Circular Motion

Necessary conditions for circular motion:

Force requirement: • Net force toward center: $F = \frac{mv^2}{r}$ • Must be perpendicular to velocity vector • Must be continuously maintained

Energy considerations: • For uniform circular motion: kinetic energy remains constant • For non-uniform motion: work done by tangential force changes kinetic energy

Constraints: • Physical constraint (like string, track, groove) OR • Field force (gravity, electromagnetic) providing centripetal force • Constraint must be able to withstand required tension/compression

Additional requirements: • Initial velocity perpendicular to radius vector • Sufficient coefficient of friction (for motion on surfaces) • For vertical circles: minimum speed at top $v_{min} = \sqrt{gr}$

Without these conditions, the particle would follow a straight-line path (Newton's 1st law).

The Coriolis force is needed when: • Working in a rotating reference frame • AND the particle is moving relative to that rotating frame • (Centrifugal force alone is sufficient only for stationary objects in rotating frames)

Calculation: $\vec{F}_{Coriolis} = -2m(\vec{\omega} \times \vec{v}_{rel})$

Where: • $m$ is particle mass • $\vec{\omega}$ is angular velocity vector of rotating frame • $\vec{v}_{rel}$ is velocity of particle relative to rotating frame

Properties: • Always perpendicular to both rotation axis and relative velocity • Magnitude: $F_{Coriolis} = 2m\omega v_{rel}\sin\phi$ (where $\phi$ is angle between vectors) • Causes apparent deflection of moving objects in rotating frames • Examples: hurricane circulation, apparent paths of projectiles on Earth

The combination of centrifugal and Coriolis forces allows application of Newton's laws in non-inertial rotating reference frames.

For a conical pendulum (mass on string making angle $\theta$ with vertical while rotating):

Force analysis: • Tension: $T$ along string • Gravity: $mg$ downward • Circular motion in horizontal plane of radius $r = L\sin\theta$

Vertical equilibrium: $T\cos\theta = mg$

Horizontal (centripetal force): $T\sin\theta = \frac{mv^2}{r} = m\omega^2r = m\omega^2L\sin\theta$

From these: $\tan\theta = \frac{\omega^2L\sin\theta}{g}$ $\omega^2 = \frac{g}{L\cos\theta}$

Period: $T = 2\pi\sqrt{\frac{L\cos\theta}{g}}$

Important relationships: • Larger angle $\theta$ → shorter period • Period depends on $\sqrt{L}$ (similar to simple pendulum) • Horizontal circle radius: $r = L\sin\theta$ • Vertical height: $h = L\cos\theta$

• Radial unit vector (êr): Points outward along the radius from the center O to the particle P
• Tangential unit vector (êt): Points perpendicular to êr in the direction of increasing θ (tangent to the circle)

These unit vectors form a local coordinate system that moves with the particle, providing a natural basis for expressing forces and motion in circular paths.

Note: Unlike fixed Cartesian unit vectors (î, ĵ), these unit vectors continuously change direction as the particle moves around the circle.

• Radial unit vector: êr = î cos θ + ĵ sin θ
• Tangential unit vector: êt = -î sin θ + ĵ cos θ

These expressions show how the circular motion unit vectors can be decomposed into components along the fixed coordinate axes, essential for analyzing motion and forces in rotating systems.

Example: At θ = 45°, êr = (î + ĵ)/√2 and êt = (-î + ĵ)/√2

The relative dominance of acceleration components in nonuniform circular motion:

Radial component (v²/r) dominates when:

• The circular path has a small radius
• The speed is very high
• The rate of speed change is small
• Example: A car taking a sharp turn at high speed (primary concern is preventing skidding outward)

Tangential component (dv/dt) dominates when:

• There is rapid acceleration or deceleration
• The radius is large
• The speed is relatively low
• Example: A train starting from rest on a curved track (primarily feels forward thrust)

Effects on motion:

• Radial-dominated: Object maintains its circular path but experiences strong inward forces
• Tangential-dominated: Object changes speed significantly while following the path
• Balanced components: Object follows a spiral-like path if unconstrained

Practical application: Vehicle designers must account for both components when designing suspension systems for vehicles that accelerate through turns.

• The ideal speed formula is: v = √(rg·tanθ) Where:

  • v = velocity
  • r = radius of the curve
  • g = acceleration due to gravity
  • θ = banking angle

• When a vehicle travels at exactly this speed:

  • No friction is needed to maintain the circular path
  • The normal force alone provides the necessary centripetal force
  • The vehicle neither slides up nor down the banking
  • The horizontal component of the normal force exactly equals mv²/r
  • The vertical component of the normal force exactly equals the weight mg

• This represents the perfect equilibrium condition for the banked turn

• True vertical points toward Earth's center (direction of gravitational force mg)
• Apparent vertical deviates by angle α, where:
  • tanα = (ω²R·sinθ·cosθ)/(g - ω²R·sin²θ)
  • α is maximum at mid-latitudes
  • α = 0 at poles and equator
• Practical implications:
  • Plumb lines align with apparent vertical, not true vertical
  • Buildings are constructed parallel to apparent vertical
  • Water surfaces at rest are perpendicular to apparent vertical
  • This deviation must be accounted for in precise surveying and astronomical measurements
  • Navigation systems must correct for this difference in high-precision applications

• Maximum weight reduction occurs at equator (θ = 90°): mg' = mg - mω²R
• At latitude λ (colatitude θ = 90° - λ), the reduction factor depends on sin²θ
• For half the maximum reduction: sin²θ = 0.5
• Therefore: sin²(90° - λ) = 0.5
• Solving: λ = 45° (occurs at 45° latitude north and south)
• Significance:
  • This demonstrates that apparent weight reduction is not linearly related to latitude
  • The effect follows a sin² curve, decreasing more rapidly near equator
  • At 45° latitude, objects weigh halfway between their equatorial and polar weights
  • This has implications for precise scientific measurements that depend on weight or the local value of g
  • Geological surveys and gravitational models must account for this variation

Colatitude is the angle (θ) between Earth's rotation axis and the radius vector from Earth's center to a specific point on its surface.

Key properties:

• Complement to the geographical latitude (90° - latitude)
• Determines the radius of the circular path a point follows during Earth's rotation
• The radius of this circular path = R·sin(θ), where R is Earth's radius

For example, at the equator, colatitude is 90° and points move in a circle of radius equal to Earth's radius. At the poles, colatitude is 0°, and points remain stationary with respect to Earth's rotation.