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?
To find the value of D that corresponds to the least tension, we need to use the principle of moments. The principle of moments states that the sum of the clockwise moments about any point must be equal to the sum of the anticlockwise moments about the same point, when an object is in equilibrium.
Let's denote the distance between the left end of the beam and the point where the cable is bolted as x.
Step 1: Determine the forces acting on the beam.
- The weight of the beam acts vertically downward with a magnitude of 520 N.
- The tension in the cable acts upward.
- The reaction force at the hinge on the left wall acts vertically upward.
Step 2: Set up the equation for the principle of moments.
Since the beam is in equilibrium, we can take moments about any point. Let's choose the left hinge as the point about which we will take moments. This means the anticlockwise moments will be positive and the clockwise moments will be negative.
Sum of anticlockwise moments = Sum of clockwise moments
Clockwise moments:
- The weight of the beam acts at its mid-point, which is 3.4/2 = 1.7 m away from the left hinge. Therefore, the clockwise moment due to the weight is -520 N * 1.7 m = -884 Nm.
- The tension in the cable acts at a distance x from the left hinge. Therefore, the clockwise moment due to the tension is -1200 N * x Nm.
Anticlockwise moment:
- The reaction force at the hinge on the left wall acts at a distance of 0 m from the left hinge. Therefore, the anticlockwise moment due to the reaction force is +R * 0 Nm = 0 Nm, where R is the magnitude of the reaction force.
So, our equation becomes: -884 - 1200x = 0.
Step 3: Solve for x.
To find the value of x, rearrange the equation: -1200x = 884.
Now divide both sides of the equation by -1200 to isolate x: x = 884 / 1200.
Calculating the value of x: x ≈ 0.737 m.
Therefore, the value of D corresponds to that tension is approximately 0.737 m.