Question

A crane is oriented so that the end of the 25 -m boom AO lies in the yz plane. At the instant shown, the tension in cable AB is 4kN. Determine the moment about each of the coordinate axes of the force exerted on A by cable AB

Answer 1

To determine the moment about each axis of the force exerted on point A by cable AB, find the components of the force in the x, y, and z directions using trigonometry.

Step-by-step explanation:

To determine the moment about each of the coordinate axes of the force exerted on point A by cable AB, we need to calculate the components of the force in the x, y, and z directions. Since the boom lies in the yz plane, the x component of the force is zero.

To find the y and z components, we can use trigonometry. Let's call the angle between the AB cable and the y-axis θ. From the given diagram, we can see that the distance from point A to the yz plane is 25 m, and the tension in cable AB is 4 kN.

The y component of the force, Fy, is given by Fy = T × sin(θ), where T is the tension in cable AB. The z component of the force, Fz, is given by Fz = T × cos(θ).

Therefore, the moment about the x-axis is 0 Nm, the moment about the y-axis is 4kN × sin(θ) Nm, and the moment about the z-axis is 4kN × cos(θ) Nm.

Answer 2

To calculate the moment about coordinate axes of the force at point A by the cable, use the vector-cross-product method with the position vector for point A, the force vector from the tension, and the corresponding components for each axis.

Step-by-step explanation:

To determine the moment about each of the coordinate axes of the force exerted at point A by the cable AB, we can use the vector-cross-product method. Assuming point A is the origin and the boom AO lies along the yz plane, we will find the position vector for point A and the force vector from cable AB. The tension in cable AB is 4kN, so the force vector's magnitude is 4kN, directed along the cable from point A to point B.

First, find the unit vector in the direction of AB, then multiply it by the magnitude of the force to get the force vector in vector form. Then, calculate the moment about each axis by taking the cross product of the position vector (which extends from the origin to the point of force application) with the force vector.

For the x-axis moment, use the components of the force vector in the yz-plane. The y-axis moment will involve the z-component of the force vector and the z-axis moment will involve the y-component of the force vector. Each of these moments gives you the tendency of the force to rotate the crane about the respective coordinate axis.