Work Energy Theorem
* The dot product of two vectors is equal to the product of the magnitude of the two vectors and cosine of the angle between the two vectors. The result of a dot product is always scalar.
* Using the dot product, the work done is measured as the dot product of the force and displacement, which is equal to Fd cos θ.
* The work done can be positive, negative or zero depending on the values of the angle between the force and displacement, that is,θ.
* The work done by a force is positive when the angle between force and displacement is acute, that is, the angle is less than 90 degree.
* The work done by a force is negative when the angle between force and displacement is obtuse, that is , greater than 90 degree and less than or equal to 180 degrees.
Work done by variable Force and Potential Energy
* A variable force is a force whose magnitude or direction or both vary during the displacement of a body on which it acts.
* The work done by a variable force can be expressed integral from as W= \int_{x1}^{xn} F dx
* The expression W= \int_{x1}^{xn} F dx proves the work- Energy Theorem for a variable force.
* The energy stored in a body by virtue of its position on configuration is called potential energy.
* Mathematically, the potential energy V(χ) is defined for any force F(χ) which can be
written as F(χ) = – \frac{dv}{dx}
* If the work done by a force is independent of the path followed, then it is a conservative force.
* If the work done by a force depends on the path followed, then it is a non-conservative force.
Conservation of Mechanical Energy
The sum of kinetic energy (K) and potential energy (V) possessed by body is called the total mechanical energy of a body.
* The principle of Conservation of Mechanical Energy states that the total Mechanical energy of a system is conserved if the forces doing the work on it are conservative.
* Work done by a conservative force will be path independent. It is equal to the difference between the potential energies between the initial and final positions and is completely recoverable.
* Work done by a conservative force in a closed path in zero.
Potential Energy of a Spring
* The spring, when stretched or compressed, comes under the action of a variable force known as the spring force.
* The spring force is the restoring force responsible for bringing the block attached to the spring back to its mean position.
* Spring force ,F= – Kχ Where K is the spring constant, represents the force law for the spring known as Hooke’s Law.
* Work done by the spring force is W_{s} = - \frac{1}{2} K. χ.^{2} m
* The Work done by the elongating force is WE = = + \frac{1}{2} K. χ.^{2} m
* The work done by the external force is against the spring force and hence is stored as potential energy in the spring.
* The spring force is a conservative force as the work done for a round trip or closed path in zero.
Law of Conservation of energy and power
* Some of the different forms of energy are heat energy, chemical energy, electrical energy and nuclear energy, electrical energy and nuclear energy.
* Chemical energy is produced when the molecules of the reactants participating in a chemical reaction, combine together to attain greater stability by forming stables compounds.
* Albert Einstein showed that mass and energy are equivalent and are related by the relation E= mc2 where c, the speed of light in vacuum is approximately 3 x 108m/s
* Nuclear energy is the energy derived from the nuclear reaction such as nuclear fusion or nuclear fission from which energy may be released in accordance with Einstein’s Mass- Energy Equivalence Relation.
* Law of Conservation of Energy states that the total energy of an isolated system is always conserved and that energy can only be transformed from one form to the other but can neither be created nor destroyed.
* Power is defined as the rate of doing work.
* Power is a scalar quantity and its SI unit is watt and its dimensional formula is { M^{1} L^{2} t-^{3} }
Collisions
* In the subatomic world, there may be collisions between various subatomic particles such as protons, electrons and neutrons.
* The total change in the momentum during a collision is zero or the total momentum of the system is conserved.
* During a collision, if the both the total momentum and total kinetic energy of the colliding bodies are conserved, it is an elastic collision.
* During a collision, if only the total momentum of the colliding bodies is conserved and the total kinetic energy is not conserved, and the bodies stick together after the collision, it is a perfectly inelastic collision.
* During a collision, if only the total momentum of the colliding bodies is conserved and not the kinetic energy, and the colliding bodies move separately after the collision, the collision is called an inelastic collision.