work done by a force definition physics class 11 by a constant force by a variable force

know work done by a force definition physics class 11 by a constant force by a variable force ?

Work Done by a Force

(a) Work done by a constant force

When a constant force F acting on a body produces a displacement S, then the work done by the force is given by

W = F.S = FS cosθ

where θ is the angle between the force vector F and the displacement vector S [see Fig. below]. F and S are the magnitudes of F and S respectively

(i) If θ is acute, cosθ is positive. Hence work is positive for acute θ. In this case the force increases the speed of the body.

(ii) If θ = 90°, W = 0, i.e. if the force is perpendicular to displacement work done by the force is zero.

(iii) If θ is obtuse, W is negative. In this case the force decreases the speed of the body.

(iv) If θ = 0, i.e. force F is in the same direction as displacement S, then W = FS.

(v) If θ =180°, force F is opposite to S (example frictional force), W = – FS. Work done by frictional and viscous force is always negative.

(b) Work done by a variable force

Suppose a force F is not constant but depends on the position vector r of the body, then the work done by the force F in moving the body from a position r1 to a position r2 is given by

Energy

Energy can be defined as the capacity or ability to do work and is measured by the amount of work a body can do. So, energy is measured in the same units as work, namely, joule. Like work, energy is a scalar quantity. Energy can exist in various forms, such as heat energy, electrical energy, sound energy, light energy, chemical energy, nuclear energy, mechanical energy, etc. We will be mainly concerned with mechanical energy. Mechanical energy is of two types, kinetic and potential.

Kinetic Energy : Energy due to Motion A moving object can do work on another object when it strikes it. In other words, an object in motion has the ability to do work and, by definition, has energy. The energy possessed by a body by virtue of its motion is called kinetic energy. An initially motionless body can move and acquire a velocity only if a force acts on it. The work done by the force in causing the body to move measures the kinetic energy (written as KE) of the moving body, i.e.

KE = W

The kinetic energy of a body of mass m, moving with a velocity v is given by

This relation holds even if the force is variable, i.e. if the force varies both in magnitude and direction.

Work-Energy Principle

Suppose a body of mass m moves with an initial velocity u. A force F acts on it, as a result of which it acquires a final velocity v. The work done by the force is given by

Thus, the work done by a force in displacing a body measures the change in its kinetic energy. This is the work energy principle. Thus, when a force does work on a body, its kinetic energy increases; the increase in kinetic energy being equal to the amount of work done. The converse of this is also true. When the kinetic energy of a body is decreased by a retarding force, the decrease is equal to the work done by the body against the retarding force. Thus kinetic energy and work are equivalent quantities and are, therefore, measured in the same units, namely, joule

Potential Energy : Energy due to Position or Configuration An object can have energy not only by virtue of its motion, but also because of its position or configuration. The energy possessed by a body owing to its position or configuration is called potential energy.

For example, a wound watch spring has potential energy on account of its wound state or configuration of the coils. As the spring unwinds, it does work to move the hands of the watch. Thus, a wound spring has the potentiality to do work. Gravitational Potential Energy An object held at a position above the surface of the earth has potential energy by virtue of its position. When it falls from that position, it can do work. The potential energy of an object held above the earth is called gravitational potential energy. To calculate the energy stored in a body which has been lifted above the earth’s surface against the gravitational force, we have to calculate the amount of work done in carrying it there. Consider a body of mass m. It is lifted vertically to a height h above the earth by applying a force F vertically upward. The force F must be just enough to overcome the gravitational attraction, i.e.

F = mg

where g is the acceleration due to gravity at that place. For bodies not too far above the surface of the earth, the value of g is practically constant. Hence the work done by a constant force F in displacing a body by a height h can be calculated by the product F x h = mgh. Thus gravitational potential energy of a body of mass m at a height h above the surface of the earth is mgh. The gravitational potential energy on the surface of the earth is taken to be zero. Gravitational

PE = mgh or U = mgh

Potential Energy of a Spring. Consider a perfectly elastic spring. One end of the spring is fixed to a rigid wall and other end is fixed to a block which is placed on a frictionless horizontal surface as shown in Fig. 4.4. We assume that the mass of the spring is negligible compared to the mass of the block. If we stretch the spring by a distance x, the spring will exert a force on us during stretching. This force is due to the reaction of the spring and is called the restoring force which is proportional to the displacement x and acts in a direction opposite to the displacement, i.e.

where k is the force constant of the spring. The negative sign indicates that the force acts in a direction opposite to displacement. To stretch a spring by a displacement x, we must exert a force F’ on it, equal but opposite to the force F exerted by the spring on us. Therefore, the applied force is

F’ = – F = kx

Notice that F’ is a variable force as it depends on x. Therefore, the work done by the applied force in stretching the spring through a distance x is given by

It is evident that the work done in compressing the spring by an amount x is also given by W = ½ kx² . The work done in stretching or compressing a spring is stored in it in the form of potential energy which is due to the changed configuration of the coils of the spring. Hence the potential energy of a massless elastic spring of force constant k when it is stretched or compressed by an amount x is given by

Conservative and Non-Conservative Forces

(a) Conservative force

A force is conservative if

(i) the work done by it on a body in moving it from one position to another depends only on the initial and final positions of the body and not on the path followed by it between the two positions. or

(ii) the net work done by the force on a body that moves through any closed path is zero. The above two conditions are equivalent.

Examples of conservative forces are gravitational force, electrostatic force and spring force.

(b) Non-conservative force
A force is non-conservative if

(i) The work done by it on a body in moving it from one position to another depends on the path followed by the body between the two positions. or

(ii) The work done by the force on a body that moves through a closed path is non zero.

Examples of non-conservative forces are frictional and viscous forces.

Conservative Force and Potential Energy

For a conservative force F that depends upon position r, there is a potential energy function U which also depends on r. When a conservative force does positive work, the potential of the system decreases, i.e.

Work done = decrease in potential energy

Hence the negative derivative of the potential energy function with respect to position gives the conservative force acting on the system.

Principle of Conservation of Energy

The total energy of an isolated system remains constant; it may change from one form to another.