A crane in a marble quarry is mounted on the rock walls of the quarry and is supporting a 2900kg slab of marble as shown in the figure. The center of mass of the 750kg boom is located one-third of the way from the pivot end of its 15-m length, as shown.

Find the tension in the horizontal cable that supports the boom.

To find the tension in the horizontal cable that supports the boom, we need to analyze the forces acting on the boom and the slab of marble.

Let's consider the forces acting on the boom:
1. The weight of the boom: This force acts downwards and can be calculated using the formula W = m*g, where m is the mass of the boom and g is the acceleration due to gravity (approximately 9.8 m/s²).
Weight of the boom = 750 kg * 9.8 m/s²

2. The tension in the horizontal cable: This force acts towards the pivot point of the boom and keeps it in equilibrium. Since the boom is not accelerating vertically, the vertical component of tension is equal to the weight of the boom (750 kg * 9.8 m/s²).
But to find the tension in the horizontal cable, we need to calculate the horizontal component of the tension.

To find the horizontal component of the tension, we can use the concept of torque. Torque is the tendency of a force to rotate an object about an axis, and it depends on the force's magnitude, direction, and point of application.

In this case, the torque due to the tension in the cable must balance the torque due to the weight of the boom. The torque due to the weight of the boom can be calculated as the product of the weight and the perpendicular distance from the pivot point to the line of action of the weight.

1. Calculate the torque due to the weight of the boom:
Torque = weight of the boom * perpendicular distance from the pivot
Torque = weight of the boom * (length of the boom - one-third of the length)

2. Calculate the torque due to the tension in the horizontal cable:
Torque = tension in the horizontal cable * one-third of the length of the boom

Now, since the boom is in equilibrium (not rotating), the torques due to the weight of the boom and the tension in the horizontal cable must be equal:

tension in the horizontal cable * one-third of the length of the boom = weight of the boom * (length of the boom - one-third of the length)

Now you have an equation with one unknown, the tension in the horizontal cable. You can solve this equation to find the value of the tension.