Question

If a rope 5m long is fastened to two hooks 4m apart on a horizontal ceiling to the rope is attached a 10kg mass so that the segment of the rope is 3m and 2m. Compute the tension in each segment

Answer 1

Solution.

$c=4m;$

$a=2m;$

$b=3m;$

$b^2=a^2+c^2-2accos\beta\implies cos\beta=\dfrac{a^2+c^2-b^2}{2ac};$

$cos\beta=\dfrac{4+16-9}{2\sdot2\sdot4}=\dfrac{11}{16};$

$sin\beta=\dfrac{3\sqrt{15}}{16};$

$a^2=b^2+c^2-2accos\alpha\implies cos\alpha=\dfrac{b^2+c^2-a^2}{2bc};$

$cos\alpha=\dfrac{21}{24};$

$sin\alpha=\dfrac{3\sqrt{15}}{24};$

$T_1cos\beta=T_2cos\alpha;$

$T_1sin\beta+T_2sin\alpha=mg;$

$T_1=\dfrac{42}{33}T_2;$

$\dfrac{42}{33}T_2sin\beta+T_2sin\alpha=mg;$

$T_2=70N;$

$T_1=\dfrac{42}{33}\sdot70=90N;$

Answer: $T_1=90N;$

$T_2=70N.$