A uniform beam of weight 520 N and length 3.4 m is suspended horizontally. On the left it is hinged to a wall; on the right is it supported by a cable bolted to the wall at distance D above the beam. The least tension that will snap the cable is 1200 N.

What value of D corresponds to that tension?

Set the cable tension T equal to 1200 N, the breaking point. Then solve for the value of D that makes the total moment about the hinge equal to zero. The moment due to the weight of the beam acts trhough the center, 1.7 m from the wall. The vertical component of the cable force is T sin A, where A = arctan D/3.4

Use the moment balance to solve for the angle A and then solve for D = 3.4 tan A

To find the value of D that corresponds to a tension of 1200 N, we can start by considering the equilibrium of the beam. Since the beam is in equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about the same point.

Let's assume the distance between the left hinge and the cable bolted to the wall is x.

At the left hinge, the clockwise moment is zero because there is no distance between the hinge and the point of application of the weight. The anticlockwise moment at the left hinge is the weight of the beam (520 N) multiplied by the distance from the hinge (x). So, the moment at the left hinge is given by 520x N.

At the support point (cable bolted to the wall), the clockwise moment is the tension in the cable (1200 N) multiplied by the distance from the hinge (D - x). The anticlockwise moment at the support point is the weight of the beam (520 N) multiplied by the distance from the hinge (3.4 - x) or (x - 3.4) (taking into account the 3.4 m length of the beam). So, the moment at the support point is given by 1200(D - x) N.

Setting up the equilibrium equation:
520x = 1200(D - x)

Expanding the equation:
520x = 1200D - 1200x

Rearranging the equation:
1720x = 1200D

Simplifying the equation:
x = (1200D) / 1720

Therefore, the value of D that corresponds to a tension of 1200 N is given by D = (1720x) / 1200.

To find the value of D that corresponds to the given tension of 1200 N, we can use the principle of moments or torque.

The principle of moments states that for an object in equilibrium, the sum of the clockwise torques about any point is equal to the sum of the counterclockwise torques about the same point.

In this case, we can take the point of consideration as the hinge on the left side of the beam.

Let's proceed step by step:

1. Draw a diagram: Draw the beam, the hinge on the left side, and the cable bolted to the wall on the right side.

2. Identify the forces: The weight of the beam acts downward at its center, and the tension in the cable acts upward at a distance D above the beam.

3. Establish the torque equation: To keep the beam in equilibrium, the sum of the clockwise torques must equal the sum of the counterclockwise torques.

Considering the hinge as the point of reference, the torque created by the weight is given by:
(W × (length of beam)/2) clockwise.

The torque created by the tension in the cable is given by:
(T × D) counterclockwise.

Since the beam is in equilibrium, the sum of these two torques is zero:
(W × (length of beam)/2) = (T × D)

4. Substitute the given values:
Weight of the beam, W = 520 N
Length of the beam = 3.4 m
Tension in the cable = 1200 N

Substituting these values into the torque equation, we get:
(520 × 3.4/2) = (1200 × D)

5. Solve for D: Rearranging the equation, we can solve for D:
D = (520 × 3.4/2) / 1200

D ≈ 1.153 m

Therefore, the value of D that corresponds to the tension of 1200 N is approximately 1.153 m.