Orbits and Satellires

General physics

🎯 Learning Objectives

Analyse circular orbits by relating gravitational force to the centripetal acceleration it provides
Derive orbital speed (v = sqrt{dfrac{GM}{r}}) for a small mass orbiting a large mass
Calculate orbital period (T = dfrac{2pi r}{v}) and understand its dependence on radius
Sketch force vs radius graphs and discuss stability of circular orbits

🗣️ Language Objectives

Use terms “centripetal acceleration,” “orbital speed,” “orbital period,” “inverse-square law” correctly
Explain relationships between force, mass, velocity, and radius in clear English
Interpret and discuss derivations and graphs of circular motion
Describe stability and perturbations of orbits using precise vocabulary

📚 Key Terms and Translations

English Term	Russian	Kazakh
Centripetal acceleration	Центростремительное ускорение	Орталыққа тартатын үдеу
Orbital speed	Орбитальная скорость	Орбиталдық жылдамдық
Orbital period	Период обращения	Орбита периоды
Gravitational constant (G)	Гравитационная постоянная	Гравитациялық тұрақты
Inverse-square law	Закон обратных квадратов	Квадраттық кері заң
Radius (r)	Радиус	Радиус

🃏 Vocabulary Study Cards

Centripetal Acceleration

Definition: (a_c = dfrac{v^2}{r}), acceleration toward centre of circle

Use: Required to maintain circular motion

Orbital Speed

Definition: (v = sqrt{dfrac{GM}{r}})

Context: Speed for a stable circular orbit

Orbital Period

Definition: (T = dfrac{2pi r}{v})

Implication: Depends on radius and central mass

Inverse-Square Law

Definition: Gravitational force ∝ (1/r^2)

Effect: Strength decreases rapidly with distance

📖 Glossary of Terms

Centripetal Acceleration

The acceleration of a body moving in a circle of radius (r) at speed (v), directed toward the centre.

Translation

Russian: Ускорение тела, движущегося по окружности радиуса (r) со скоростью (v), направленное к центру.
Kazakh: Радиусы (r) және жылдамдығы (v) бар доға бойымен қозғалатын дененің орталыққа бағытталған үдеуі.
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Orbital Speed

The constant speed a small mass must have to maintain a circular orbit under gravity.

Translation

Russian: Постоянная скорость, необходимая малой массе для круговой орбиты под действием гравитации.
Kazakh: Гравитация әсерінен сақиналық орбитаны ұстап тұру үшін қажетті тұрақты жылдамдық.
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🔬 Theory: Circular Orbits in Gravitational Fields

Relating Gravity to Centripetal Force

For a mass (m) orbiting a much larger mass (M) at radius (r), the gravitational force
(F_g = G dfrac{M m}{r²}
must equal the centripetal force
(F_c = m dfrac{v^2}{r}).

Equate them:
(;G dfrac{M m}{r^2} = m dfrac{v^2}{r})

Cancel (m) and solve for orbital speed:

(v = sqrt{dfrac{G M}{r}})

Then the orbital period is:

(T = dfrac{2pi r}{v} = 2pi sqrt{dfrac{r^3}{G M}})

Translation

Russian: Приравнивая (G frac{Mm}{r^2}) и (m frac{v^2}{r}), получаем (v = sqrt{frac{GM}{r}}) и (T = 2pisqrt{frac{r^3}{GM}}).
Kazakh: (G frac{Mm}{r^2}) және (m frac{v^2}{r}) теңестіріп, (v = sqrt{frac{GM}{r}}) және (T = 2pisqrt{frac{r^3}{GM}}) алынады.
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Theory Questions

Easy: Write the equation that equates gravitational and centripetal forces for circular orbit.

Answer

(G dfrac{M m}{r^2} = m dfrac{v^2}{r}).

Medium: Derive the expression for orbital speed (v) from (G dfrac{M m}{r^2} = m dfrac{v^2}{r}).

Answer

Cancel (m), multiply by (r): (Gfrac{M}{r}=v^2), so (v=sqrt{frac{GM}{r}}).

Medium: Show that (T = 2pisqrt{dfrac{r^3}{GM}}) by substituting (v) into (T=2pi r/v).

Answer

Substitute (v): (T=2pi r/sqrt{frac{GM}{r}}=2pisqrt{frac{r^3}{GM}}).

Hard (Critical Thinking): Discuss how non-uniform mass distributions or perturbations affect circular orbit stability.

Answer

Deviations from spherical symmetry cause precession and orbital decay; perturbations (e.g. third bodies) lead to elliptical or unstable orbits.

💪 Memorization Exercises

Complete the Formulas

Centripetal acceleration: (a_c = dfrac{v^2}{__}).
Orbital speed: (v = sqrt{dfrac{__,M}{r}}).
Orbital period: (T = 2pisqrt{dfrac{r^3}{__,M}}).
Gravitational force: (F = G dfrac{M m}{r^__}).

Answer

1. (r)
2. (G)
3. (G)
4. (2)

🎥 Video Lesson

Additional Video Resources:

• Orbital Motion & Kepler’s Laws

• Gravity and Circular Motion

📐 Worked Examples

Example 1: Low Earth Orbit Speed

Calculate speed of satellite orbiting Earth (M=5.97×10²⁴ kg) at r=7.0×10⁶ m.

Orbital speed diagram

Solution

Answer

(v=sqrt{frac{6.67times10^{-11}times5.97times10^{24}}{7.0times10^6}}approx7.5times10^3text{ m/s}.)

Example 2: Geostationary Orbit Period

Find period for satellite at r=4.23×10⁷ m around Earth.

Solution

Answer

(T=2pisqrt{frac{(4.23times10^7)^3}{6.67times10^{-11}times5.97times10^{24}}}approx8.64times10^4text{ s}approx24text{ h}.)

🧪 Interactive Investigation

Explore orbits and Kepler’s laws with PhET:

Investigation Answers

Demonstrate (vpropto r^{-1/2}) and (Tpropto r^{3/2}) by measuring orbital parameters.

👥 Collaborative Group Activity

Use a Quizizz challenge on circular orbits:

📝 Individual Assessment

Solve these structured questions:

Derive (v=sqrt{frac{GM}{r}}) from force balance.
Calculate speed for Mars satellite at r=4.0×10⁶ m (M=6.42×10²³ kg).
Show how period scales with radius and mass of central body.
Discuss effect of small radial perturbation on circular orbit stability.
Compare energy (kinetic + potential) of circular vs elliptical orbit at same perigee.

🔗 Additional Learning Resources

🤔 Lesson Reflection

How does radius affect orbital speed and period?
Which assumption (negligible mass, perfect circle) is most limiting?
What real-world factors (atmosphere, perturbations) alter ideal orbits?
How can you apply circular orbit concepts to satellite design?