This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The resultant of two vectors is a single vector that produces the same effect as the original two vectors combined. It is the vector sum of the two vectors. Given forces: F_1 = 7 N and F_2 = 10 N. The magnitude of the resultant force R of two forces F_1 and F_2 with an angle between them is given by the formula: R = sqrt(F_1^2 + F_2^2 + 2F_1F_2 ) a) The forces are parallel and act in the same direction. Step 1: Determine the angle between the forces. When forces are parallel and act in the same direction, = 0^. Step 2: Substitute the values into the formula. R &= sqrt((7 N))^2 + (10 N)^2 + 2(7 N)(10 N) (0^) \\ &= sqrt(49 N)^2 + 100 N^2 + 140 N^2 (1) \\ &= sqrt(289 N)^2 \\ &= 17 N Alternatively, for forces in the same direction, R = F_1 + F_2. R = 7 N + 10 N = 17 N The resultant force is 17 N. b) The forces are parallel and act in opposite directions. Step 1: Determine the angle between the forces. When forces are parallel and act in opposite directions, = 180^. Step 2: Substitute the values into the formula. R &= sqrt((7 N))^2 + (10 N)^2 + 2(7 N)(10 N) (180^) \\ &= sqrt(49 N)^2 + 100 N^2 + 140 N^2 (-1) \\ &= sqrt(149 N)^2 - 140 N^2 \\ &= sqrt(9 N)^2 \\ &= 3 N Alternatively, for forces in opposite directions, R = |F_2 - F_1|. R = |10 N - 7 N| = 3 N The resultant force is 3 N. c) The two forces are inclined at an angle of 60^ to each other. Step 1: Determine the angle between the forces. = 60^. Step 2: Substitute the values into the formula. R &= sqrt((7 N))^2 + (10 N)^2 + 2(7 N)(10 N) (60^) \\ &= sqrt(49 N)^2 + 100 N^2 + 140 N^2 (0.5) \\ &= sqrt(149 N)^2 + 70 N^2 \\ &= sqrt(219 N)^2 \\ &≈ 14.80 N The resultant force is 14.80 N. d) The two forces are inclined at an angle of 160^. Step 1: Determine the angle between the forces. = 160^. Step 2: Substitute the values into the formula. R &= sqrt((7 N))^2 + (10 N)^2 + 2(7 N)(10 N) (160^) \\ &= sqrt(49 N)^2 + 100 N^2 + 140 N^2 (-0.9397) \\ &= sqrt(149 N)^2 - 131.558 N^2 \\ &= sqrt(17.442 N)^2 \\ &≈ 4.18 N The resultant force is 4.18 N. Send me the next one 📸