Circular Motion

Although the speed is constant in uniform circular motion, the velocity vector is continuously changing direction, which requires acceleration.

Key points: • Velocity is a vector quantity (magnitude and direction) • In circular motion, the direction changes continuously • Acceleration is defined as the rate of change of velocity • This acceleration is directed toward the center (centripetal) • Magnitude: $a_r = \frac{v^2}{r} = \omega^2r$

This is why objects in circular motion require a force directed toward the center to maintain their path.

The complete acceleration vector has two perpendicular components:

1. Radial (centripetal) component: $a_r = -\omega^2r = -\frac{v^2}{r}$ (directed toward center)

2. Tangential component: $a_t = \frac{dv}{dt} = r\frac{d\omega}{dt} = r\alpha$ (tangent to circle)

Total acceleration magnitude: $a = \sqrt{a_r^2 + a_t^2} = \sqrt{\left(\frac{v^2}{r}\right)^2 + \left(\frac{dv}{dt}\right)^2}$

Direction: Makes angle $\alpha$ with radius where: $\tan\alpha = \frac{a_t}{a_r} = \frac{dv/dt}{v^2/r}$

The radial component changes direction while tangential component changes speed.

Earth's rotation causes apparent gravity ($g'$) to differ from true gravity ($g$) due to centrifugal effects.

At colatitude $\theta$ (angle from polar axis): $g' = \sqrt{g^2 + \omega^4R^2\sin^2\theta - 2g\omega^2R\sin^2\theta}$

Where: • $g$ is true gravitational acceleration • $\omega$ is Earth's angular velocity (7.27×10⁻⁵ rad/s) • $R$ is Earth's radius (6.37×10⁶ m) • $\theta$ is colatitude (90° - latitude)

Special cases: • At poles (θ = 0°): $g' = g$ (no effect) • At equator (θ = 90°): $g' = g - \omega^2R$ (maximum effect)

The apparent vertical direction also deviates from true vertical by angle $\alpha$: $\tan\alpha = \frac{\omega^2R\sin\theta\cos\theta}{g-\omega^2R\sin^2\theta}$

This explains why: • Plumb lines don't point exactly to Earth's center • Building walls aren't exactly radial to Earth's center • Water surfaces aren't perfectly spherical

Inertial (laboratory) frame analysis: • Forces: weight, normal force, and friction • Particle moves in straight line after leaving turntable • Acceleration exists only while on turntable: $a = \frac{v^2}{r}$ toward center • Conservation of linear momentum applies after particle leaves • Maximum speed without slipping: $v_{max} = \sqrt{\mu_s gr}$

Rotating frame analysis: • Additional pseudo-forces included: centrifugal and Coriolis • Centrifugal force: $F_{cf} = m\omega^2r$ (outward) • Coriolis force: $F_{cor} = 2m\omega v_{rel}$ (perpendicular to relative motion) • Particle appears to follow curved path even after leaving turntable • Appears to "deflect" in direction opposite to rotation

Same phenomenon, different descriptions: • Inertial frame: particle continues in straight line (Newton's 1st law) • Rotating frame: particle curves outward due to "centrifugal force"

The physical outcome is identical, but the mathematical description and apparent forces differ based on reference frame choice.

The acceleration in circular motion decomposes into:

• Radial component: ar = -ω²r = -v²/r (points toward center, along -êr)

  • Represents the centripetal acceleration
  • Always present in circular motion, even at constant speed
  • Changes the direction of velocity without changing its magnitude

• Tangential component: at = dv/dt (points along êt)

  • Represents change in speed of the particle
  • Only present when speed is changing (non-uniform circular motion)
  • Changes the magnitude of velocity without changing its direction

Example: A car rounding a curve while accelerating experiences both components: centripetal acceleration keeping it on the circular path and tangential acceleration increasing its speed.

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.

• Banking roads provides the necessary centripetal force for vehicles to navigate curves safely without depending solely on friction
• It utilizes the component of the normal force to generate an inward force toward the center of the curve
• The banking angle creates a horizontal component of the normal force that points toward the center of curvature
• This allows vehicles to maintain higher speeds through turns with reduced risk of skidding
• Real-world application: Highway and racetrack curves are banked to allow vehicles to navigate turns at higher speeds than would be possible on flat roads

• 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

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.