Questions: Particle Dynamics and Accelerated Motion

Question 1

A car rounds a horizontal circular curve at constant speed. A student concludes that the net force on the car must be zero because the speed is not changing. What is wrong with this reasoning?

• A. Nothing — constant speed means zero acceleration, which by ΣF = ma means zero net force
• B. Speed is constant, but velocity is not — direction is changing, so there is centripetal acceleration directed toward the center of the curve, requiring a net inward force
• C. The engine force is nonzero, so ΣF cannot be zero even at constant speed
• D. Friction is the only force acting, and friction is not counted in the net force

Answer: (B)

Acceleration is the rate of change of the velocity vector — not just its magnitude. When a car rounds a curve at constant speed, the velocity direction changes continuously, producing centripetal acceleration a_c = v²/ρ directed toward the center of curvature. By ΣF = ma, a nonzero acceleration requires a nonzero net force. The net force is directed inward (centripetal), provided by the road's friction on the tires. Zero net force would mean zero acceleration — straight-line, constant-speed motion.

Question 2

A 5 kg block is placed on a frictionless incline angled at 30°. It is released from rest. Choosing axes parallel and perpendicular to the incline, which equation correctly gives the acceleration along the incline?

• A. ΣF = 0 along the incline, because the normal force balances the component of gravity
• B. mg sin30° = ma, giving a = g sin30° directed down the incline
• C. mg cos30° = ma, giving acceleration perpendicular to the incline
• D. mg = ma, because gravity is the only real force acting on the block

Answer: (B)

Along the incline, the only force component is gravity's parallel component: mg sin30°. The normal force is perpendicular to the incline and contributes zero to the along-incline equation. Applying ΣF_t = ma_t: mg sin30° = ma, so a = g sin30° ≈ 4.9 m/s². Option A mistakes the perpendicular equilibrium (N − mg cos30° = 0) for the full picture. Options C and D apply the wrong force component.

Question 3

If a particle moves in a straight line at constant speed, Newton's second law tells us that no forces are acting on it.

• T. True
• F. False

Answer: False

ΣF = ma requires the NET force to be zero for zero acceleration — not that no forces exist. A book sliding at constant speed across a frictionless table has gravity and normal force both acting, but they cancel. A satellite in a straight-line escape trajectory has a gravitational force acting the whole time. Zero net force (ΣF = 0) is the correct condition for zero acceleration; the individual forces acting on the object may be large and non-zero.

Question 4

D'Alembert's principle is algebraically equivalent to Newton's second law — it reformulates ΣF = ma as ΣF − ma = 0 so that dynamics problems can be treated using static equilibrium techniques.

• T. True
• F. False

Answer: True

D'Alembert adds a fictitious 'inertial force' of −ma (equal in magnitude to ma, but opposite in direction to acceleration) to convert the dynamic equation into a formal equilibrium: ΣF + (−ma) = 0. This is algebraically identical to ΣF = ma — no new physics is introduced. The practical benefit is that engineers already know how to solve equilibrium problems (balancing forces and moments), so d'Alembert converts unfamiliar dynamics into familiar statics. Critics note the inertial force is not physical, but the algebra is identical.

Question 5

Explain why the direction of a particle's acceleration is not necessarily the same as the direction of its velocity, and give a concrete example where they differ.

Model answer: Acceleration is the rate of change of the velocity vector. Velocity can change in direction without changing in magnitude — when this happens, the acceleration is perpendicular to the velocity. A particle moving in a circle at constant speed has velocity tangent to the circle but acceleration pointing radially inward (centripetal). Another example: a ball thrown horizontally has horizontal velocity at release, but acceleration is purely vertical (downward gravity), perpendicular to the initial velocity direction.

The confusion between velocity direction and acceleration direction is the most common error in dynamics problems. It matters because Newton's second law says ΣF = ma — forces sum to the mass times the acceleration vector, not the velocity vector. Setting up the right coordinate system aligned with the acceleration (not the velocity) is the first step in any curvilinear dynamics problem. For circular motion in normal-tangential coordinates: the normal axis is always aligned with the centripetal acceleration, regardless of the velocity direction.