Askiitians Tutor Team
Last Activity: 4 Months ago
To solve this problem, we need to analyze the situation involving the uniform rod, the pivot point, and the impulse delivered. The key here is to understand how the impulse affects the rod and how the pivot reacts to that impulse.
Understanding the Setup
We have a uniform rod of length \( L = 1 \) that is pivoted at point A. When a horizontal force delivers an impulse \( J \) at a distance \( x \) from point A, it creates a torque about the pivot point. The impulse delivered by the pivot will be zero when the net torque about the pivot is balanced, meaning that the forces acting on the rod do not create any rotational effect at that point.
Torque and Impulse
Torque (\( \tau \)) is calculated as the product of the force and the distance from the pivot point where the force is applied. In this case, the torque due to the impulse \( J \) can be expressed as:
- Torque due to impulse \( J \) at distance \( x \): \( \tau = J \cdot x \)
Since the pivot is at point A, we need to consider the direction of the torque. If the impulse is applied horizontally, it will create a rotational effect about the pivot point A.
Conditions for Zero Impulse at the Pivot
The impulse delivered by the pivot will be zero when the torque created by the impulse \( J \) at distance \( x \) is countered by the torque due to the weight of the rod acting at its center of mass. The center of mass of a uniform rod is located at its midpoint, which is at \( L/2 \) from point A.
Finding the Critical Distance
For the impulse at distance \( x \) to not create any net torque about the pivot, the distance \( x \) must be such that:
- The torque due to the impulse \( J \) at distance \( x \) must equal the torque due to the weight of the rod acting at \( L/2 \).
Mathematically, we can express this condition as:
- Torque due to impulse: \( J \cdot x \)
- Torque due to weight: \( mg \cdot (L/2) \) (where \( mg \) is the weight of the rod)
Setting these equal gives us:
\( J \cdot x = mg \cdot (L/2) \)
Evaluating the Options
To find the distance \( x \) where the impulse delivered by the pivot is zero, we can analyze the given options:
- (A) \( L/2 \)
- (B) \( L/3 \)
- (C) \( 2L/3 \)
- (D) \( 3L/4 \)
Since the center of mass is at \( L/2 \), the torque due to the impulse will be balanced when \( x = L/2 \). Thus, the impulse delivered by the pivot will be zero at this point.
Final Answer
The correct answer is (A) \( L/2 \). This is the distance from point A where the impulse delivered by the pivot becomes zero, as it balances the torque created by the impulse applied to the rod.
