Answer to Question #171711 in Mechanics | Relativity for Lea Marie Depalubos

Question #171711

2. A 30 foot ladder weighing 100 lbs having its center of mass one-third of the way up from the bottom rests against a smooth wall so that it makes an angle of 60 degrees with the ground. If the coefficient of friction between the ground and the ladder is 0.4, how high can a 150-lb man go before the ladder slips. 


1
Expert's answer
2021-03-16T11:36:37-0400

F+N0+mg+Mg+Nl=0,\vec{F}+\vec{N_0}+m\vec{g}+M\vec{g}+\vec{N_l}=\vec{0},

{F+Nl=0mgMg+N0=0\begin{cases} -F+N_l=0 \\ -mg-Mg+N_0=0 \end{cases}

{F=Nl(m+M)g=N0\begin{cases} F=N_l\\ (m+M)g=N_0 \end{cases}

FμN0F\leqslant \mu N_0

{NlμN0(m+M)g=N0\begin{cases} N_l\leqslant \mu N_0\\ (m+M)g=N_0 \end{cases}

Nlμ(m+M)gN_l\leqslant \mu(m+M)g


mgl3cosα+Mgxcosα=Nllsinαmg\frac l3 cos \alpha +Mgxcos\alpha=N_l lsin\alpha

gcosα(Mx+ml3)=Nllsinαgcos\alpha (Mx+m\frac l3)=N_l lsin \alpha

Mx+ml3=Nlltanαgμ(m+M)ltanαMx+m\frac l3=\frac{N_l ltan\alpha}{g} \leqslant\mu(m+M)l tan\alpha

xl(μtanα(1+mM)m3M)=27.97 ft.x\leqslant l(\mu tan \alpha(1+\frac mM)-\frac{m}{3M})=27.97~ft.



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