Question #124776

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

Expert's answer

**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."

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