Work, Energy and Power — Revision Notes

⚡ 30-Second Revision

Work: — $W = Fd \cos\theta$ (constant force), $W = \int F(x)\,dx$ (variable force). Unit: Joule (J).
Kinetic Energy: — $E_k = \frac{1}{2}mv^2$. Unit: Joule (J).
Gravitational Potential Energy: — $U_g = mgh$. Unit: Joule (J).
Elastic Potential Energy: — $U_e = \frac{1}{2}kx^2$. Unit: Joule (J).
Work-Energy Theorem: — $W_{net} = \Delta E_k$.
Conservation of Mechanical Energy: — $E_{k,i} + U_i = E_{k,f} + U_f$ (if only conservative forces).
Work by Non-Conservative Forces: — $W_{nc} = \Delta E_{mech}$.
Power: — $P = \frac{W}{t}$ (average), $P = \vec{F} \cdot \vec{v}$ (instantaneous). Unit: Watt (W) = J/s.
Conservative Forces: — Path-independent work, zero work in closed loop, potential energy defined (e.g., gravity, spring).
Non-Conservative Forces: — Path-dependent work, non-zero work in closed loop, dissipate mechanical energy (e.g., friction, air resistance).

2-Minute Revision

Work, Energy, and Power are interconnected concepts describing force-induced motion and energy transformations. Work is the energy transferred when a force causes displacement ($W = Fd \cos\theta$). It can be positive, negative, or zero, and for variable forces, it's calculated by integration.

Energy is the capacity to do work, existing as kinetic energy ($E_k = \frac{1}{2}mv^2$) due to motion, and potential energy (gravitational $mgh$, elastic $\frac{1}{2}kx^2$) due to position or configuration.

The Work-Energy Theorem ($W_{net} = \Delta E_k$) links net work to kinetic energy change. Mechanical energy ($E_k + U$) is conserved only if conservative forces (like gravity) are at play; non-conservative forces (like friction) do negative work, dissipating mechanical energy.

Power is the rate of doing work ($P = W/t$) or energy transfer, also expressed as $P = \vec{F} \cdot \vec{v}$. All are scalar quantities, with Joules for work/energy and Watts for power. Focus on identifying force types and applying the correct energy conservation principle.

5-Minute Revision

Work, Energy, and Power are central to mechanics. Work ($W$) is the transfer of energy when a force ($\vec{F}$) causes a displacement ($\vec{d}$). For a constant force, $W = Fd \cos\theta$. If the force varies with position, $W = \int F(x)\,dx$. Work is positive if the force aids motion, negative if it opposes, and zero if perpendicular. Its SI unit is the Joule (J).

Energy is the capacity to do work. Kinetic energy ($E_k = \frac{1}{2}mv^2$) is due to motion. Potential energy ($U$) is stored energy due to position or configuration, associated with conservative forces.

Gravitational potential energy is $U_g = mgh$, and elastic potential energy for a spring is $U_e = \frac{1}{2}kx^2$. The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy ($W_{net} = \Delta E_k$).

This is a powerful tool, often simplifying problems by avoiding direct use of acceleration.

Conservative forces (e.g., gravity, spring force) have path-independent work and allow for potential energy definition; mechanical energy ($E_k + U$) is conserved in their presence. Non-conservative forces (e.g., friction, air resistance) have path-dependent work, dissipate mechanical energy, and thus, mechanical energy is not conserved if they are present ($W_{nc} = \Delta E_{mech}$). Total energy, however, is always conserved.

Power ($P$) is the rate at which work is done or energy is transferred. Average power is $P_{avg} = W/\Delta t$. Instantaneous power is $P = dW/dt$ or $P = \vec{F} \cdot \vec{v}$. Its SI unit is the Watt (W), equivalent to J/s. Understanding these concepts is crucial for solving problems involving motion, forces, and energy transformations. Practice identifying force types, applying the correct energy conservation principle, and using the Work-Energy Theorem effectively.

Prelims Revision Notes

1
Work (W): — Scalar quantity. $W = Fd \cos\theta$. If $\theta=0^\circ$, $W=Fd$. If $\theta=90^\circ$, $W=0$. If $\theta=180^\circ$, $W=-Fd$. For variable force, $W = \int F(x)\,dx$. Unit: Joule (J). $1\text{ J} = 1\text{ N}\cdot\text{m}$.
2
Energy: — Capacity to do work. Scalar quantity. Unit: Joule (J).

* **Kinetic Energy ($E_k$):** Energy of motion. $E_k = \frac{1}{2}mv^2$. Always positive. Relation with momentum $p$: $E_k = \frac{p^2}{2m}$. * **Potential Energy ($U$):** Stored energy due to position/configuration. Defined for conservative forces. * Gravitational Potential Energy: $U_g = mgh$. Reference level is arbitrary; change in $U_g$ is significant. * Elastic Potential Energy: $U_e = \frac{1}{2}kx^2$ for a spring with constant $k$ stretched/compressed by $x$.

1
Work-Energy Theorem: — $W_{net} = \Delta E_k = E_{k,f} - E_{k,i}$. Net work done by all forces equals change in kinetic energy.
2
Conservative Forces: — Work done is path-independent. Work done in a closed loop is zero. Potential energy can be defined. Examples: gravity, spring force, electrostatic force.
3
Non-Conservative Forces: — Work done is path-dependent. Work done in a closed loop is non-zero. Dissipate mechanical energy (e.g., friction, air resistance).
4
Conservation of Mechanical Energy: — If only conservative forces do work, $E_{k,i} + U_i = E_{k,f} + U_f = \text{constant}$.
5
Work Done by Non-Conservative Forces: — $W_{nc} = \Delta E_{mech} = (E_{k,f} + U_f) - (E_{k,i} + U_i)$. This accounts for energy loss/gain due to non-conservative forces.
6
Power (P): — Rate of doing work or transferring energy. Scalar quantity. Unit: Watt (W). $1\text{ W} = 1\text{ J/s}$.

* Average Power: $P_{avg} = \frac{W}{\Delta t}$. * Instantaneous Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} = Fv \cos\theta$.

1
Efficiency ($\eta$): — Ratio of output power (or energy) to input power (or energy). $\eta = \frac{P_{out}}{P_{in}} = \frac{E_{out}}{E_{in}}$. Always less than or equal to 1 (or 100%).

Vyyuha Quick Recall

W-E-P: Work is Energy's Path.

Work: Force Does Cos ($\mathbf{F} \cdot \mathbf{d} \cos\theta$). Energy: Kinetic Potential ($\frac{1}{2}mv^2$, $mgh$, $\frac{1}{2}kx^2$). Power: Fast Velocity ($\mathbf{F} \cdot \mathbf{v}$) or Work Time ($\frac{W}{t}$).

Remember: Conservative forces Conserve Mechanical Energy. Non-conservative forces Negate Mechanical Energy.