Question-214248

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

Commented by mr W last updated on 05/Dec/24

$${the}\:{ladder}\:{or}\:{this}\:{drawing}? \\ $$

Commented by 0202391905769 last updated on 05/Dec/24

$$\mathrm{comment}\:\mathrm{as}\:\mathrm{tu}\:\mathrm{construire}\:\mathrm{sa}? \\ $$

Answered by mr W last updated on 04/Dec/24

Commented by mr W last updated on 05/Dec/24

h=l sin θ let ω=−(dθ/dt) u_(My) =−(dh/dt)=ωl cos θ a_(My) =ω^2 l sin θ u_(mx) =((d(0.5l cos θ))/dt)=((ωl sin θ)/2) u_(my) =−((d(0.5l sin θ))/dt)=((ωl cos θ)/2) h_0 =l cos 22.5° Mg(h_0 −h)+2mg((h_0 /2)−(h/2)) =((Mω^2 l^2 cos^2 θ)/2)+2×((mω^2 l^2 )/2)(((sin^2 θ+cos^2 θ)/4))+2×((ml^2 )/(12))×ω^2 12(M+m)g(cos 22.5°−sin θ)=(6M cos^2 θ+5m)ω^2 l with μ=(m/M) ⇒ω^2 =((12(1+μ)g(cos ((3π)/8)−sin θ))/((6 cos^2 θ+5μ)l)) T=(√((2h_0 )/g))∫_0 ^((3π)/8) (√((6 cos^2 θ+5μ)/(24(1+μ)sin ((3π)/8)(cos ((3π)/8)−sin θ)))) dθ u_P =ωl cos θ=cos θ(√((12(1+μ)gl(cos ((3π)/8)−sin θ))/(6 cos^2 θ+5μ))) example: μ=0.1 T≈1.055212(√((2h_0 )/g)) u_P =(√((1+μ)/(1+((5μ)/6))))(√(2gh_0 ))≈1.008(√(2gh_0 )) Mg−N=Mω^2 l sin θ (N/(Mg))=1−((ω^2 l sin θ)/g)=1−((12(1+μ) sin θ(sin ((3π)/8)−sin θ))/(6 cos^2 θ+5μ))

$${h}={l}\:\mathrm{sin}\:\theta \\ $$$${let}\:\omega=−\frac{{d}\theta}{{dt}} \\ $$$${u}_{{My}} =−\frac{{dh}}{{dt}}=\omega{l}\:\mathrm{cos}\:\theta \\ $$$${a}_{{My}} =\omega^{\mathrm{2}} {l}\:\mathrm{sin}\:\theta \\ $$$${u}_{{mx}} =\frac{{d}\left(\mathrm{0}.\mathrm{5}{l}\:\mathrm{cos}\:\theta\right)}{{dt}}=\frac{\omega{l}\:\mathrm{sin}\:\theta}{\mathrm{2}} \\ $$$${u}_{{my}} =−\frac{{d}\left(\mathrm{0}.\mathrm{5}{l}\:\mathrm{sin}\:\theta\right)}{{dt}}=\frac{\omega{l}\:\mathrm{cos}\:\theta}{\mathrm{2}} \\ $$$${h}_{\mathrm{0}} ={l}\:\mathrm{cos}\:\mathrm{22}.\mathrm{5}° \\ $$$${Mg}\left({h}_{\mathrm{0}} −{h}\right)+\mathrm{2}{mg}\left(\frac{{h}_{\mathrm{0}} }{\mathrm{2}}−\frac{{h}}{\mathrm{2}}\right) \\ $$$$\:\:\:=\frac{{M}\omega^{\mathrm{2}} {l}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \:\theta}{\mathrm{2}}+\mathrm{2}×\frac{{m}\omega^{\mathrm{2}} {l}^{\mathrm{2}} }{\mathrm{2}}\left(\frac{\mathrm{sin}^{\mathrm{2}} \:\theta+\mathrm{cos}^{\mathrm{2}} \:\theta}{\mathrm{4}}\right)+\mathrm{2}×\frac{{ml}^{\mathrm{2}} }{\mathrm{12}}×\omega^{\mathrm{2}} \\ $$$$\mathrm{12}\left({M}+{m}\right){g}\left(\mathrm{cos}\:\mathrm{22}.\mathrm{5}°−\mathrm{sin}\:\theta\right)=\left(\mathrm{6}{M}\:\mathrm{cos}^{\mathrm{2}} \:\theta+\mathrm{5}{m}\right)\omega^{\mathrm{2}} {l} \\ $$$${with}\:\mu=\frac{{m}}{{M}} \\ $$$$\Rightarrow\omega^{\mathrm{2}} =\frac{\mathrm{12}\left(\mathrm{1}+\mu\right){g}\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{8}}−\mathrm{sin}\:\theta\right)}{\left(\mathrm{6}\:\mathrm{cos}^{\mathrm{2}} \:\theta+\mathrm{5}\mu\right){l}} \\ $$$${T}=\sqrt{\frac{\mathrm{2}{h}_{\mathrm{0}} }{{g}}}\int_{\mathrm{0}} ^{\frac{\mathrm{3}\pi}{\mathrm{8}}} \sqrt{\frac{\mathrm{6}\:\mathrm{cos}^{\mathrm{2}} \:\theta+\mathrm{5}\mu}{\mathrm{24}\left(\mathrm{1}+\mu\right)\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{8}}\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{8}}−\mathrm{sin}\:\theta\right)}}\:{d}\theta \\ $$$${u}_{{P}} =\omega{l}\:\mathrm{cos}\:\theta=\mathrm{cos}\:\theta\sqrt{\frac{\mathrm{12}\left(\mathrm{1}+\mu\right){gl}\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{8}}−\mathrm{sin}\:\theta\right)}{\mathrm{6}\:\mathrm{cos}^{\mathrm{2}} \:\theta+\mathrm{5}\mu}} \\ $$$${example}: \\ $$$$\mu=\mathrm{0}.\mathrm{1} \\ $$$${T}\approx\mathrm{1}.\mathrm{055212}\sqrt{\frac{\mathrm{2}{h}_{\mathrm{0}} }{{g}}} \\ $$$${u}_{{P}} =\sqrt{\frac{\mathrm{1}+\mu}{\mathrm{1}+\frac{\mathrm{5}\mu}{\mathrm{6}}}}\sqrt{\mathrm{2}{gh}_{\mathrm{0}} }\approx\mathrm{1}.\mathrm{008}\sqrt{\mathrm{2}{gh}_{\mathrm{0}} } \\ $$$$ \\ $$$${Mg}−{N}={M}\omega^{\mathrm{2}} {l}\:\mathrm{sin}\:\theta \\ $$$$\frac{{N}}{{Mg}}=\mathrm{1}−\frac{\omega^{\mathrm{2}} {l}\:\mathrm{sin}\:\theta}{{g}}=\mathrm{1}−\frac{\mathrm{12}\left(\mathrm{1}+\mu\right)\:\mathrm{sin}\:\theta\left(\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{8}}−\mathrm{sin}\:\theta\right)}{\mathrm{6}\:\mathrm{cos}^{\mathrm{2}} \:\theta+\mathrm{5}\mu} \\ $$

Answered by a.lgnaoui last updated on 04/Dec/24

la situation { ((x_0 =Lsin (π/8) t=0)),((y_0 =Lcos (π/8))) :} { ((x=L(1−sin (π/8)) (t_0 +t))),((y=o)) :} { ((ax=o x=vt)),((ay=+g v=gt y=(1/2)gt^2 +Lsin (π/8))) :} a t t=(x/v)=((L(1−sin (π/8)))/v)=(v/g) v=(√(gL(1−sin (π/8)))) y=(x^2 /(2v^2 ))gL(1−sin (π/8))+Lsin (π/8) { ((x=L(1−sin (π/8)) ;y=0)),(((x^2 /(2v^2 ))g(1−sin (𝛑/8))+sin (π/8)=0)) :} g(1−sin )x^2 =−2v^2 sin ((π/8)) ∣v∣=x(√((g(1−sin (π/8)))/(2sin (π/8)))) v=L(√((g(1−sin (π/8))^3 )/(2sin (π/8)))) t=(x/v) ⇒ x=L(1−sin (𝛑/8)) t=(√((2sin (π/8))/(g(1−sin (π/8))))) soit v=gt v= (√((2gsin (𝛑/8))/(1−sin (𝛑/8))))

$$\mathrm{la}\:\mathrm{situation}\: \\ $$$$\:\begin{cases}{\boldsymbol{\mathrm{x}}_{\mathrm{0}} =\boldsymbol{\mathrm{L}}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\:\:\mathrm{t}=\mathrm{0}}\\{\boldsymbol{\mathrm{y}}_{\mathrm{0}} =\boldsymbol{\mathrm{L}}\mathrm{cos}\:\frac{\pi}{\mathrm{8}}}\end{cases} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{L}}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{t}}_{\mathrm{0}} +\boldsymbol{\mathrm{t}}\right)}\\{\boldsymbol{\mathrm{y}}=\mathrm{o}}\end{cases} \\ $$$$\:\:\begin{cases}{\mathrm{ax}=\mathrm{o}\:\:\:\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{vt}}}\\{\boldsymbol{\mathrm{ay}}=+\boldsymbol{\mathrm{g}}\:\:\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{gt}}\:\:\:\:\:\boldsymbol{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{\mathrm{gt}}^{\mathrm{2}} +\boldsymbol{\mathrm{L}}\mathrm{sin}\:\frac{\pi}{\mathrm{8}}}\end{cases} \\ $$$$\:\:\mathrm{a}\:\boldsymbol{\mathrm{t}}\:\:\:\boldsymbol{\mathrm{t}}=\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{v}}}=\frac{\boldsymbol{\mathrm{L}}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)}{\mathrm{v}}=\frac{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{g}}} \\ $$$$\:\:\:\boldsymbol{\mathrm{v}}=\sqrt{\boldsymbol{\mathrm{gL}}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)} \\ $$$$\:\:\:\boldsymbol{\mathrm{y}}=\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{2}\boldsymbol{\mathrm{v}}^{\mathrm{2}} }\boldsymbol{\mathrm{gL}}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)+\boldsymbol{\mathrm{L}}\mathrm{sin}\:\frac{\pi}{\mathrm{8}} \\ $$$$\begin{cases}{\mathrm{x}=\mathrm{L}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)\:\:\:;\boldsymbol{\mathrm{y}}=\mathrm{0}}\\{\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{2}\boldsymbol{\mathrm{v}}^{\mathrm{2}} }\boldsymbol{\mathrm{g}}\left(\mathrm{1}−\mathrm{sin}\:\frac{\boldsymbol{\pi}}{\mathrm{8}}\right)+\mathrm{sin}\:\frac{\pi}{\mathrm{8}}=\mathrm{0}}\end{cases} \\ $$$$\:\:\mathrm{g}\left(\mathrm{1}−\mathrm{sin}\:\right)\mathrm{x}^{\mathrm{2}} =−\mathrm{2v}^{\mathrm{2}} \mathrm{sin}\:\left(\frac{\pi}{\mathrm{8}}\right) \\ $$$$\:\:\:\mid\mathrm{v}\mid=\mathrm{x}\sqrt{\frac{\mathrm{g}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)}{\mathrm{2sin}\:\frac{\pi}{\mathrm{8}}}} \\ $$$$\:\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{L}}\sqrt{\frac{\mathrm{g}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)^{\mathrm{3}} }{\mathrm{2sin}\:\frac{\pi}{\mathrm{8}}}} \\ $$$$\:\:\:\boldsymbol{\mathrm{t}}=\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{v}}}\:\Rightarrow\:\:\:\:\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{L}}\left(\mathrm{1}−\mathrm{sin}\:\frac{\boldsymbol{\pi}}{\mathrm{8}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{t}}=\sqrt{\frac{\mathrm{2sin}\:\frac{\pi}{\mathrm{8}}}{\mathrm{g}\left(\mathrm{1}−\mathrm{sin}\:\frac{\pi}{\mathrm{8}}\right)}} \\ $$$$\:\:\: \\ $$$$ \\ $$$$\boldsymbol{\mathrm{soit}}\:\:\:\:\mathrm{v}=\mathrm{gt}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{v}}=\:\:\sqrt{\frac{\mathrm{2}\boldsymbol{\mathrm{g}}\mathrm{sin}\:\frac{\boldsymbol{\pi}}{\mathrm{8}}}{\mathrm{1}−\mathrm{sin}\:\frac{\boldsymbol{\pi}}{\mathrm{8}}}} \\ $$$$ \\ $$

Commented by mr W last updated on 04/Dec/24