Work Energy Power Formulas

1. Work done by constant force

W = Force × Component of displacement along force
= displacement × Component of force along displacement
W = FS cos ╬╕ ( ╬╕ is angle between \(\overrightarrow{\mathrm{F}} \text { and } \overrightarrow{\mathrm{S}}\))
W = \(\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{S}}\) S=r2-r(,
\(\overrightarrow{\mathrm{S}}=\overrightarrow{\mathrm{r}_{2}}-\overrightarrow{\mathrm{r}_{1}}\), displacement vector \(\Delta \overrightarrow{\mathrm{r}}\)

2. Work done by a variable force

W = \(\int_{r_{1}}^{r_{2}} \vec{F} \cdot d \vec{r}\) = Area under the F – r curve and position axis
W = \(\int_{x_{1}}^{x_{2}} F_{x} d x+\int_{y_{1}}^{y_{2}} F_{y} d y+\int_{z_{1}}^{z_{2}} F_{z} d z\)

3. Work done by spring force

W = –[\(\frac{1}{2}\)kx₂² – \(\frac{1}{2}\)kx₁²] If x₁ = 0 and x₂ = x. Then W = \(\frac{1}{2}\) kx┬▓

4. Energy

Kinetic Energy = \(\frac{1}{2}\)mv┬▓ = \(\frac{p^{2}}{2 m}\)

5. Work – energy theorem

The sum of the work done by all the forces acting on a particle is equal to the change in its K,E.
╬öW= (K. E.)_f – (K. E.)_i = ╬öK = W_C + W_NC + W_Other

6. Conservative force

W_{AB, 1} = W_{AB, 2}
means work done by conservative force does not depends on path, depends only on initial and final position.
Ex. Gravitational force, Electrostatics, etc.

7. Non conservative force
W_{AB, 1} Γëá W_{AB, 2}
means work done by non conservative force depends on path.
Ex. Frictional force, viscous force, air resistance etc.

8. Potential energy

• Potential energy of a system is always defined corresponding to a conservative internal force.
• Change in P.E. = – Work done by the internal conservative force on the system. W_C = – ╬öU = – W_ext
  ╬öU = U_f – U_i = – W_C = –\(\int \overrightarrow{\mathrm{F}}_{\mathrm{c}} \cdot \overrightarrow{\mathrm{d}} \mathrm{r}\)
• It is a respective quantity.

9. Gravitational P.E.

P.E. near the earth surface w.r.t. ground = mgh
spring potential energy = \(\frac{1}{2}\)kx┬▓
ΓçÆ F_c = – \(\vec{\nabla}\) U;
\(\vec{\nabla}=\frac{\partial}{\partial x} \hat{i}+\frac{\partial}{\partial y} \hat{j}+\frac{\partial}{\partial z} \hat{k}\)

10. Power

P = \(\frac{\mathrm{dW}}{\mathrm{dt}}=\overrightarrow{\mathrm{F}} \cdot \frac{\overrightarrow{\mathrm{d}} \mathrm{r}}{\mathrm{dt}}=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{v}}\)

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