Video: Finding the Unknown Components of Three Forces in Equilibrium Acting on a Point

The forces ๐นโ‚ = 2๐‘– + 7๐‘—, ๐นโ‚‚ = ๐‘Ž๐‘– โˆ’ 6๐‘— and ๐นโ‚ƒ = 6๐‘– + (๐‘ + 8)๐‘— act on a particle, where ๐‘– and ๐‘— are two perpendicular unit vectors. Given that the system is in equilibrium, determine the values of ๐‘Ž and ๐‘.

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Video Transcript

The forces ๐น one, which is equal to two ๐‘– plus seven ๐‘—; ๐น two, which is equal to ๐‘Ž๐‘– minus six ๐‘—; and ๐น three, which is equal to six ๐‘– plus ๐‘ plus eight ๐‘—, act on a particle, where ๐‘– and ๐‘— are two perpendicular unit vectors. Given that the system is in equilibrium, determine the values of ๐‘Ž and ๐‘.

If the system is in equilibrium, then the resultant force must equal zero. This means that the ๐‘–-components โ€” two ๐‘–, ๐‘Ž๐‘–, and six ๐‘– โ€” must equal zero. The coefficients are two, ๐‘Ž, and six. Therefore, two plus ๐‘Ž plus six equals zero. Two plus six is equal to eight. Therefore, ๐‘Ž plus eight is equal to zero. Subtracting eight from both sides of the equation gives us a value of ๐‘Ž of negative eight. This means that the force ๐น two is negative eight ๐‘– minus six ๐‘—.

As the ๐‘—-components must also equal zero, seven ๐‘— minus six ๐‘— and ๐‘ plus eight ๐‘— must equal zero. Once again, the coefficients are seven, negative six, and ๐‘ plus eight. Seven minus six plus ๐‘ plus eight equals zero. Seven take away six is one. One plus eight is equal to nine. Therefore, ๐‘ plus nine equals zero. Subtracting nine from both sides of this equation gives us a value of ๐‘ equal to negative nine.

This means that the force ๐น three is equal to six ๐‘– plus negative nine plus eight ๐‘—. Negative nine plus eight is negative one. Therefore, ๐น three is six ๐‘– minus one ๐‘—. If the three forces ๐น one, ๐น two, and ๐น three are acting on a particle where the system is in equilibrium, then the value of ๐‘Ž is negative eight and the value of ๐‘ is negative nine.

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