Answer to Question #210999 in Classical Mechanics for Syed

Question #210999

A uniform ladder of length 50 feet, rests against a vertical wall with which it makes an angle of

60 degree, the coefficient of friction between wall and ladder and ground respectively being 1/2 and 3/5. If a man whose weight is one-fourth to the ladder ascends the ladder, how high will

be he when ladder slips?

Expert's answer

"\\begin{cases}\n \\mu_2mg\\cos a\\cdot 0=mg\\sin a\\frac l2+Mg\\sin a d=\\mu_1 mg \\sin a l, \\\\\n \\mu_1 mg \\sin a\\cdot 0=Mg\\sin a(l-d)+mg\\sin a\\frac l2+\\mu_2 mg\\cos ag,\n\\end{cases}"

"\\begin{cases}\n m\\frac l2+Md=\\mu_1ml,\\\\\n 0=M\\sin a(l-d)+m\\sin a\\frac l2+\\mu_2m\\cos al,\n\\end{cases}"

"d=l-\\frac{ml-2\\mu_1ml+m\\sin a\\frac l2-\\mu_2 m\\cos al}{2M+m\\cos a}\\approx 0.23l=11.5~ft."