Questions: Curvilinear Motion of Particles

Question 1

A car rounds a circular track at perfectly constant speed. What is true about its acceleration?

• A. Its acceleration is zero because speed is constant
• B. It has only tangential acceleration, directed forward along the path
• C. It has only normal acceleration, directed toward the center of the circle
• D. It has both tangential and normal acceleration because the car is moving

Answer: (C)

Acceleration in curvilinear motion has two components: tangential (aₜ = dv/dt, rate of speed change) and normal (aₙ = v²/ρ, rate of direction change). Constant speed means aₜ = 0, but the velocity direction is continuously changing as the car follows the circular track. That directional change is captured by the normal component, which is nonzero and directed toward the center of curvature. Option A is the classic misconception: zero change in speed ≠ zero acceleration.

Question 2

In normal-tangential (n-t) coordinates, a particle travels along a curved path while its speed is increasing. How is the total acceleration vector oriented?

• A. Purely in the normal direction, pointing toward the center of curvature
• B. Purely in the tangential direction, along the path
• C. With both a tangential component (along the path) and a normal component (toward the center of curvature)
• D. Radially outward from the center of curvature

Answer: (C)

When both speed and direction are changing, both acceleration components are nonzero. Tangential acceleration aₜ = dv/dt handles the speed increase (points forward along the path since the particle is speeding up). Normal acceleration aₙ = v²/ρ handles the direction change (always points inward toward the center of curvature). The total acceleration is the vector sum of these two perpendicular components. Only in special cases (constant speed or straight path) does one component vanish.

Question 3

A particle moving at constant speed along a curved path still has a nonzero acceleration.

• T. True
• F. False

Answer: True

Velocity is a vector quantity with both magnitude (speed) and direction. A particle on a curved path continuously changes its velocity direction even if speed is constant. This change in direction constitutes acceleration — specifically, the normal (centripetal) acceleration aₙ = v²/ρ, directed toward the center of curvature. Acceleration being zero would require both zero speed change and zero direction change, which only occurs for straight-line motion at constant speed.

Question 4

Normal acceleration (aₙ) represents the rate at which a particle's speed is changing along its path.

• T. True
• F. False

Answer: False

Normal acceleration aₙ = v²/ρ represents the rate of direction change — it steers the particle around the curve by pointing toward the center of curvature. It is tangential acceleration aₜ = dv/dt that represents the rate of speed change (speeding up or slowing down). This distinction is fundamental: you can have normal acceleration with zero tangential acceleration (constant speed on a curve), tangential with zero normal (changing speed on a straight line), or both simultaneously.

Question 5

A particle travels along a circular arc at constant speed. Explain why it is still accelerating, and identify the direction of that acceleration.

Model answer: The particle accelerates because its velocity direction continuously changes as it follows the circular arc. Even though the speed |v| is constant, the velocity vector rotates, and any rate of change of velocity — whether in magnitude or direction — is acceleration. The acceleration is purely normal (centripetal): aₙ = v²/ρ, directed toward the center of the circle. There is no tangential component because speed is constant (dv/dt = 0).

This is the conceptual core of curvilinear motion: confusing 'constant speed' with 'no acceleration' is the most common error. In Newtonian mechanics, this normal acceleration requires a net centripetal force (e.g., tension in a string, friction on a road surface, gravity for orbital motion). Without that force, the particle would travel in a straight line. The circular path is maintained by continuous force redirecting the particle inward.