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Questions and Answers
What does Kepler's third law imply regarding the relationship between the period of orbit and the radius of the orbit?
What does Kepler's third law imply regarding the relationship between the period of orbit and the radius of the orbit?
What represents gravitational potential energy at a height 'h' above the ground?
What represents gravitational potential energy at a height 'h' above the ground?
In the context of gravitational forces, why is the work done independent of the path taken?
In the context of gravitational forces, why is the work done independent of the path taken?
What is considered the reference point for gravitational potential energy?
What is considered the reference point for gravitational potential energy?
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According to the equation for the gravitational potential energy, which of the following variables increases as a mass is raised to a greater height?
According to the equation for the gravitational potential energy, which of the following variables increases as a mass is raised to a greater height?
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What is the formula for gravitational potential energy at a distance 'r' from a source mass?
What is the formula for gravitational potential energy at a distance 'r' from a source mass?
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When a test mass moves from a point at a distance greater than 'r' to a distance 'r', which of the following describes the change in potential energy if $r_{i} > r_{r}$?
When a test mass moves from a point at a distance greater than 'r' to a distance 'r', which of the following describes the change in potential energy if $r_{i} > r_{r}$?
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What happens to gravitational potential energy when height 'h' is much smaller than the radius of the Earth?
What happens to gravitational potential energy when height 'h' is much smaller than the radius of the Earth?
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What is the gravitational potential at a distance 'r' from a point mass M?
What is the gravitational potential at a distance 'r' from a point mass M?
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What is the gravitational potential inside a thin uniform spherical shell with radius R?
What is the gravitational potential inside a thin uniform spherical shell with radius R?
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Study Notes
Kepler's Laws of Planetary Motion
- Motion is relative
- Divided into bounded and unbounded motion
- Bounded Motion: Total energy is negative (E < 0). Particle has two or more extreme points where kinetic energy is zero. Eccentricity (e) is 0 ≤ e < 1
- Unbounded Motion: Total energy is positive (E > 0). Particle has one extreme point where kinetic energy is zero. Eccentricity (e) is e ≥ 1
- Kepler's First Law: Planets revolve around the sun in elliptical orbits with the sun at one focus. Perihelion is the point closest to the sun and aphelion is the farthest. The sum of the distances from any planet to the two foci is constant.
Kepler's Second Law
- Radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time.
- Kinetic energy is not constant; more near perihelion, less near aphelion.
- Speed is more at perihelion and less at aphelion
Kepler's Third Law
- The square of the period of revolution (T²) of a planet is directly proportional to the cube of its semi-major axis (r³).
- T² ∝ r³
- Equation: P² = 4π² /{G(Ma₁ + Ma₂)} × r³
Gravitational Potential Energy
- Work done in moving a body of mass (m) from infinity to a point within a gravitational field (without acceleration).
- Stored as potential energy.
- Represented by GPE.
- Equation: GPE = mgh
Derivation of Gravitational Potential Energy
- Detailed derivation of the equation for gravitational potential energy using calculus.
- Formula developed to account for force being applied.
- Work is done on the body and the displacement is in the same direction
Gravitational Potential
- Work done in moving a unit test mass from infinity to a point in the gravitational field of the source mass.
- Gravitational potential energy possessed by a unit test mass
- Equation: V = -GM/r
Gravitational Potential of a Spherical Shell
- Potential inside a thin uniform spherical shell is constant, and potential outside the shell is -GM/r.
Gravitational Potential of a Uniform Solid Sphere
- Potential inside a uniform solid sphere varies as V = -GM(3R²-r²)/(2R³).
- Potential on the surface of uniform sphere is V = -GM/R.
- Potential outside a uniform solid sphere is V = -GM/r
Escape Velocity
- Minimum velocity required for an object to escape a gravitational field.
- Independent of the mass of the object
- Equation: Ve = √(2GM/R)
Orbital Velocity
- Velocity required for an object to orbit a source mass in a circular path.
- Depends on the mass of the source and the radius of the orbit.
- Equation: Vo = √(GM/r)
Relationship Between Escape and Orbital Velocity
- Escape velocity is √2 times greater than orbital velocity
Motion of Satellites Around Earth
- Satellites orbit in a circular path.
- Orbital velocity is the speed needed for the satellite to stay in orbit.
- Time period is the time taken for one complete revolution
- Equation: T = 2π√(r³/GM)
Kinetic Energy of a Satellite
- Kinetic energy of a satellite revolving around a planet.
- Equation: K = 1/2 mr²ω²
Potential Energy of a Satellite
- Potential energy of a satellite at a distance / from the center of the earth.
- Equation: U=-GMm/r
Angular Momentum of a Satellite
- Angular momentum of a satellite in orbit.
- Equation: L=mr²ω
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Description
Test your understanding of Kepler's Laws of Planetary Motion, which describe how planets move around the sun. Covering concepts like bounded and unbounded motion, eccentricity, and the specific laws governing planetary orbits, this quiz will reinforce your knowledge of celestial mechanics.
