Podcast
Questions and Answers
What is the primary concept illustrated by Kepler's Second Law of Planetary Motion?
What is the primary concept illustrated by Kepler's Second Law of Planetary Motion?
- Planetary orbits are perfect circles with the sun at the center.
- A line connecting a planet to the sun sweeps out equal areas during equal intervals of time. (correct)
- The gravitational force between a planet and the sun is constant.
- Planets move at a constant speed throughout their orbit.
Given two spherical balls of equal mass placed a certain distance apart, how would increasing the mass of both balls and halving the distance between them affect the gravitational force between them?
Given two spherical balls of equal mass placed a certain distance apart, how would increasing the mass of both balls and halving the distance between them affect the gravitational force between them?
- The gravitational force would decrease by a factor of 4.
- The gravitational force would remain the same.
- The gravitational force would increase by a factor of 2.
- The gravitational force would increase by a factor of 8. (correct)
Using Kepler's Third Law, if the period of revolution of a planet around a star is doubled, how does its mean distance from the star change?
Using Kepler's Third Law, if the period of revolution of a planet around a star is doubled, how does its mean distance from the star change?
- The mean distance is squared.
- The mean distance is multiplied by the square root of 2.
- The mean distance is multiplied by the cube root of 4. (correct)
- The mean distance is halved.
If the mass of a planet is doubled, and its orbital radius remains the same, how does the force exerted on the planet change, assuming that the angular speed remains constant?
If the mass of a planet is doubled, and its orbital radius remains the same, how does the force exerted on the planet change, assuming that the angular speed remains constant?
Consider two planets orbiting a star. Planet A has a shorter orbital period than Planet B. What can be inferred about their mean distances from the star?
Consider two planets orbiting a star. Planet A has a shorter orbital period than Planet B. What can be inferred about their mean distances from the star?
A satellite orbiting Earth maintains its orbit due to a balance between which two forces?
A satellite orbiting Earth maintains its orbit due to a balance between which two forces?
What happens to the required speed of a satellite in a circular orbit if the orbital radius r increases?
What happens to the required speed of a satellite in a circular orbit if the orbital radius r increases?
What is the escape velocity of a satellite from Earth's gravitational pull?
What is the escape velocity of a satellite from Earth's gravitational pull?
How does the mass of a satellite affect its orbital speed at a given radius?
How does the mass of a satellite affect its orbital speed at a given radius?
If a satellite's orbital radius is doubled, how is its orbital speed affected?
If a satellite's orbital radius is doubled, how is its orbital speed affected?
What is the relationship between the gravitational force ($F_g$) on a satellite and its acceleration ($a$)?
What is the relationship between the gravitational force ($F_g$) on a satellite and its acceleration ($a$)?
What is the effect of increasing a planet's mass on the orbital speed of a satellite at a constant radius?
What is the effect of increasing a planet's mass on the orbital speed of a satellite at a constant radius?
The period T of a satellite's orbit is related to its orbital radius r. If the radius r increases, what happens to the period T?
The period T of a satellite's orbit is related to its orbital radius r. If the radius r increases, what happens to the period T?
If vectors $\vec{A}$ and $\vec{B}$ are oriented such that their vector product $\vec{A} \times \vec{B} = 0$, which of the following statements must be true?
If vectors $\vec{A}$ and $\vec{B}$ are oriented such that their vector product $\vec{A} \times \vec{B} = 0$, which of the following statements must be true?
Given two vectors, $\vec{P}$ and $\vec{Q}$, where $|\vec{P}| = 5$, $|\vec{Q}| = 10$, and the angle between them is 30 degrees, what is the magnitude of the vector product $|\vec{P} \times \vec{Q}|$?
Given two vectors, $\vec{P}$ and $\vec{Q}$, where $|\vec{P}| = 5$, $|\vec{Q}| = 10$, and the angle between them is 30 degrees, what is the magnitude of the vector product $|\vec{P} \times \vec{Q}|$?
Torque is calculated using the formula $\vec{\tau} = \vec{r} \times \vec{F}$. If the position vector $\vec{r}$ is doubled and the force vector $\vec{F}$ is halved, what happens to the magnitude of the torque?
Torque is calculated using the formula $\vec{\tau} = \vec{r} \times \vec{F}$. If the position vector $\vec{r}$ is doubled and the force vector $\vec{F}$ is halved, what happens to the magnitude of the torque?
If $\vec{A} \times \vec{B} = \vec{C}$, which of the following statements is always true?
If $\vec{A} \times \vec{B} = \vec{C}$, which of the following statements is always true?
Which of the following is a direct consequence of the anti-commutative property of the vector product?
Which of the following is a direct consequence of the anti-commutative property of the vector product?
A force $\vec{F} = 2\hat{i} - 3\hat{j}$ N is applied at a point with position vector $\vec{r} = \hat{i} + \hat{j}$ m. What is the resulting torque $\vec{\tau}$?
A force $\vec{F} = 2\hat{i} - 3\hat{j}$ N is applied at a point with position vector $\vec{r} = \hat{i} + \hat{j}$ m. What is the resulting torque $\vec{\tau}$?
If the period of a planet's orbit is doubled, how does this affect the radius of its orbit, assuming the relationship $T^2 ∝ r^3$ holds?
If the period of a planet's orbit is doubled, how does this affect the radius of its orbit, assuming the relationship $T^2 ∝ r^3$ holds?
For three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$, if $\vec{A} \times \vec{B} = \vec{A} \times \vec{C}$ , which of the following can be correctly inferred?
For three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$, if $\vec{A} \times \vec{B} = \vec{A} \times \vec{C}$ , which of the following can be correctly inferred?
Given the equation $g = \frac{GM_e}{r_e^2}$, how would an increase in the Earth's radius ($r_e$) affect the acceleration due to gravity ($g$) at the surface, assuming the Earth's mass ($M_e$) remains constant?
Given the equation $g = \frac{GM_e}{r_e^2}$, how would an increase in the Earth's radius ($r_e$) affect the acceleration due to gravity ($g$) at the surface, assuming the Earth's mass ($M_e$) remains constant?
A rigid body is subjected to two torques: $\vec{\tau}_1$ with magnitude 10 Nm and $\vec{\tau}_2$ with magnitude 15 Nm. If the angle between $\vec{\tau}_1$ and $\vec{\tau}_2$ is 90 degrees, what is the magnitude of the net torque acting on the body?
A rigid body is subjected to two torques: $\vec{\tau}_1$ with magnitude 10 Nm and $\vec{\tau}_2$ with magnitude 15 Nm. If the angle between $\vec{\tau}_1$ and $\vec{\tau}_2$ is 90 degrees, what is the magnitude of the net torque acting on the body?
Suppose a planet has twice the mass of Earth and twice the radius. How does the acceleration due to gravity on the planet's surface compare to that on Earth's surface?
Suppose a planet has twice the mass of Earth and twice the radius. How does the acceleration due to gravity on the planet's surface compare to that on Earth's surface?
If a hypothetical planet has the same density as Earth but twice the radius, how would the acceleration due to gravity at its surface compare to Earth's?
If a hypothetical planet has the same density as Earth but twice the radius, how would the acceleration due to gravity at its surface compare to Earth's?
A satellite orbits Earth at a certain radius. If the Earth's mass were to suddenly double, what immediate effect would this have on the satellite's orbital period, assuming the radius remains constant?
A satellite orbits Earth at a certain radius. If the Earth's mass were to suddenly double, what immediate effect would this have on the satellite's orbital period, assuming the radius remains constant?
Two planets have the same mass, but planet A has half the radius of planet B. How does the gravitational force on the surface of planet A compare to that on planet B?
Two planets have the same mass, but planet A has half the radius of planet B. How does the gravitational force on the surface of planet A compare to that on planet B?
Given $T^2 = \frac{4π^2}{k}r^3$, if a new planet is discovered with an orbital radius three times that of Earth, how will its orbital period compare, assuming k is the same?
Given $T^2 = \frac{4π^2}{k}r^3$, if a new planet is discovered with an orbital radius three times that of Earth, how will its orbital period compare, assuming k is the same?
A spacecraft is located twice the Earth's radius away from the center of the Earth. What is the gravitational force acting on the spacecraft, compared to the gravitational force on the Earth's surface?
A spacecraft is located twice the Earth's radius away from the center of the Earth. What is the gravitational force acting on the spacecraft, compared to the gravitational force on the Earth's surface?
A car is moving around a curve with a constant speed. According to the principles of centripetal acceleration, which statement accurately describes what's happening?
A car is moving around a curve with a constant speed. According to the principles of centripetal acceleration, which statement accurately describes what's happening?
An object is swung in a circle by a string. If the string suddenly breaks, what path will the object take immediately after the break, assuming no other forces are acting on it?
An object is swung in a circle by a string. If the string suddenly breaks, what path will the object take immediately after the break, assuming no other forces are acting on it?
Two objects with masses $m_1$ and $m_2$ are separated by a distance $r$. If the distance between them is doubled, what happens to the gravitational force between them?
Two objects with masses $m_1$ and $m_2$ are separated by a distance $r$. If the distance between them is doubled, what happens to the gravitational force between them?
An object is moving in a circular path with a constant angular speed. Which of the following statements accurately describes the relationship between its angular speed ($\omega$) and its period ($T$)?
An object is moving in a circular path with a constant angular speed. Which of the following statements accurately describes the relationship between its angular speed ($\omega$) and its period ($T$)?
A planet orbits a star in a circular path. If the mass of the star were to suddenly double, what immediate effect would this have on the gravitational force between the star and the planet, assuming the planet's orbit remains the same?
A planet orbits a star in a circular path. If the mass of the star were to suddenly double, what immediate effect would this have on the gravitational force between the star and the planet, assuming the planet's orbit remains the same?
A car is navigating a curve with a radius $r$ at a speed $v$. If the road is icy, reducing the maximum frictional force, what is the most direct consequence regarding the car's ability to stay on its circular path?
A car is navigating a curve with a radius $r$ at a speed $v$. If the road is icy, reducing the maximum frictional force, what is the most direct consequence regarding the car's ability to stay on its circular path?
Two satellites of equal mass are orbiting Earth. Satellite A orbits at a distance $r$ from Earth's center, while Satellite B orbits at a distance $2r$ from Earth's center. What is the ratio of the gravitational force on Satellite A to the gravitational force on Satellite B?
Two satellites of equal mass are orbiting Earth. Satellite A orbits at a distance $r$ from Earth's center, while Satellite B orbits at a distance $2r$ from Earth's center. What is the ratio of the gravitational force on Satellite A to the gravitational force on Satellite B?
Given vectors $\vec{A}$ and $\vec{B}$, if $\vec{A} \times \vec{B} = 7\hat{k}$, what is the result of $\vec{B} \times \vec{A}$?
Given vectors $\vec{A}$ and $\vec{B}$, if $\vec{A} \times \vec{B} = 7\hat{k}$, what is the result of $\vec{B} \times \vec{A}$?
A force $\vec{F} = (5.00\hat{i} + 2.00\hat{j})$ N is applied at a point $\vec{r} = (1.00\hat{i} - 3.00\hat{j})$ m relative to a fixed axis. What is the torque $\vec{\tau}$ applied?
A force $\vec{F} = (5.00\hat{i} + 2.00\hat{j})$ N is applied at a point $\vec{r} = (1.00\hat{i} - 3.00\hat{j})$ m relative to a fixed axis. What is the torque $\vec{\tau}$ applied?
An object moves in a circle of radius $r$ with a speed $v$. Now, both the radius and the speed are doubled. What happens to the centripetal acceleration?
An object moves in a circle of radius $r$ with a speed $v$. Now, both the radius and the speed are doubled. What happens to the centripetal acceleration?
Under what conditions is angular momentum conserved?
Under what conditions is angular momentum conserved?
What is the SI unit of angular momentum?
What is the SI unit of angular momentum?
If a particle's position vector $\vec{r}$ changes direction but its linear momentum $\vec{p}$ remains constant, how does its angular momentum $\vec{L}$ change relative to a fixed origin?
If a particle's position vector $\vec{r}$ changes direction but its linear momentum $\vec{p}$ remains constant, how does its angular momentum $\vec{L}$ change relative to a fixed origin?
A particle moves with constant velocity $\vec{v}$ parallel to the x-axis. How does its angular momentum with respect to the origin change over time?
A particle moves with constant velocity $\vec{v}$ parallel to the x-axis. How does its angular momentum with respect to the origin change over time?
A force is applied at a point $\vec{r}$ relative to a pivot. If the angle between the force vector $\vec{F}$ and the position vector $\vec{r}$ is 0 degrees, what is the magnitude of the torque?
A force is applied at a point $\vec{r}$ relative to a pivot. If the angle between the force vector $\vec{F}$ and the position vector $\vec{r}$ is 0 degrees, what is the magnitude of the torque?
Two particles have the same mass and speed, but particle A moves in a circle with twice the radius of particle B. What is the ratio of the angular momentum of particle A to that of particle B, $L_A / L_B$?
Two particles have the same mass and speed, but particle A moves in a circle with twice the radius of particle B. What is the ratio of the angular momentum of particle A to that of particle B, $L_A / L_B$?
Flashcards
What is a Vector Product?
What is a Vector Product?
A mathematical operation that multiplies two vectors, resulting in another vector.
Torque Vector Formula
Torque Vector Formula
Torque (τ) is the vector product of the distance (r) from the axis of rotation and the force (F) applied.
Magnitude of Vector Product
Magnitude of Vector Product
The magnitude of the vector product A × B is AB sin θ, where θ is the angle between A and B.
Is Vector Product Commutative?
Is Vector Product Commutative?
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Vector Product Order
Vector Product Order
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Parallel Vectors & Vector Product
Parallel Vectors & Vector Product
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A × A = ?
A × A = ?
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Direction of Resultant Vector
Direction of Resultant Vector
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A × B
A × B
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B × A
B × A
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A × B = −B × A
A × B = −B × A
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Force A = (2.00î + 3.00ĵ) N
Force A = (2.00î + 3.00ĵ) N
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Position r = (4.00î + 5.00ĵ) m
Position r = (4.00î + 5.00ĵ) m
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Torque (τ)
Torque (τ)
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τ = r × F
τ = r × F
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Angular Momentum (L)
Angular Momentum (L)
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Centripetal Acceleration
Centripetal Acceleration
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Centripetal Force
Centripetal Force
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Angular Speed (ω)
Angular Speed (ω)
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Period (T)
Period (T)
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Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation
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Gravitational Constant (G)
Gravitational Constant (G)
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r (in Gravitation Formula)
r (in Gravitation Formula)
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m1 and m2 (in Gravitation Formula)
m1 and m2 (in Gravitation Formula)
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Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation
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Kepler's First Law
Kepler's First Law
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Kepler's Second Law
Kepler's Second Law
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Kepler's Third Law
Kepler's Third Law
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Escape Velocity
Escape Velocity
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Earth's Escape Velocity
Earth's Escape Velocity
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Satellite Motion
Satellite Motion
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Satellite
Satellite
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Gravitational Force
Gravitational Force
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Orbit Radius (r)
Orbit Radius (r)
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Orbital Period (T)
Orbital Period (T)
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Inverse-Square Law
Inverse-Square Law
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T² ∝ r³
T² ∝ r³
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Newton's Law of Gravitation
Newton's Law of Gravitation
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Acceleration Due to Gravity (g)
Acceleration Due to Gravity (g)
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g=Gme/(re)2
g=Gme/(re)2
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Calculate Earth's Mass
Calculate Earth's Mass
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T and r Formula
T and r Formula
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g and G Equation
g and G Equation
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Study Notes
Vector Product and Torque
- The vector product is a way of multiplying two vectors, relating to the vector nature of torque.
- The torque vector ((\vec{\tau})) relates to the vectors (\vec{r}) and (\vec{F}), connected by the vector product: (\vec{\tau} = \vec{r} \times \vec{F}).
- For vectors (\vec{A}) and (\vec{B}), the vector product (\vec{A} \times \vec{B}) results in a vector (\vec{C}) with magnitude (AB\sin\theta), where (\theta) is the angle between (\vec{A}) and (\vec{B}).
- The magnitude of (\vec{C}) is given by (C = AB\sin\theta).
- The vector product is not commutative: (\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}).
- If (\vec{A}) is parallel to (\vec{B}) ((\theta = 0) or (180^\circ)), then (\vec{A} \times \vec{B} = 0) and thus (\vec{A} \times \vec{A} = 0).
- If (\vec{A}) is perpendicular to (\vec{B}), the magnitude of (\vec{A} \times \vec{B}) is (|\vec{A} \times \vec{B}| = AB).
- Vector product follows the distributive law: (\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}).
- The derivative of the vector product with respect to a variable (t) is: (\frac{d}{dt}(\vec{A} \times \vec{B}) = \frac{d\vec{A}}{dt} \times \vec{B} + \vec{A} \times \frac{d\vec{B}}{dt}).
- Cross products of unit vectors (\hat{i}), (\hat{j}), and (\hat{k}) follow these rules:
- (\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0)
- (\hat{i} \times \hat{j} = -\hat{j} \times \hat{i} = \hat{k})
- (\hat{j} \times \hat{k} = -\hat{k} \times \hat{j} = \hat{i})
- (\hat{k} \times \hat{i} = -\hat{i} \times \hat{k} = \hat{j})
- The cross product (\vec{A} \times \vec{B}) can be expressed in determinant form using components (A_x, A_y, A_z) and (B_x, B_y, B_z).
- The expanded determinant form is: (\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}).
Angular Momentum
- Angular momentum measures an object's rotational motion around a point or axis.
- It is a conserved quantity in closed systems, remaining constant without external torque.
- For a particle with mass (m), velocity (\vec{v}), and position (\vec{r}), angular momentum (\vec{L}) is (L=rxp=rxmv).
- The SI unit for angular momentum is (\text{kg} \cdot \text{m}^2/\text{s}).
- Both magnitude and direction depend on the choice of axis.
- Magnitude of (L) is (L = mvr \sin\phi), where (\phi) is the angle between (r) and (p).
- (L) is zero when (r) is parallel to (p) ((\phi = 0) or (180^\circ)).
- If (r) is perpendicular to (p) ((\phi = 90^\circ)), then (L = mvr).
Conservation of Angular Momentum
- If no external torque acts on a system, its total angular momentum remains constant.
- Mathematically expressed as (\frac{dL}{dt} = 0), where (L) is the system's total angular momentum.
- Examples include planetary orbits, spinning tops, gyroscopes, neutron stars, and figure skaters pulling in their arms.
Circular Motion
- Circular motion occurs when an object moves along a circular path.
- A force continuously acts perpendicular to the object's velocity, changing direction but not speed.
- Uniform Circular Motion: The object maintains constant speed, but velocity changes due to continuous change in direction.
- Non-uniform Circular Motion: The object's speed changes, resulting in both tangential and centripetal acceleration.
Concepts in Circular Motion
- Angular Displacement ((\theta)): The angle in radians, through which an object rotates around the circle's center, with (\theta = 2\pi) radians for a full revolution.
- Angular Velocity ((\omega)): The rate of change of angular displacement, constant in uniform circular motion, given by (\omega = \frac{\theta}{t}), measured in radians per second (rad/s).
- Tangential Velocity ((v)): The object's linear speed along the circular path, related to angular velocity by (v = r\omega), where (r) is the radius.
- Centripetal Acceleration ((a_c)): The acceleration directed towards the circle's center, causing a change in direction, with (a_c = \frac{v^2}{r} = r\omega^2).
- Centripetal Force ((F_c)): The net force required to maintain circular motion, directed towards the center, given by (F_c = ma_c = \frac{mv^2}{r} = mr\omega^2).
Newton's Law of Universal Gravitation
- Every mass attracts every other mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
- Mathematically, (F = \frac{Gm_1m_2}{r^2}), where:
- (F) is the gravitational force.
- (m_1) and (m_2) are the masses of the objects.
- (G) is the gravitational constant ((\approx 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2)).
- (r) is the distance between the centers of the masses.
Kepler's Laws of Planetary Motion
- Law 1: Planets orbit the sun in ellipses, with the sun at one focus.
- Law 2: A line joining the sun and a planet sweeps out equal areas during equal intervals of time.
- Law 3: The square of the orbital period is proportional to the cube of the semi-major axis of the orbit.
- The force acting on a planet of mass (m) is (mr\omega^2), with (\omega = \frac{2\pi}{T}), thus (F = mr(\frac{2\pi}{T})^2 = \frac{4\pi^2mr}{T^2}).
Gravitational Constant ‘G' and Acceleration of Gravity ‘g'
- The relationship between (g) and (G) is derived from Newton's law of gravitation: (F = \frac{Gm_em}{r_e^2} = mg).
- The force per unit mass is: (\frac{F}{m} = \frac{Gm_e}{r_e^2} = g).
- Acceleration due to gravity: (g = \frac{Gm_e}{r_e^2}).
- The mass of the earth can be calculated: (m_e = \frac{gr_e^2}{G}).
Gravitation Potential Energy
- Gravitational potential energy is the work done in moving a unit mass from infinity to a point, measured in J/kg.
- Given by (V = -\frac{Gm}{r}), where (m) is the mass creating the field and (r) is the distance.
- The potential decreases as (r) increases, approaching zero at infinity.
Escape Velocity
- Escape velocity ((v_e)) is the minimum velocity to escape a body's gravitational influence.
- Given by (v_e = \sqrt{2gR}), where (R) is the radius of the body.
Satellite Motion and Orbits
- Satellite motion is influenced by gravitational forces.
- A satellite's orbit is sustained by a balance between gravitational force and centripetal force.
- Gravitational Force: Pulls the satellite towards the planet's center.
- Centripetal Force: Maintains the satellite's circular or elliptical orbit.
- With (a_{rad} = \frac{v^2}{r}) and (F_g = \frac{GM_Em}{r^2}), it follows that (v = \sqrt{\frac{GM_E}{r}}).
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