Two light inextensible strings are attached to a horizontal surface and suspended a 10-kilogram object as shown in the diagram below
The tension in the strings are \(T_1\) newtons and \(T_2\) newtons.
- Express \(T_1\) in terms of \(T_2\). (2 marks)
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- If the acceleration due to gravity is 9.8 ms\(^{-2}\), determine the exact values of \(T_1\) and \(T_2\), in newtons. (2 marks)
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Show Worked Solution
a. \(\text{Resolve forces into horizontal/vertical components:}\)
\(\text{Horizontal forces are equal.}\)
| \(T_1 \cos 60^{\circ}\) | \(=T_2 \cos 30^{\circ}\) |
| \(T_1 \times \dfrac{1}{2}\) | \(=T_2 \times \dfrac{\sqrt{3}}{2}\) |
| \(T_1\) | \(=\sqrt{3}\, T_2\) |
b. \(\text{Vertical forces are equal.}\)
| \(T_1 \sin 60^{\circ}+T_2 \sin 30^{\circ}\) | \(=10 \times 9.8\) |
| \(T_1 \times \dfrac{\sqrt{3}}{2}+T_2 \times \frac{1}{2}\) | \(=98\) |
\(\text {Substitute} \ \ T_1=\sqrt{3}\, T_2 :\)
| \(\sqrt{3}\, T_2 \times \dfrac{\sqrt{3}}{2}+T_2 \times \dfrac{1}{2}\) | \(=98\) |
| \(2\, T_2\) | \(=98\) |
| \(T_2\) | \(=49 \ \text{newtons}\) |
\(\therefore T_1=49 \sqrt{3}, \ T_2=49\)