"Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the particles."
or
The force by which the Earth attracts objects towards its centre is called gravitational force.
"Every particle of matter attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."
The gravitational constant is the proportionality constant G in Newton's Law of Universal Gravitation, which relates the force of gravitational attraction between two masses and the distance separating them.
The gravitational force due to the earth on a body results in its acceleration. This is called acceleration due to gravity and is denoted by ‘g’.
Acceleration due to gravity at depth d is the rate at which an object accelerates toward the Earth when placed at a distance d below the Earth's surface. It is defined as the gravitational force per unit mass acting on a body at that depth.
Latitude is an angle made by the radius vector of any point from the centre of the Earth with the equatorial plane. Obviously it ranges from 0° at the equator to 90° at the poles.
"Weight of an object is the force with which the Earth attracts that object."
"Potential energy is the work done against conservative force (or forces) in achieving a certain position or configuration of a given system."
OR
The energy stored in an object because of its position or state is called potential energy.
What is meant by the gravitational potential energy?
The gravitational potential energy refers to the potential energy that a body possesses as a result of the Earth's force of attraction to it.
The potential difference (p.d.) between two points is equal to the work done per unit charge in moving a positive test charge from one point to the other.
OR
The work done per unit positive charge in moving a charge from one point to another in an electric field, is called potential difference between those two points.
Define the following:
Potential difference
Potential difference: The potential difference between two points may be defined as the work done in moving a unit positive charge from one point to the other.
Define Electric potential.
Electric potential is a measure of work done on the unit's positive charge to bring it to that point against all electrical forces. It is represented as ‘V’.
The potential at a point is defined as the amount of work done per unit charge in bringing a positive test charge from infinity to that point.
"The minimum velocity with which a body should be thrown vertically upwards from the surface of the Earth so that it escapes the Earth’s gravitational field, is called the escape velocity (ve) of the body."
Answer the following question in detail.
What is a critical velocity?
The exact horizontal velocity of projection that must be given to a satellite at a certain height so that it can revolve in a circular orbit round the Earth is called the critical velocity or orbital velocity (vc).
Answer the following question.
Define the binding energy of a satellite.
The minimum energy required by a satellite to escape from Earth’s gravitational influence is the binding energy of the satellite.
The objects that revolve around the Earth are called Earth satellites.
The exact horizontal velocity of projection that must be given to a satellite at a certain height so that it can revolve in a circular orbit around the Earth is called the critical velocity or orbital velocity (vc).
The feeling of weightlessness is the state where "there will not be any feeling of weight," and the weighing machine will record zero.
"The minimum energy required by a satellite to escape from Earth’s gravitational influence is the binding energy of the satellite."
Newton’s Universal Law of Gravitation:
F = \[G\frac{m_1m_2}{r^2}\]
where:
Kepler’s Third Law relates the time period T of a planet’s revolution to the semi-major axis a of its elliptical orbit:
T2 ∝ a3
where,
The area swept by the planet of mass m in a given interval Δt is:
\[\Delta\vec{A}=\frac{1}{2}(\vec{r}\times\vec{v}\Delta t)\]
The gravitational force of attraction (F) between two bodies of mass m1 and m2 separated by a distance r is:
F: Gravitational Force of attraction (in Newtons, N).
\[m_1, m_2\]: Masses of the two objects (in kilograms, kg).
r (or d in the first part): Distance between the two objects (in meters, m).
G: The constant of proportionality, called the Universal gravitational constant.
Value in SI units: \[G=6.67\times10^{-11}\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2\]
Dimensions: \[[G]=[\mathrm{L}^3\mathrm{M}^{-1}\mathrm{T}^{-2}]\]
The magnitude of the force of attraction (F) between a big sphere (Mass M) and a neighbouring small sphere (Mass m) separated by distance r is:
The torque (τ) generated by the force of attraction is:
\[\tau = F \cdot L = G \frac{mM}{r^2} L\]
| G | Gravitational constant (the value to be calculated). |
| m | Mass of the small spheres (s1 and s2). |
| M | Mass of the large spheres (L1 and L2). |
| r | Initial distance of separation between the centres of the big and the neighbouring small sphere. |
| L | Length of the light rigid rod. |
| τ | Gravitational torque. |
| K | Restoring torque per unit angle of the suspension wire. |
| θ | Angle of twist of the suspension wire. |
The value of the acceleration due to gravity (g) on the surface of the Earth is given by the formula:
Where:
The formulas for acceleration due to gravity (g) are provided below:
Definition of Terms:
Where:
The effective acceleration due to gravity (g') at a point on the Earth's surface at latitude θ is given by:
Where:
Based on the relationship between work and energy, the change in potential energy is given by:
\[\vec F\] · d\[\vec x\] = dU
U(r) = -\[\frac {GMm}{r}\]
Where:
V = \[\frac {W}{Q}\]
or
W = QV
\[v_e=\sqrt{\frac{2GM}{R}}\]
vc = \[\sqrt{\frac{GM}{R+h}}\]
Where:
According to Newton's second law of motion:
Where:
For a passenger in a lift, the net force in the downward direction is:
Where:
T = \[2\pi\sqrt{\frac{(R+h)^3}{GM}}\]
Where:
Where:
The Earth can be thought of as many concentric spherical shells. According to the Shell Theorem:
Starting with the equation for acceleration due to gravity at the surface:
g = \[\frac {GM}{R^2}\]
If we assume Earth has uniform density ρρ, then:
M = \[\frac {4}{3}\]πR³ρ
Therefore:
g = \[\frac{G\times\frac{4}{3}\pi R^{3}\rho}{R^{2}}=\frac{4}{3}\pi RG\rho\]
At depth d, only the inner sphere of radius (R - d) matters:
M′ = \[\frac{4}{3}\pi(R-d)^3\rho\]
gd = \[\frac{G\times\frac{4}{3}\pi(R-d)^{3}\rho}{(R-d)^{2}}=\frac{4}{3}\pi(R-d)G\rho\]
Dividing the equation at depth d by the equation at the surface:
\[\frac{g_d}{g}=\frac{\frac{4}{3}\pi(R-d)G\rho}{\frac{4}{3}\pi RG\rho}=\frac{R-d}{R}=1-\frac{d}{R}\]
This gives us:
gd = g[1 − \[\frac {d}{R}\]]
