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Circular Motion and Gravity

Subject: Physics
Topic: 4
Cambridge Code: 0625

Circular Motion Fundamentals​

Angular Displacement​

Angular displacement (θ) - Angle swept in radians

θ=srθ = \frac{s}{r}

Where:

  • s = arc length
  • r = radius
  • 1 radian = 57.3°
  • 2π radians = 360°

Angular Velocity​

Angular velocity (ω) - Rate of angular change

ω=θt=2πTω = \frac{θ}{t} = \frac{2π}{T}

Where:

  • T = period (time for one revolution)
  • Unit: rad/s

Relationship to linear velocity: v=ωrv = ωr

Centripetal Acceleration​

Centripetal acceleration - Acceleration toward center

a=ω2r=v2ra = ω^2r = \frac{v^2}{r}

Key points:

  • Always toward center
  • Magnitude constant (uniform circular motion)
  • Direction continuously changes

Centripetal Force​

Force providing centripetal acceleration:

F=ma=mv2r=mω2rF = ma = m\frac{v^2}{r} = mω^2r

Where:

  • m = mass
  • v = speed
  • r = radius

Source of centripetal force varies:

  • String tension (horizontal)
  • Normal force (vertical)
  • Friction
  • Gravity

Circular Motion Examples​

Horizontal Circle (String)​

Tension provides centripetal force: T=mv2rT = \frac{mv^2}{r}

If tension increases:

  • Speed increases or radius decreases
  • Or string may break

Vertical Circle​

At top of circle: mg+T=mv2rmg + T = \frac{mv^2}{r}

  • Weight and tension both toward center
  • Minimum speed to maintain contact: T = 0
  • mg=mvmin2rmg = \frac{mv_{\text{min}}^2}{r} → vmin=grv_{\text{min}} = \sqrt{gr}

At bottom of circle: N−mg=mv2rN - mg = \frac{mv^2}{r}

  • Normal force away from center
  • Normal force larger than weight

Period and Frequency​

Period (T) - Time for one revolution

  • Unit: seconds
  • T=2πrv=2πωT = \frac{2πr}{v} = \frac{2π}{ω}

Frequency (f) - Revolutions per second

  • Unit: Hertz (Hz)
  • f=1T=ω2πf = \frac{1}{T} = \frac{ω}{2π}

Relationship: ω=2πfω = 2πf

Gravitational Field​

Gravitational field - Region where gravity acts

Field strength (g) - Force per unit mass

g=Fm=GMr2g = \frac{F}{m} = \frac{GM}{r^2}

Where:

  • G = gravitational constant = 6.67 × 10⁻¹¹ N·m²/kg²
  • M = mass creating field
  • r = distance from mass
  • Unit: N/kg or m/s²

On Earth​

g=GMERE2≈9.8 m/s2g = \frac{GM_E}{R_E^2} ≈ 9.8 \text{ m/s}^2

Increases with depth initially, decreases with altitude

Newton's Law of Universal Gravitation​

F=Gm1m2r2F = G\frac{m_1m_2}{r^2}

Where:

  • F = gravitational force
  • m₁, m₂ = masses
  • r = separation
  • G = 6.67 × 10⁻¹¹ N·m²/kg²

Properties:

  • Always attractive
  • Acts between any two masses
  • Decreases with distance squared
  • Independent of medium

Orbital Motion​

Satellite Orbits​

For circular orbit, gravity provides centripetal force:

GMmr2=mv2r\frac{GMm}{r^2} = \frac{mv^2}{r}

v=GMrv = \sqrt{\frac{GM}{r}}

Orbital velocity:

  • Decreases with altitude (larger r)
  • Independent of satellite mass
  • Faster orbits are lower

Escape Velocity​

Minimum velocity to escape gravitational field:

ve=2GMr=2grv_e = \sqrt{\frac{2GM}{r}} = \sqrt{2gr}

(At planet surface: ve=2gRv_e = \sqrt{2gR})

Orbital Period​

From cirular orbit condition: T2=4π2r3GMT^2 = \frac{4π^2r^3}{GM}

Kepler's Third Law: T2∝r3T^2 ∝ r^3

Geostationary Orbits​

Geostationary satellite:

  • Period = 24 hours (Earth's rotation period)
  • Remains above same point on Earth
  • Used for communications

Orbital radius: rgeo≈42,000 km from Earth centerr_{\text{geo}} ≈ 42,000 \text{ km from Earth center}

≈36,000 km altitude≈ 36,000 \text{ km altitude}

Simple Harmonic Motion Basics​

SHM - Motion with restoring force ∝ displacement

a=−ω2xa = -ω^2x

Displacement: x=Acos⁡(ωt+φ)x = A\cos(ωt + φ)

Where:

  • A = amplitude
  • ω = angular frequency
  • φ = phase constant
  • t = time

Energy in SHM​

Total energy constant: E=12kA2=Ek+EpE = \frac{1}{2}kA^2 = E_k + E_p

Maximum KE: At equilibrium (v maximum) Maximum PE: At maximum displacement

Key Points​

  1. Centripetal force always toward center
  2. Centripetal acceleration = v²/r
  3. Angular velocity: ω = 2π/T
  4. Gravitational force ∝ 1/r²
  5. Orbital velocity independent of mass
  6. Escape velocity = √(2gR)
  7. Period T² ∝ r³ (Kepler's law)
  8. Geostationary orbit: T = 24 hours
  9. Circular motion requires net inward force
  10. SHM: a = -ω²x

Practice Questions​

  1. Calculate centripetal force
  2. Find orbital velocity
  3. Determine escape velocity
  4. Apply Newton's gravitation law
  5. Analyze vertical circular motion
  6. Calculate periods and frequencies
  7. Determine orbital altitudes
  8. Analyze gravitational fields
  9. Solve SHM problems
  10. Complex orbital mechanics

Revision Tips​

  • Understand centripetal direction always toward center
  • Apply F = mv²/r correctly
  • Know gravitational force law
  • Practice orbital problems
  • Understand field concept
  • Remember Kepler's laws
  • Distinguish escape vs orbital velocity
  • Visualize circular motion
  • Use energy conservation for orbits