Normal Force at Point Contact on Circular Hole

Normal Force and Point Contact

• Normal force is defined as the contact force perpendicular to the surface: $\mathbf{N} \perp \text{surface}$ at the point of contact.
• For a circle (a circular hole), the contact is effectively at a single point because of curvature; despite that, the normal direction points along the line perpendicular to the tangent, i.e., toward the center of the circle.
• In a frictionless contact, the normal force does no work because the displacement is tangential to the surface; energy transfer occurs via tangential components (gravity along the surface).

Contact Geometry: Circle and Hole

• The contact occurs with one point; The normal line passes through the centers (for smooth circles) when the ball touches the inside of the circular boundary.
• The ball is constrained to move along the circular path inside the hole; the 'hole' imposes a curved constraint.

Forces on the Ball and Decomposition

• Gravity: $\mathbf{F}<em>g = m \mathbf{g}, \quad |\mathbf{F}</em>g| = m g$ downward.

• Normal force: $\mathbf{N}$, directed perpendicular to the surface (toward the interior of the hole along the radius).

• If frictionless, there is no tangential contact force.

• Decomposition in terms of angle (\theta) (measured from the bottom of the circle along the circumference):

  • Let (\theta) be the angle from the bottom ((\theta = 0) at the bottom, increasing as you move along the circle).
  • The normal direction is radial; gravity makes angle (\theta) with the normal.
  • Components:
  • Normal component: $F_{g\perp} = m g \cos \theta$
  • Tangential component: $F_{g\parallel} = - m g \sin \theta$
  • Note: The sign indicates the tangential component tends to decrease or increase (\theta) depending on convention.

• Kinematics:

  • Tangential velocity: $v = R \dot{\theta}$
  • Tangential acceleration: $a_t = R \ddot{\theta}$

• Equations of motion for a frictionless bead on a vertical circular hoop:

  • Tangential (along the circle): $m R \ddot{\theta} = - m g \sin \theta \Rightarrow \ddot{\theta} = - \dfrac{g}{R} \sin \theta$
  • Radial (toward center): $N - m g \cos \theta = m \dfrac{v^2}{R} \Rightarrow N = m \left( \dfrac{v^2}{R} + g \cos \theta \right)$

• Energy considerations (frictionless case):

  • Height relative to bottom: (h = R (1 - \cos \theta))
  • Potential energy: $U(\theta) = m g h = m g R (1 - \cos \theta)$
  • Kinetic energy (translational): $K = \tfrac{1}{2} m v^2 = \tfrac{1}{2} m (R \dot{\theta})^2$
  • Energy conservation: (E = K + U = \text{const} )

• Small-amplitude motion:

  • For small (\theta), (\sin \theta \approx \theta) and (\cos \theta \approx 1 - \theta^2/2)
  • Bead executes simple harmonic motion about (\theta = 0) with angular frequency (\omega = \sqrt{g/R})
  • Thus the bottom is a stable equilibrium point.

Bottom vs Angle: Equilibrium and Motion

• At (\theta = 0) (bottom), tangential component of gravity is zero; if initially at rest, the ball remains at rest.
• If displaced slightly ((\theta \approx) small), gravity provides a restoring tangential force, leading to oscillations around the bottom (SHM for small angles).
• In a frictionless scenario, the energy transfers between potential and kinetic as the ball slides down and up along the circle.
• The normal force at the bottom provides the centripetal requirement when there is motion; its magnitude increases with speed:
  • $N = m \left( \dfrac{v^2}{R} + g \cos \theta \right)$; at (\theta = 0), (N = m (\dfrac{v^2}{R} + g))

Real-World Considerations and Connections

• If there is friction:
  • A tangential friction force (f \leq \mu N) acts opposite the motion along the surface, converting some energy into heat.
  • Eventually the ball will settle at the bottom due to energy dissipation.
• If the ball is rolling rather than sliding:
  • Rotational kinetic energy must be included: $K = \tfrac{1}{2} m v^2 + \tfrac{1}{2} I \omega^2$ with (I) the moment of inertia and (\omega = v/r) for rolling without slipping.
• Connections to foundational principles:
  • Newton's laws in constrained motion: forces decompose into normal and tangential components relative to a constraint.
  • Energy conservation and energy methods for constrained systems.
  • Potential energy landscapes: the bottom of a circular well is a stable equilibrium due to decreasing potential energy with height.
  • Real surfaces and engineering relevance: contact mechanics, friction, wear, and how curvature affects contact forces.
• Practical takeaways:
  • The normal force points against the hole, perpendicular to the surface, and for a circular boundary points along the radius toward the center.
  • A ball inside a circular hole on Earth tends to move toward the bottom under gravity when friction is negligible.
  • The contact is a point contact for a circle; nonetheless, the normal force acts as if it were acting along the line of centers for two smooth curved surfaces.