Questions: Newton's Second Law Applied to Particle Dynamics

Question 1

A particle moves in a circle at constant speed. Which statement correctly describes the free-body diagram (FBD)?

• A. The FBD includes a centripetal force mv²/r directed inward as one of the applied forces
• B. The FBD shows only real applied forces; ΣFn = mv²/r is an equation relating those forces to acceleration, not an additional force
• C. The FBD includes an outward inertia force ma to balance the net inward force
• D. No FBD is needed because the particle travels at constant speed

Answer: (B)

The FBD contains only real physical forces (gravity, tension, normal force, etc.). The term mv²/r is the kinetic resultant — it appears on the right side of ΣFn = maₙ. Placing it on the FBD as an applied force would double-count it. The kinetic diagram showing the ma vector is drawn separately.

Question 2

In polar coordinates, if a particle moves at constant radial distance r from the origin (pure rotation), the radial equation simplifies to ΣFr = −mrθ̇².

• T. True
• F. False

Answer: True

The general radial equation is ΣFr = m(r̈ − rθ̇²). When r is constant, ṙ = 0 and r̈ = 0, leaving ΣFr = −mrθ̇². This negative radial term represents the centripetal acceleration directed inward toward the origin, consistent with the particle being held on a circular path.

Question 3

Why is it incorrect to include a 'centrifugal force' pointing outward on the free-body diagram when applying Newton's second law to a particle rounding a curve?

Model answer: Newton's second law ΣF = ma is valid in inertial (non-accelerating) reference frames. In an inertial frame only real contact and body forces act on the particle; centrifugal force is a fictitious force that appears only when the equations are written in a rotating (non-inertial) reference frame.

Adding a fictitious centrifugal force to the FBD while also computing the real net force would incorrectly cancel the centripetal acceleration. In an inertial frame the net inward force — whatever combination of normal, tension, and gravity provides it — equals mv²/r, which is the centripetal requirement derived from the particle's curved path.