Kepler’s Law of Gravitation
* To resolve shortcomings of the models propounded by Ptolemy and Copernicus,
* Kepler propounded three laws on planetary motion.
* Kepler’s first law, the law of orbits, states that the orbit of a planet is an ellipse with the sun at one of the foci.
* An ellipse is a closed, curved shape that is defined by two points called foci representing an elongated circle.
* The perihelion is the closest point on a planet’s orbit from the sun.
* The aphelion is the farthest point on a planet’s orbit from the sun.
* Kepler’s second law, the law of areas, states that the line joining a planet too the sun sweeps out equal areas in equal intervals at the planet travels in its orbit.
* Kepler’s third law or the law of periods states that square of orbital period, T, of a planet is proportional to the cube of its mean distance, R, from the sun (T2 ∝ R3)
* The orbital period of a planet, denoted by T, is the time taken by the planet to make complete revolution around the sun along its orbit.
Scientists use Kepler’s law to:
* Determine the positions and the orbital periods of various planets
*Perform calculations involved in determining orbital periods of satellites
Universal Law of Gravitation
* The force of attraction exerted by earth on any other body is called gravity.
* The force with which all bodies in the universe attract each other is called gravitational force.
* A force exerted on a body, moving in a circular path, acting towards the centre of the circular path, is known as centripetal force.
* Newton’s universal law of gravitation states that every object in the universe attracts every other object with a force called gravitational force, which is directly proportional to the product of their masses and inversely proportional to the square of distance between them.
F = G\frac{ m_{1} m_{2} }{ r^{2} }
Where:
G = Universal gravitational constant = 6.67 x 10-11 N m2 kg-2
m_{1} = Mass of first object
m_{2} = Mass of second object
r = Distance between the objects
* Newton’s inverse square law states that the gravitational force between two bodies is inversely proportional to the square of distance between them, r.
F ∝ \frac{ 1 }{ r^{2} }
* An object is in free fall, when it falls to the earth under the sole influence of gravity.
* The acceleration of a free falling body due to gravitational force of the earth is known as acceleration due to gravity.
* Mass is the amount of matter contained in a body.
* Weight is the force with which a body is pulled towards the centre of the earth.
* Weight of an object on the moon would be approximately one-sixth of its weight on earth.
* For a body under the influence of gravitational field, equations for motion are:
| Equations of Motion | Body Dropped from a Height u = 0, a = g |
Body Projected Downwards u ≠ 0, a = g |
Body Projected Upwards u ≠ 0, a = – g |
|
v = u + at |
v = gt |
v = u + gt |
v = u – gt |
| s = ut + \frac{1}{2} at² |
h = \frac{1}{2} gt² |
h = ut + \frac{1}{2} gt² | h = ut – \frac{1}{2} gt² |
| v² = u² + 2as | v² = 2gh | v² = u² + 2gh | v² = u² – 2gh |
Thrust and Pressure
* Thrust (F) is the force acting normally on a surface.
* Pressure (P) is thrust acting per unit area.
* The thrust exerted by a body remains constant placed in any position, whereas the pressure exerted by the body changes with the change in its position.
* The mathematical equation for pressure in fluids is: P = hdg
Where:
P = Pressure at the given point
H = height of the fluid
d = density of the fluid
g = acceleration due to gravity
* Pascal’s Law states that pressure applied at any point in a confined liquid is transmitted equally and undiminished to all parts of liquid and acts normally on the sides of the container.
* Application of Pascal’s law is seen in air brakes and hydraulic brakes.
Archimedes’ Principle and Buoyancy
* Archimedes’ Principle states that “when a body is immersed fully or partially in a fluid, displaced by it”.
* A hydrometer uses Archimedes’ Principle to determine the density of any liquid.
* Relative density is the ratio of the density of a substance to the density of water.
* The relative density of a substance is calculated using the formula:
* Relative density = \frac{Density \thinspace \thinspace of \thinspace \thinspace substance }{Density \thinspace \thinspace of \thinspace \thinspace water}
* Applications of relative density are seen in instruments like lactometer, alcoholmeter and saccharometer.
* Buoyancy is the upward force that a fluid exerts on an object.
* Upthrust or buoyant force is the force in the upword direction, experienced by a body when it is partially or completely immersed in a fluid.
* According to the principle of buoyancy:
* Objects with a density less than that of a given liquid float on that liquid.
* Objects with a density greater than that of a given liquid sink when placed in that liquid.