Answer to Question #124776 in Mechanics | Relativity for John Yokk'a

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
1
Expert's answer
2020-07-01T15:42:05-0400

Solution.

c=4m;c=4m;

a=2m;a=2m;

b=3m;b=3m;

b2=a2+c22accosβ    cosβ=a2+c2b22ac;b^2=a^2+c^2-2accos\beta\implies cos\beta=\dfrac{a^2+c^2-b^2}{2ac};

cosβ=4+169224=1116;cos\beta=\dfrac{4+16-9}{2\sdot2\sdot4}=\dfrac{11}{16};

sinβ=31516;sin\beta=\dfrac{3\sqrt{15}}{16};

a2=b2+c22accosα    cosα=b2+c2a22bc;a^2=b^2+c^2-2accos\alpha\implies cos\alpha=\dfrac{b^2+c^2-a^2}{2bc};

cosα=2124;cos\alpha=\dfrac{21}{24};

sinα=31524;sin\alpha=\dfrac{3\sqrt{15}}{24};

T1cosβ=T2cosα;T_1cos\beta=T_2cos\alpha;

T1sinβ+T2sinα=mg;T_1sin\beta+T_2sin\alpha=mg;

T1=4233T2;T_1=\dfrac{42}{33}T_2;

4233T2sinβ+T2sinα=mg;\dfrac{42}{33}T_2sin\beta+T_2sin\alpha=mg;

T2=70N;T_2=70N;

T1=423370=90N;T_1=\dfrac{42}{33}\sdot70=90N;

Answer: T1=90N;T_1=90N;

T2=70N.T_2=70N.



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