The scalar product or dot product of any two vectors A and B, denoted as A.B is defined as
where θ is the angle between the two vectors. Since A, B and are scalars, the dot product of A and B is a scalar quantity. Each vector, A and B, has a direction but their scalar product does not have a direction.
Work is done by a force on the body over a certain displacement.
The work-energy theorem states that the change in kinetic energy of a body is the work done by the net force on the body.
The change in kinetic energy of a particle is equal to thework done on it by the net force.
The work done by the force is defined to be the product of component of the force in the direction of the displacement and the magnitude of this displacement. Thus
If an object of mass m has velocity v, its kinetic energy K is
Gravitational potential energy of an object, as a function of the height h, is denoted by
and it is the negative of work done by the gravitational force in raising the object to that height.
The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.
If the total energy of the reactants is more than the products of the reaction, heat is released and the reaction is said to be an exothermic reaction. If the reverse is true, heat is absorbed and the reaction is endothermic.
Mass and energy are equivalent and are related by the relation
where c, the speed of light in vacuum is approximately . Thus, a staggering amount of energy is associated with a mere kilogram of matter
This is equivalent to the annual electrical outputof a large (3000 MW) power generating station.
Energy may be transformed from one form to another but the total energy of anisolated system remains constant. Energy can neither be created, nor destroyed.
Power is defined as the time rate at which work is done or energy is transferred.
The average power of a force is defined as the ratio of the work, W, to the total time t taken
The instantaneous power is defined as the limiting value of the average power as time interval approaches zero,
The work dW done by a force F for a displacement dr is The instantaneous power can also be expressed as
where v is the instantaneous velocity when the force is F.
Consider a drop of mass1.00 g falling from a height 1.00 km. It hits the ground with a speed of 50.0 m s-1. (a) What is the work done by the gravitational force? What is the work done by the unknown resistive force?
(a) The change in kinetic energy of the drop is where we have assumed that the drop is initially at rest.Assuming that g is a constant with a value 10 m/s2, the work done by the gravitational force is,
(b) From the work-energy theorem
where Wr is the work done by the resistive force on the raindrop. Thus
= 1.25 −10
= − 8.75 J
An elevator can carry a maximum load of 1800 kg (elevator +passengers) is moving up with a constant speed of 2 m s–1. The frictional force opposing the motion is 4000 N. Determine the minimum power delivered by the motor to the elevator in watts as well as in horsepower.
The downward force on the elevator is
The motor must supply enough power to balance this force. Hence,