Questions: Angular Impulse and Momentum for Rigid Bodies

Question 1

During a brief impact between a bat and a ball, which reference point should you choose to eliminate the unknown impulsive contact force from your angular impulse-momentum equation?

• A. The mass center G of the bat, because I_G is always known
• B. The contact point itself, because the impulsive force acts there and creates a zero moment arm
• C. A fixed point far from the impact, to minimize moment arms of all forces
• D. The center of mass of the ball, to treat both objects symmetrically

Answer: (B)

A moment equals force times perpendicular distance. If you sum moments about the contact point, the impulsive contact force — acting at exactly that point — has zero moment arm and thus contributes zero angular impulse. This eliminates the unknown from the equation, leaving something you can solve. This is the same strategy used in statics: choose a moment center that passes through unknown forces to remove them. Summing about G (option A) does not eliminate the contact force, because the contact point is generally not at G.

Question 2

A rigid body slides across a surface while also rotating (general planar motion). You want to compute its angular momentum about a fixed floor point O. Which formula applies?

• A. H_O = I_O × ω, using mass moment of inertia about O
• B. H_O = I_G × ω, using mass moment of inertia about the mass center G
• C. H_O = I_G × ω + m × v_G × d, where d is the perpendicular distance from O to v_G
• D. H_O = m × v_G × d only, because the spinning contribution cancels for a sliding body

Answer: (C)

For a body in general planar motion (not rotating about a fixed point), angular momentum about any point O has two contributions: (1) the spin about the mass center (I_G × ω) and (2) the orbital contribution of the mass center moving around O (m × v_G × d). Omitting the cross-term (option B) is the most common error in rigid-body impact problems. H_O = I_O × ω is valid only when O is a fixed point the body rotates about — it cannot be applied to a body that simultaneously slides and spins.

Question 3

During a very brief impact, gravitational impulse is negligible compared to the impulsive contact forces and can be ignored in the impulse-momentum equations.

• T. True
• F. False

Answer: True

Impulse = force × time. During an impact, the contact force is enormous (often thousands of newtons over microseconds) while gravity is modest (weight × small Δt). Because the impact time interval is so short, gravity's impulse contribution is negligible compared to the impulsive contact force. This approximation is standard in impact mechanics and is what allows 'finite' forces like gravity, springs, and friction to be ignored during the impact duration while treating contact forces as dominant.

Question 4

The formula H_O = I_O × ω can be used to compute angular momentum about any chosen reference point O for a rigid body in planar motion.

• T. True
• F. False

Answer: False

H_O = I_O × ω is only valid when O is a fixed point about which the body rotates. For a body in general planar motion — simultaneous translation and rotation — the correct expression is H_O = I_G × ω + m × v_G × d. Using H_O = I_O × ω for a body that is also translating would be incorrect because I_O implicitly assumes all motion is rotational about O. This is one of the most common errors in rigid-body impact analysis.

Question 5

Explain why the location of the impact point on a bat (near the barrel vs. near the handle) affects post-impact rotation, even if the same magnitude of force is applied.

Model answer: Angular impulse equals the contact force times its perpendicular distance (moment arm) from the bat's mass center. Even for the same contact force, a point near the barrel (far from G) creates a large moment arm, delivering more angular impulse and producing greater post-impact angular velocity. A contact point near G creates little moment arm and barely spins the bat. This is why the 'sweet spot' exists: at a specific location, the angular and linear impulses combine so that the reaction force at the grip point is zero, eliminating the sting in the batter's hands.

This connects the abstract formula (angular impulse = ΣM × Δt) to the physical intuition of torque. Moment arm is the lever in angular impulse-momentum, just as in statics. Understanding this makes clear why impact problems require both linear and angular impulse-momentum equations simultaneously — they are coupled through the geometry of where the force is applied relative to G.