A force,
F = (6x̂ + 7y hat) N,
is applied to an object at a point whose position vector with respect to the pivot point is
r = (7x̂ + 7y hat + 7ẑ) m.
Calculate the torque created by the force about that pivot point.
To calculate the torque created by the force about the pivot point, we can use the cross product between the force vector and the position vector. The formula for calculating torque is given by:
τ = r × F
In this case, the force vector, F, is given as (6x̂ + 7y hat) N, and the position vector, r, is given as (7x̂ + 7y hat + 7ẑ) m.
Using the cross product formula, we can calculate the torque as follows:
τ = (7x̂ + 7y hat + 7ẑ) × (6x̂ + 7y hat)
To expand this cross product, we can use the determinant method:
τ = (7i + 7j + 7k) × (6i + 7j)
= (7i + 7j + 7k) × 6i + (7i + 7j + 7k) × 7j
= 42(i × i) + 42(i × j) + 42(i × k) + 49(j × i) + 49(j × j) + 49(j × k)
Now, we can use the cross product rules to calculate each term:
(i × i) = (j × j) = (k × k) = 0
(i × j) = k
(j × i) = -k
Substituting these values back into the equation:
τ = 0 + 0 + 0 + 49(-k) + 0 + 0
= -49k
Therefore, the torque created by the force about the pivot point is -49k N·m.