The work done by the conservative force for a closed path is

1. Dot product of two vector:

(i) Dot product or scalar product of two vectors A and B is defined as $A.B=A\times B\times \cos \theta$, where θ is the angle between A and B

(ii) A=Axi^+Ayj^+Azk^ and B=Bxi^+Byj^+Bzk^ then, $A.B=A_x{B}_x+A_y{B}_y+A_z{B}_z$

(iii) Work done by a constant force, W=F→.s→

(iv) Work done by a variable force dW=F→.ds→

2. Relation between kinetic energy and momentum: 

Relation between momentum and kinetic energy, $KE=\frac{P^2}{2m}$ and $P=\sqrt{2m\times KE}$, where $P$ is linear momentum

3. Potential energy:

(i) Potential energy $\left(U\right)$ and force are related as, ∫U1U2dU=-∫r1r2F→.dr→ i.e., U2-U1=-∫r1r2F→.dr→=-W where $W$ is the work done by the conservative force.

(ii) Conservative force can be found from potential energy using the equation, F(x)=-dUdx

3. Work energy theorem:

According to work-energy theorem, the net work done on a body is equal to the change in the kinetic energy, $W_{net}=\Delta KE$

4. Power:

(i) Power is the rate of work done, $P=\frac{dW}{dt}$

(ii) The average power delivered by an agent is given by Pav=Wt

(iii) Instantaneous power, P=FdSdt=FdSdt=F.V

5. Conservation of energy:

According to conservation of mechanical energy, the total mechanical energy of an isolated system is conserved if the internal forces are conservative.

6. Collisions:

(i) The total momentum of a system remains constant during a collision.

(ii) Gravitational force and spring force are always non-impulsive during collision.

(iii) Total energy is conserved in an elastic collision.

(iv) The energy lost is maximum in a perfectly inelastic collision.

(v) The two equations which we can use in a collision are: $m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$ and $e(u_1-u_2)=v_2-v_1$.

(vi) $e$ is the coefficient of restitution. It is $1$ for elastic collision, $0$ for perfectly inelastic collision and for inelastic collision it will be between $0$ and $1$.