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Answer :
Final answer:
The center of gravity of the irregularly shaped object placed between scales on either end is 7.4 meters from the left. This is determined by setting up a torque balance equation and solving for the distance.
Explanation:
In order to identify the center of gravity, we need to consider the whole system as a balance system with torques caused by gravity on either side. Here, the center of gravity can be determined by setting up a torque balance equation. The forces on the left and right scales are equivalent to the torques around the center of gravity because the force times distance to the center of gravity is equal on both sides. We can express this as:
71N * x = 96N * (10 - x)
Solving for x, we find x equals 7.4 m when you solve the above equation. Therefore, the center of gravity is 7.4 meters from the left.
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The center of gravity is found using the principle of moments, with calculations showing it to be roughly [tex]5.7 m[/tex] from the left end. Thus, the correct answer is option 1.
To find the center of gravity of an irregularly shaped object, we can use the principle of moments, which states that the object is in equilibrium when the clockwise moments equal the counterclockwise moments.
Given:
Length of the object, [tex]L = 10 m[/tex]
Force on the right scale, [tex]F_r = 96 N[/tex]
Force on the left scale, [tex]F_l = 71 N[/tex]
Let x be the distance from the left end to the center of gravity.
We set up the equation for moments around one end (let's choose the left end):
[tex]96 \, \text{N} \times (10 \, \text{m} - x) = 71 \, \text{N} \times x[/tex]
Solving for x:
[tex]960 \, \text{N} \cdot \text{m} - 96 \, \text{N} \cdot x = 71 \, \text{N} \cdot x\\960 \, \text{N} \cdot \text{m} = 167 \, \text{N} \cdot x\\x = \frac{960 \, \text{N} \cdot \text{m}}{167 \, \text{N}}\\x \approx 5.75 \, \text{m}[/tex]
Therefore, the center of gravity is approximately [tex]5.75m[/tex] from the left end, matching option 1.
