a machine weighing 1500N is supported bt two chains attached to some point on the machine.one of these ropes goes to the eye bolt in the well & is inclined 30 degrees to the horizontal & the other goes to the hook in ceiling & is inclined at 45 degrees to the horizontal. find the tension in the two chains
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To find the tension in the two chains, we need to resolve the weight of the machine into its components along the direction of each chain.
Let's start by breaking down the weight of the machine along the chain attached to the eye bolt in the well. This chain is inclined at 30 degrees to the horizontal.
Step 1: Resolve the weight along the direction of the chain attached to the eye bolt:
The weight component along the direction of the chain = Weight of the machine * cos(angle of inclination)
Weight component along the direction of the chain = 1500N * cos(30°)
Step 2: Break down the weight along the direction of the chain attached to the hook in the ceiling. This chain is inclined at 45 degrees to the horizontal.
Step 2: Resolve the weight along the direction of the chain attached to the hook:
The weight component along the direction of the chain = Weight of the machine * cos(angle of inclination)
Weight component along the direction of the chain = 1500N * cos(45°)
Once you calculate these two values, you will have the tension in each chain.
To find the tension in the two chains, we can break down the forces acting on the machine into horizontal and vertical components. Then we can use trigonometry to find the tensions in each chain.
Let's label the tension in the chain attached to the eye bolt as T1 and the tension in the chain attached to the hook in the ceiling as T2.
1. Chain attached to the eye bolt:
Since the chain is inclined at 30 degrees to the horizontal, we need to find the horizontal and vertical components of the tension T1.
The horizontal component (T1x) can be found using the formula:
T1x = T1 * cos(30°)
The vertical component (T1y) can be found using the formula:
T1y = T1 * sin(30°)
2. Chain attached to the hook in the ceiling:
Since the chain is inclined at 45 degrees to the horizontal, we need to find the horizontal and vertical components of the tension T2.
The horizontal component (T2x) can be found using the formula:
T2x = T2 * cos(45°)
The vertical component (T2y) can be found using the formula:
T2y = T2 * sin(45°)
Now let's set up the equations using Newton's second law:
In the horizontal direction,
T1x + T2x = 0 (since there is no horizontal acceleration)
In the vertical direction,
T1y + T2y - mg = 0 (where 'm' is the mass of the machine and 'g' is the acceleration due to gravity)
Given that the weight of the machine is 1500N, we can substitute 'mg' with 1500N.
Now we can solve these equations to find T1 and T2.