A uniform rod of length 1 is pivoted at point a. it is struck by a horizontal forcewhich delivries an impulse J at a distance x from point A. impulse delivered by pivot is zero if x is equal to

(A) L/2 (B)L/3 (C)2L/3 (D)3L/4

To tackle this problem, we need to analyze the dynamics of the uniform rod when it is subjected to an impulse at a certain distance from the pivot point. The key here is understanding how the impulse affects the rod's motion and the role of the pivot in this scenario.

Understanding Impulse and Torque

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. In this case, the impulse \( J \) is applied horizontally at a distance \( x \) from the pivot point \( A \). The pivot point itself does not move, but it can exert a reaction force to maintain equilibrium.

Torque Calculation

When the impulse is applied at a distance \( x \), it generates a torque about the pivot point. Torque (\( \tau \)) is calculated using the formula:

• \( \tau = r \times F \)

Where \( r \) is the distance from the pivot to the point of force application (in this case, \( x \)), and \( F \) is the force applied. Since we are dealing with an impulse, we can relate it to the change in momentum:

• \( J = F \Delta t \)

However, for our purposes, we can consider the effect of the impulse directly on the rod's motion.

Condition for Zero Impulse at the Pivot

The impulse delivered by the pivot is zero when the net torque about the pivot point is zero. This occurs when the applied impulse does not create any rotational effect. For a uniform rod, this happens when the distance \( x \) from the pivot is such that the torque produced by the impulse is balanced out.

Finding the Critical Point

For a uniform rod of length \( L \), the center of mass is located at \( L/2 \). If the impulse is applied at the center of mass, the torque generated will be zero because the force acts directly through the center of mass, resulting in no rotational effect. However, we need to find the specific distance \( x \) from point \( A \) where the impulse at that point results in zero torque about the pivot.

Evaluating the Options

Let's evaluate the given options:

• (A) \( L/2 \) - This is the center of mass, and here the torque is zero.
• (B) \( L/3 \) - This point would create a torque.
• (C) \( 2L/3 \) - This point would also create a torque.
• (D) \( 3L/4 \) - This point would create a torque as well.

Conclusion

From our analysis, the only point where the impulse delivered by the pivot is zero is at the center of mass, which corresponds to option (A) \( L/2 \). Thus, the correct answer is:

(A) L/2