. In many applications a bolt is required to be tightened to a specific moment (torque), this is measured with a torque wrench. A bolt is located at the origin and is rotated so that the axis of rotation should be the z axis. A 0.5 m wrench is attached to the bolt at a 30◦ angle to the x axis such that
d = 0.86i + 0.25j + 0k
If the magnitude of the moment of the force at the bolt is required to be ∥M∥ = 100N·m, find the normal force (i.e. at right angles) applied at the handle of the wrench that will achieve this value.
If I understand your geometry the wrench is in the x y plane perpendicular to the z axis. In that case moment about the z axis due to force normal to the wrench = force * length * 1
or
F * 0.5 = 100
To find the normal force applied at the handle of the wrench in order to achieve the desired magnitude of the moment, we can use the concept of torque.
Torque is the product of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force. In this case, the force is the normal force and the perpendicular distance is the distance vector d.
The equation for torque is given by:
M = r x F
Where M is the moment of the force, r is the distance vector, and x represents the cross product.
Given that the magnitude of the moment, ∥M∥, is 100 N·m, we can find the force F by rearranging the torque equation:
F = M / ∥r∥
Where ∥r∥ is the magnitude of the distance vector, which can be calculated using the formula:
∥r∥ = sqrt(d_x^2 + d_y^2 + d_z^2)
In this case, d = 0.86i + 0.25j + 0k, so ∥r∥ = sqrt(0.86^2 + 0.25^2 + 0^2) = 0.897 m (rounded to three decimal places).
Now we can calculate the normal force F:
F = 100 N·m / 0.897 m = 111.42 N (rounded to two decimal places).
Therefore, the normal force applied at the handle of the wrench that will achieve a moment magnitude of 100 N·m is approximately 111.42 N.