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Question
Read the following passage and answer the questions given below:
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Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area. Team A pulls with force F1 = `6hati + 0hatj kN`, Team B pulls with force F2 = `-4hati + 4hatj kN`, Team C pulls with force F3 = `-3hati - 3hatj kN`,
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- What is the magnitude of the force of Team A ?
- Which team will win the game?
- Find the magnitude of the resultant force exerted by the teams.
OR
In what direction is the ring getting pulled?
Solution
We have,
`|vecF_1| = sqrt(6^2 + 0^2)` = 6 kN,
`|vecF_2| = sqrt((-4)^2 + 4^2) = sqrt(32) = 4sqrt(2) kN`,
`|vecF_3| = sqrt((-3)^2 + (-3)^2) = sqrt(18) = 3sqrt(2) kN`.
i. Magnitude of force of Team A = 6 kN.
ii. Since `veca + vecc = 3(hati - hatj)` and `vecb = -4(hati - hatj)`
So, `vecb` and `veca + vecc` are unlike vectors having same intial point
and `|vecb| = 4sqrt(2)` and `|veca + vecc| = 3sqrt(2)`
Thus `|vecF_2| > |vecF_1 + vecF_3|` also `vecF_2` and `vecF_1 + vecF_3` are unlike
Hence B will win the game
iii. `vecF = vecF_1 + vecF_2 + vecF_3`
= `6hati + 0hatj - 4hati + 4hatj - 3hati - 3hatj`
= `-hati + hatj`
∴ `|vecF| = sqrt((-1)^2 + (1)^2) = sqrt(2) kN`.
OR
`vecF = -hati + hatj`
∴ θ = `π - tan^-1 (1/1)`
= `π - π/4`
= `(3π)/4`; where 'θ' is the angle made by the resultant force with the +ve direction of the x-axis.
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`veca = 4hati - hatj + hatk` and `vecb = -2hati + hatj - 2hatk`
