Questions: Constrained Particle Motion and Constraint Forces

Question 1

A bead slides along a smooth circular wire loop in a vertical plane. You apply Newton's second law in normal-tangential (n-t) coordinates. In which equation does the wire's normal force N appear?

• A. In both the normal and tangential equations, because it keeps the bead on the loop and accelerates it
• B. Only in the normal equation (ΣFₙ = mv²/r), not in the tangential equation
• C. Only in the tangential equation, because the normal force changes the bead's speed
• D. It does not appear at all in either equation for a smooth (frictionless) constraint

Answer: (B)

The constraint force (normal force from the wire) acts perpendicular to the bead's motion — in the normal direction. In n-t coordinates, the normal direction carries the centripetal acceleration mv²/r, and N appears there as the primary unknown. In the tangential direction, only forces with a tangential component drive tangential acceleration (dv/dt). Since N is purely normal, it has zero tangential projection and vanishes from the tangential equation entirely. This separation is the computational payoff of aligning coordinates with the constraint geometry.

Question 2

A particle is sliding along a curved ramp and at some point the normal force N from the ramp becomes zero. What happens next?

• A. The particle continues along the ramp surface, now supported entirely by friction
• B. The particle instantaneously stops and reverses direction
• C. The particle leaves the ramp surface and follows a free trajectory (e.g., projectile motion)
• D. The particle accelerates along the ramp because the constraint force no longer opposes motion

Answer: (C)

N = 0 is the leaving condition. A surface can only push (not pull) — the normal force keeps the particle on the surface by pushing inward toward the center of curvature. When the geometry and speed require a 'pulling' normal force, the surface cannot provide it, and the particle separates. Setting N = 0 in the normal equation and solving for speed or angle gives the critical departure condition. After leaving, the particle follows a free trajectory governed only by gravity and other applied forces.

Question 3

Constraint forces such as normal forces and string tensions do no work on the particles they constrain.

• T. True
• F. False

Answer: True

Work equals force times displacement in the direction of the force. Constraint forces are always perpendicular to the direction of motion (that is precisely what it means to 'constrain' a path — the force enforces the geometry without driving motion along it). Since the angle between force and displacement is 90°, the dot product F·ds = 0, and no work is done. This is why work-energy methods can ignore constraint forces entirely — they cancel out of the energy accounting.

Question 4

A particle constrained to move along a surface will generally remain on the surface as long as the constraint force is acting.

• T. True
• F. False

Answer: False

Constraint forces maintain contact only when they are compressive (surface pushing) or tensile in the correct direction (string pulling). When the required constraint force reverses sign — a surface would need to pull the particle, or the speed exceeds what centripetal balance allows — the physical constraint fails. The particle leaves the surface at the instant N = 0. This is why checking the sign of N throughout the motion is essential: if it ever goes negative, the particle has already left the constrained path.

Question 5

Why can work-energy methods bypass the need to solve for constraint forces, even though those forces appear as unknowns in Newton's second law equations?

Model answer: Constraint forces are always perpendicular to the particle's displacement (velocity direction). Work equals F·Δs, and a force perpendicular to displacement does zero work. Since constraint forces do no work, they contribute nothing to the work-energy equation W_net = ΔKE. You can therefore compute kinetic energy changes from applied forces alone, finding speeds directly without ever calculating the constraint force.

This is why work-energy methods are so powerful for curved-path problems — you often want the speed at a particular location and don't need the normal force. Newton's second law gives you the normal force (useful if you need to know whether the particle stays on the surface or want the reaction magnitude); work-energy gives you the speed directly. The two methods are complementary: use Newton's law when you need forces, use work-energy when you need kinematics.