Question

2. If a force (7i+5j + 3k)N acts at a point P (2, 4, 2) meter, find the torque
(1) about the origin (ii) about a point (2,-1, 3).​

Answer 1

(i) force, F = (7 i + 5j + 3k)

position of the point P(2, 4,2) with respect to origin , r = (2i + 4j + 2k) - (0i + 0j + 0k) = 2i + 4j + 2k

now, torque, τ = r × F

= (2i + 4j + 2k) × (7i + 5j + 3k)

= (4 × 3 - 2 × 5)i - (2 × 3 - 7 × 2) j + (2 × 5 - 4 × 7) k

= (12 - 10)i - (6 - 14)j + (10 - 28)k

= 2i + 8j - 18k

(ii) position of the point (2, 4, 2) with respect to (2, -1, 3) , r = (2i + 4j + 2k) - (2i - j + 3k) = 0i + 5j - k

now torque , τ = r × F

= (0i + 5j - k) × (7i + 5j + 3k)

= (5 × 3 - 5 × -1)i - (0 × 3 - 7 × -1)j + (0 × 5 - 7 × 5)k

= (15 + 5)i - (7)j + (0 - 35)k

= 20i - 7j - 35k

Answer 2

$\huge\bold\purple{Answer:-}$

(i) force, F = (7 i + 5j + 3k)

position of the point P(2, 4,2) with respect to origin , r = (2i + 4j + 2k) - (0i + 0j + 0k) = 2i + 4j + 2k

now, torque, τ = r × F

= (2i + 4j + 2k) × (7i + 5j + 3k)

= (4 × 3 - 2 × 5)i - (2 × 3 - 7 × 2) j + (2 × 5 - 4 × 7) k

= (12 - 10)i - (6 - 14)j + (10 - 28)k

= 2i + 8j - 18k

(ii) position of the point (2, 4, 2) with respect to (2, -1, 3) , r = (2i + 4j + 2k) - (2i - j + 3k) = 0i + 5j - k

now torque , τ = r × F

= (0i + 5j - k) × (7i + 5j + 3k)

= (5 × 3 - 5 × -1)i - (0 × 3 - 7 × -1)j + (0 × 5 - 7 × 5)k

= (15 + 5)i - (7)j + (0 - 35)k

= 20i - 7j - 35k