`text((a))` Work done by normal reaction can be positive, negative or zero.

Here work done by normal reaction between `A` & `B`.

(i) on `A=(W_N)_A=NS` (Negative)
(ii) On `B = (W_N )_B = NS` (Positive )

`text((b))` Work done by normal reaction on the whole system is zero

`(W_N)_(system)=(W_N)_A + (W_N)_B`

`(W _N ) s =-NS+NS=` zero

`text((a))` Work done by tension force can be positive, negative or zero.

Here work done by tension force `T`

(i) On `A =(W_T) = TS`

(ii) On `B = (W_T) = -TS`

`text((b))` Work done by tension on the whole system is always zero

`(W_T)_s =(W_T)_A + (W_T)_B

(W_T )_s = TS -TS =0`

`text((a))` Work done by gravity can be positive, negative or zero.

(i) When the object moves vertically downwards, work done by gravity is positive.
(ii) When the object moves vertically upwards, work done by gravity is negative.
(iii) for horizontal displacement work done by gravity is zero.

`text((b))` Work done by gravity is independent of the path, it depends on initial and final vertical coordinates of the object.

`W_g=vecF.vecS=-mghatj.(Deltaxhati+Deltayhatj+Deltazhatk)`

`W_g=-mgDeltay=mg(y_i - y_f)`

`text((a))` Work done by spring force can be positive, negative or zero.
`text((b))` Spring can do work at its both ends.
`text((c))` Spring force is a variable force so work done by spring force is given by

Here `x` is extension or compression in the spring from its natural length.

`W` = `1/2` `k` `(x_i)^2` `-` `1/2` `k` `(x_f)^2`

`text((a))` Work done by pseudo force can be positive, negative or zero.
`text((b))` Point of application of pseudo force is taken as center of mass of the object for translatory reference frame.

`text((a))` Internal force exists for a system of particles. For a single particle, there is no internal force.

`text((b))` Internal forces will do work when there is deformation within the system. In the case of rigid body net work done by internal forces is zero.

`text((c))` Work done by internal force can be positive, negative or zero.

When the magnitude and direction of a force vary in three dimensions, it can be expressed as a function of the position vector `vecF(r)`, or in terms of the coordinates `F(x,y,z)`. The work done by such a force in an infinitesimal displacement `dvecs` is

`dW=vecF.dvecs......(1)`

The total work done in going from point A to point B as shown in the figure i.e.

`W_(A->B)= int_A^B vecF.dvecs=int_A^B(Fcostheta)ds`

In terms of rectangular components,

`vecF=vecF_xhati+vecF_yhatj+vecF_zhatk`

and `dvecs=dxhati+dyhatj+dzhatk`

`therefore` `W_(A->B)=int^(x_B)F_xdx+int^(y_B)F_ydy+int^(z_B)F_zdz`

Graphically , the work done by a variable force `F(x)` from an initial point `x_i` to final point `x_f` is given by the area under the force- displacement curve as shown in the figure.

The done by the spring when the displacement of its free end changes from `x_i` to `x_f` is the area of the trapezoid;

`W_s=1/2 k(x_f^2 - x_i^2)`

Area (work) above the x - axis is taken as positive , and vice-versa.