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Correct Answer: 60 (n)
the question appears to involve a scenario in which a massless rod is on a balance, with forces acting on it. to determine the force required at one end of the rod to maintain equilibrium, one must understand the concept of torque and how it applies to a balance.
torque is a measure of the force causing the rotation of an object about an axis. every force applied has a distance from the pivot point, known as the lever arm. the product of the force and the lever arm gives the magnitude of the torque. in the context of a balance or see-saw, for equilibrium to be maintained, the sum of the clockwise torques must equal the sum of the counterclockwise torques.
in the question, it is mentioned that the massless rod has a length \( l \) and sits on a balance. assuming the balance pivot is at the midpoint of the rod, and given forces \( f \) and \( mg \) are applied at different points on the rod, let’s analyze the situation:
1. \( mg \) is a force acting at a distance of \( \frac{3l}{4} \) from the pivot point. this force could either be due to a mass \( m \) hung at this point or some other downward force. the direction of this force contributes to clockwise torque because it tends to rotate the rod clockwise about the pivot.
2. \( f \) is the unknown force we need to find, and it acts at a distance of \( \frac{l}{4} \) from the pivot. the placement of \( f \) is such that it would generate counterclockwise torque, opposing the torque generated by \( mg \).
according to the principle of moments (also known as the principle of lever or torque balance), for the rod to be in rotational equilibrium (i.e., not rotating), the total clockwise torque must equal the total counterclockwise torque. mathematically, this is represented as:
\[ mg \cdot \left( \frac{3l}{4} \right) = f \cdot \left( \frac{l}{4} \right) \]
to find \( f \), we rearrange the equation:
\[ f = \frac{mg \cdot \left( \frac{3l}{4} \right)}{\left( \frac{l}{4} \right)} \]
\[ f = 3mg \]
it appears there was a specific value of \( mg \) given as 80 n in the problem setup (though not explicitly mentioned in the explanation). substituting \( 80 \, \text{n} \) for \( mg \) gives:
\[ f = 3 \times 80 \, \text{n} \]
\[ f = 240 \, \text{n} \]
however, the provided answers and explanations suggest that \( f = 60 \, \text{n} \). this discrepancy could be due to either a misunderstanding of the question or an error in the provided details. to resolve this confusion, one would need to verify the exact positions and magnitudes of the forces acting on the rod.
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