Question

A ladder, whose length is 10 m and whose mass is 40 kg rests against a frictionless, vertical wall. Its upper end is a distance of 7.7 m above the ground. The center of mass of the ladder is 1/3 of the way up the ladder. The coefficient of static friction between the ladder and the ground is 0.53. If a carpenter climbs 85% of the way up the ladder before it starts to slip, find the mass of the carpenter.

Answer 1

Gives

L=10 m

$d_1=7.7m\\L'=\frac{L}{3}$

Mass(m)=40kg

$\mu_k=0.53$

$\theta=cos^{-1}(\frac{7.7}{10})=15°$

$\sum{F_y}=0$

0=-Mg-mg+N

N=(m+M)g=(40+M)g

$F_R=\mu_kN$

$F=0.53\times(40+M)$

According to daigram

$0+Mgd_1+Mgd_2-Fd_3$

$d_1=\frac{L}{3},d_2=\frac{Lsin\theta}{2},d_3=Lcos\theta$

Put value

$$F(cos\theta)=\frac{mgsin\theta}{3}+mg\frac{L}{2}sin\theta-F(Lcos\theta)$$

$$F=\frac{\frac{mgsin15}{3}+mg\frac{L}{2}sin15-F(Lcos15)}{cos15}=109.4N$$

$109.4=0.53\times{(40+M)}\\M=65.5kg$