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Difficulty: HardNewton's Laws of Motion and Linear Momentum

For an object of constant mass moving at constant speed along a complete circular path, the net vector impulse imparted to the object over one full revolution is zero, even though a continuous centripetal force acts on the object throughout the motion.

Answer: Answer

Answer

The statement is true because the impulse-momentum theorem states that net impulse equals the change in momentum (J⃗=Δp⃗\vec{J} = \Delta \vec{p}). After one full revolution, the object's initial and final velocity vectors are identical, resulting in zero change in momentum.
The statement is correct because linear momentum is a vector quantity. Over a complete circular revolution, the initial and final velocity vectors are identical in both magnitude and direction, making the change in momentum—and therefore the net vector impulse—equal to zero.

Step-by-Step Solution

1
Recall the vector definition of impulse and the impulse-momentum theorem.
The net impulse J⃗\vec{J} acting on a body equals its change in linear momentum: J⃗=Δp⃗=p⃗f−p⃗i=mv⃗f−mv⃗i\vec{J} = \Delta \vec{p} = \vec{p}_f - \vec{p}_i = m\vec{v}_f - m\vec{v}_i.
Impulse is a vector quantity dependent on the initial and final states of momentum over the time interval.
2
Evaluate the velocity vector of the object after one complete circular revolution at constant speed.
Since the speed is constant and the trajectory completes a closed loop, the final velocity vector v⃗f\vec{v}_f has the exact same magnitude and direction as the initial velocity vector v⃗i\vec{v}_i.
A complete revolution returns the object to its starting point with its velocity pointing in the initial direction.
3
Calculate the change in momentum Δp⃗\Delta \vec{p}.
Δp⃗=m(v⃗f−v⃗i)=m(0)=0⃗\Delta \vec{p} = m(\vec{v}_f - \vec{v}_i) = m(0) = \vec{0}. Thus, net impulse J⃗=0⃗\vec{J} = \vec{0}.
Subtracting identical vectors yields zero vector magnitude.

Key Concept

Impulse-Momentum Theorem and Vector Nature of Linear Momentum
Estimated Time:1m 30s
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Topics

Newton's Laws of Motion and Linear Momentum

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