In the glass atrium at the entrance to the city aquarium, a designer wants to suspend a 2400-N sculpture of a dolphin. It will be secured by three chains, each of length 4 m. The chains are anchored to the ceiling at three points, spaced
3 m apart, that form an equilateral triangle. a) Determine the magnitude of the tension in each chain.
To determine the magnitude of the tension in each chain, we can use the concept of vector forces and equilibrium. We'll follow these steps:
Step 1: Identify the forces acting on the sculpture
In this case, the only force acting on the sculpture is its weight, which is given as 2400 N.
Step 2: Understand the geometry of the system
The three chains are anchored to the ceiling at three points forming an equilateral triangle. The distance between the anchor points is given as 3 m, and each chain has a length of 4 m.
Step 3: Analyze the equilibrium condition
For the sculpture to be in equilibrium, the net force acting on it must be zero. Therefore, the vertical forces acting on the sculpture must balance out the weight.
Step 4: Resolve the forces
We can resolve the 2400 N weight force into its vertical component. Since the sculpture is suspended, this vertical component must be equal to the sum of the tensions in the three chains.
Each chain is pulling at an angle of 120 degrees (due to the equilateral triangle) relative to the vertical direction. To calculate the vertical component, we can use trigonometry.
Step 5: Use trigonometry to find the vertical component of the tension
We can use the formula: Vertical Component = Tension x cos(angle)
Angle = 120 degrees
Vertical Component = 2400 N
The vertical component of tension in one chain is given by:
Vertical Component = Tension x cos(120 degrees)
Step 6: Calculate the tension in each chain
Since the vertical component of the tension in each chain must balance the weight, we can write an equation:
Vertical Component in Chain 1 + Vertical Component in Chain 2 + Vertical Component in Chain 3 = Weight of sculpture
Tension x cos(120 degrees) + Tension x cos(120 degrees) + Tension x cos(120 degrees) = 2400 N
Simplifying the equation:
3 x Tension x cos(120 degrees) = 2400 N
Step 7: Solve for the tension in each chain
To solve for the tension, we divide both sides of the equation by 3 x cos(120 degrees):
Tension = 2400 N / (3 x cos(120 degrees))
Using a scientific calculator, we can calculate the value of cos(120 degrees) as -0.5:
Tension = 2400 N / (3 x -0.5)
Tension ≈ 1600 N
So, the magnitude of the tension in each chain is approximately 1600 N.