Work, Energy and Power

The One Thing

Work is the transfer of energy by a force acting through displacement, while energy conservation links changes between kinetic, potential, thermal and other forms. The work–energy theorem, power, and conservation laws provide the main methods for analysing motion, energy transfer and collisions.

Reactions, Processes and Experiments

Key Terms

• Work: Work is done when a force causes displacement. For a constant force, $W=Fs\cos \theta$, where $\theta$ is the angle between force and displacement.
• Joule: The SI unit of work and energy. One joule is the work done by a force of one newton producing a displacement of one metre in its direction.
• Kinetic Energy: The energy possessed by an object because of its motion, given by $K=1/2m{v}^2$.
• Potential Energy: Stored energy due to position, configuration, or deformation. Near Earth's surface, gravitational potential energy is $U=mgh$.
• Work-Energy Theorem: The net work done on a body equals the change in its kinetic energy: $W_{\text{net}}=\Delta K=K_{\text{final}}-K_{\text{initial}}$.
• Conservative Force: A force for which work depends only on the initial and final positions, not on the path followed. Gravity and spring force are examples.
• Non-Conservative Force: A force for which work depends on the path. Friction and air resistance are common examples, and they usually convert mechanical energy into thermal energy.
• Mechanical Energy: The sum of kinetic and potential energies: $E_{\text{mechanical}}=K+U$.
• Conservation of Mechanical Energy: When only conservative forces act, the sum of kinetic and potential energies remains constant.
• Power: The rate at which work is done or energy is transferred. Average power is $P=W/t$, and instantaneous power is $P=dW/dt=F\text{ dot }v$.
• Watt: The SI unit of power. One watt equals one joule per second.
• Linear Momentum: The product of mass and velocity: $p=mv$. Momentum is a vector quantity.
• Impulse: The product of force and the time interval for which it acts. Impulse equals the change in momentum: $J=F\Delta t=\Delta p$.
• Collision: A short-duration interaction between bodies during which large forces may act and momentum is transferred.
• Elastic Collision: A collision in which both total momentum and total kinetic energy are conserved.
• Inelastic Collision: A collision in which total momentum is conserved but kinetic energy is not conserved.
• Perfectly Inelastic Collision: A collision in which the bodies stick together after impact and move with a common velocity.
• Coefficient of Restitution: The ratio of relative speed of separation to relative speed of approach, expressed as $e=(v_2-v_1)/(u_1-u_2)$ for a suitable one-dimensional sign convention.
• Efficiency: The useful output energy or power divided by the input energy or power, multiplied by $100$ percent.

Easily Confused

• Conservative force and non-conservative force: Work by a conservative force depends only on initial and final positions, whereas work by a non-conservative force depends on the path.
• Mechanical energy and total energy: Mechanical energy is $K+U$; total energy also includes thermal energy and other transferred or transformed forms.
• Elastic and inelastic collision: Both conserve momentum in an isolated system, but only an elastic collision conserves total kinetic energy.
• Inelastic and perfectly inelastic collision: In every inelastic collision kinetic energy is not conserved, but only in a perfectly inelastic collision do the bodies stick together.
• Average power and instantaneous power: Average power is $P_{\text{average}}=W/t$, whereas instantaneous power is $P_{\text{instantaneous}}=dW/dt=F\text{ dot }v$.
• Work and power: Work measures energy transfer; power measures the rate of that transfer.
• Zero work and zero force: Work can be zero when the point of application has no displacement or when the force is perpendicular to displacement, even if the force is non-zero.
• Momentum conservation and kinetic-energy conservation: Momentum is conserved in an isolated collision, but kinetic energy is conserved only in an elastic collision.
• Perfectly elastic and perfectly inelastic collisions: For a perfectly elastic collision $e=1$; for a perfectly inelastic collision $e=0$.

What Gets Asked

• Calculate work using $W=Fs\cos \theta$, identifying whether the force and displacement are parallel, perpendicular or opposite. A common mark-losing error is treating perpendicular forces as doing non-zero work.
• Use the work–energy theorem, $W_{\text{net}}=\Delta K$, to determine changes in speed. The key slip is using the work done by one force instead of the net work.
• Apply energy conservation using $K=1/2m{v}^2$, $U=mgh$, and $U=1/2k{x}^2$. Mechanical energy may be conserved only when non-conservative forces do no net work.
• Determine average or instantaneous power using $P_{\text{average}}=W/t$ and $P_{\text{instantaneous}}=dW/dt=F\text{ dot }v$. The distinction between energy transferred and the rate of transfer is essential.
• Solve collision problems using momentum conservation and, where appropriate, kinetic-energy conservation or the coefficient of restitution. The main error is assuming kinetic energy is conserved in every collision.
• Analyse perfectly inelastic collisions using $v=(m_1{u}_1+m_2{u}_2)/(m_1+m_2)$. The required condition is that the bodies stick together and share a common final velocity.