos (π/8))) :} v=(√(2gx)) =(√(gLsin (𝛑/8))) t=(√(((Lsin (π/8))/g) )) ?
$$\mathrm{mouvement}\:\mathrm{de}\:\mathrm{l}'\:\mathrm{escalief}: \\ $$$$\mathrm{force}\:\mathrm{apliques}\:\mathrm{au}\:\mathrm{centre}\:\mathrm{de}\:\mathrm{chaque} \\ $$$$\mathrm{escalier}\:\:\:\mathrm{mg}+\mathrm{T}=\mathrm{ma} \\ $$$$\mathrm{escslier}\:\mathrm{gisse}\:\mathrm{de}\:\:\boldsymbol{\mathrm{x}}\:/\mathrm{axe}\:\mathrm{x} \\ $$$$\theta=\mathrm{90}−\frac{\pi}{\mathrm{8}}\:\:\boldsymbol{\mathrm{x}}=\frac{\boldsymbol{\mathrm{L}}}{\mathrm{2}}\mathrm{cos}\:\left(\mathrm{90}−\frac{\boldsymbol{\pi}}{\mathrm{8}}\right) \\ $$$$\boldsymbol{\mathrm{x}}=\frac{\boldsymbol{\mathrm{L}}}{\mathrm{2}}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\:\:\: \\ $$$$\mathrm{d}\:\mathrm{autre}\:\mathrm{psrr}\:\mathrm{h}\:\mathrm{passe}\:\mathrm{a}\:\boldsymbol{\mathrm{y}}=\frac{\boldsymbol{\mathrm{L}}}{\mathrm{2}}\mathrm{cos}\:\frac{\boldsymbol{\pi}}{\mathrm{8}} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{x}}} =\mathrm{0}\:\:\:\:\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{cte}}\:\:\:\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{vt}}\:=\frac{\mathrm{L}}{\mathrm{2}}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}}\\{\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{y}}} =+\boldsymbol{\mathrm{g}}\:\:\:\:\:\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{gt}}\:\:\:\boldsymbol{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{gt}}^{\mathrm{2}} =\frac{\mathrm{L}}{\mathrm{2}}\mathrm{cos}\:\frac{\pi}{\mathrm{8}}}\end{cases} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{v}}=\sqrt{\mathrm{2}\boldsymbol{\mathrm{gx}}}\:=\sqrt{\boldsymbol{\mathrm{g}}\mathrm{Lsin}\:\frac{\boldsymbol{\pi}}{\mathrm{8}}} \\ $$$$\:\:\:\boldsymbol{\mathrm{t}}=\sqrt{\frac{\boldsymbol{\mathrm{L}}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}}{\boldsymbol{\mathrm{g}}}\:}\:\:\:\:\:? \\ $$$$\: \\ $$
Commented by a.lgnaoui last updated on 03/Dec/24

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