What is the torque about the origin on a particle located at = 3 m + 4 m - 2 m if a force = 5 N - 2 N + 3 N acts on the particle? A)(8 + - 26 )N ∙ m B)(8 - 19 - 26 )N ∙ m C)(8 + 2 + )N ∙ m D)(16 - 19 - 26 )N ∙ m E)(8 - 2 + )N ∙ m

Question

Quizplus · Accepted Answer

To find the torque about the origin, we need to find the perpendicular distance from the origin to the line of action of the force and multiply it by the magnitude of the force.

First, we need to find the line of action of the force. This is defined by the points where the force is applied:

S1U1B1 = (3, 4, -2)
S1U1B0 = (4, -2, -2)
S1U1B1 = (-2, 3, 4)
S1U1B0 = (-2, -2, 3)

We can find two points on the line of action by choosing any two of these points. Let's choose S1U1B1 and S1U1B0:

Direction vector = (4-(-2), -2-4, -2-(-2)) = (6, -6, -4)
Point on the line = S1U1B1 = (3, 4, -2)

So the line of action is given by the vector equation:

r = (3, 4, -2) + t(6, -6, -4)

Next, we need to find the perpendicular distance from the origin to this line. The direction vector of the line is perpendicular to any vector that lies in the plane containing the origin and the line.

Let's find two vectors that lie in this plane:

Vector 1 = (3, 4, -2)
Vector 2 = (6, -6, -4)

The cross product of these vectors gives us a vector that is perpendicular to both:

Vector perpendicular = Vector 1 x Vector 2 = (8, 11, 11)

The magnitude of this vector is the area of the parallelogram formed by Vector 1 and Vector 2, which is also the perpendicular distance from the origin to the line:

Perpendicular distance = |Vector perpendicular| = sqrt(8^2 + 11^2 + 11^2) ≈ 16.55

Finally, we can find the torque about the origin by multiplying the force by the perpendicular distance and taking the cross product of the force and the vector from the origin to the point of application:

Torque = r x F = (3i + 4j - 2k) x (5i - 2j + 3k)
       = 8i - 19j - 26k

(Note that we could also have used the other two points to find the same line of action, but we would have ended up with a different point on the line and a different perpendicular distance, which would have cancelled out in the final calculation.)

What Is the Torque About the Origin on a Particle