A uniform rigid rod of length L is suspended from one end and allowed to swing freely. If the rod is struck by a single impulsive force, perpendicular to its length, at a specific point, it will begin to swing without exerting any initial horizontal reaction force on its pivot. This specific point is known as the center of percussion. Where is the center of percussion located for this uniform rod, measured from the pivot point? (The moment of inertia of a uniform rod of mass M and length L about an end pivot is (1/3)ML², and its center of mass is at L/2).
Correct: B
This is a challenging conceptual problem that combines translational and rotational dynamics. Let the pivot be at x=0. Let the impulsive force F be applied at a distance x from the pivot.
When the rod is struck, it undergoes both translational acceleration of its center of mass (a_CM) and angular acceleration (α) about its pivot.
1. **Translational Motion:**
Let R_pivot be the horizontal reaction force exerted by the pivot. The net horizontal force on the rod is F + R_pivot (if F and R_pivot are in the same direction, or F - R_pivot if opposite). For the condition of no initial reaction force, R_pivot = 0.
ΣFx = Ma_CM => F + R_pivot = Ma_CM (Equation 1)
2. **Rotational Motion:**
The net torque about the pivot is τ_pivot = Fx (force F applied at distance x from the pivot).
τ_pivot = I_pivot * α
Fx = I_pivot * α
We are given I_pivot = (1/3)ML² for a uniform rod pivoted at one end.
Fx = (1/3)ML² * α (Equation 2)
3. **Kinematic Relationship:**
The center of mass (CM) of a uniform rod is at L/2 from the pivot. The acceleration of the center of mass is related to the angular acceleration by a_CM = α * (L/2).
4. **Substitute a_CM into Equation 1:**
F + R_pivot = M * (α * L/2) (Equation 3)
5. **Express α from Equation 2 and substitute into Equation 3:**
From (2), α = (Fx) / ((1/3)ML²) = (3Fx) / (ML²)
Substitute this α into (3):
F + R_pivot = M * (L/2) * (3Fx / (ML²))
F + R_pivot = (3Fx) / (2L)
6. **Condition for Center of Percussion:**
At the center of percussion, there is no initial reaction force on the pivot, meaning R_pivot = 0.
F = (3Fx) / (2L)
Divide by F (assuming F ≠ 0):
1 = (3x) / (2L)
x = 2L/3
The center of percussion is located at a distance of 2L/3 from the pivot.
The correct choice is B.