A suspension bridge supported by a cable 100ft long has a sag of 10ft at its center. If the bridge has a weight of 20,000 lbs and its center of mass is the center of the bridge, what is the tension on the cable?
To find the tension on the cable of the suspension bridge, we can use the concept of equilibrium. In order for the bridge to remain in equilibrium, the net force acting on it must be zero.
The weight of the bridge can be considered as acting downwards from its center of mass, through the midpoint of the cable. The tension in the cable acts both horizontally and vertically to support the weight of the bridge.
We can break down the weight of the bridge into horizontal and vertical components. The vertical component is equal to the weight itself, which is 20,000 lbs. The horizontal component is zero, as the weight acts vertically downwards.
Now, let's analyze the forces acting on the bridge at its highest point (where the sag is 0). At the highest point, the tension in the cable is purely vertical and counteracts the weight of the bridge. Therefore, the tension and the weight are equal in magnitude but opposite in direction.
The weight of the bridge is 20,000 lbs downwards, so the tension in the cable at the highest point is also 20,000 lbs upwards.
Next, let's consider the forces acting on the bridge at the center (where the sag is at its maximum). The tension in the cable acts both horizontally and vertically.
The vertical component of the tension is responsible for counteracting the weight of the bridge. As the sag is 10 ft at the center, the vertical component of the tension is reduced by 10,000 lbs (10 ft * 1,000 lbs/ft). Therefore, the vertical component of the tension is 20,000 lbs - 10,000 lbs = 10,000 lbs upwards.
The horizontal component of the tension is responsible for balancing out the horizontal forces. Since the bridge is symmetric, the horizontal component of the tension on one side is equal in magnitude but opposite in direction to the horizontal component of the tension on the other side. Therefore, the horizontal component of the tension is zero.
As a result, the tension in the cable at the center of the bridge is purely vertical and is 10,000 lbs upwards.
To summarize, the tension in the cable of the suspension bridge is 20,000 lbs upwards at the highest point and 10,000 lbs upwards at the center (where the sag is 10 ft).