6.2 Work, Power, Energy & Conservation of Linear Momentum

6.2 Work, Power, Energy & Conservation of Linear Momentum

This section covers work, mechanical energy, power, conservation of momentum, rocket propulsion, Kinetic Theory of Gases, and First Law Thermodynamics. These topics represent major content areas tested on the AMC initial examination.

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1. Work, Power & Mechanical Energy

Work Done by Constant & Variable Forces

Work $W$ is the scalar product of force $\mathbf{F}$ and displacement $\mathbf{d}$: $W = \mathbf{F} \cdot \mathbf{d} = F d \cos\theta$

• SI Unit: Joule ($\text{J} = \text{N}\cdot\text{m} = \text{kg}\cdot\text{m}^2/\text{s}^2$)
• Positive Work ($0^\circ \le \theta < 90^\circ$): Force has a component in the direction of motion (e.g., engine accelerating car).
• Zero Work ($\theta = 90^\circ$): Force is perpendicular to displacement (e.g., centripetal force in circular motion, gravity on horizontal walking path).
• Negative Work ($90^\circ < \theta \le 180^\circ$): Force opposes displacement (e.g., friction, braking force).

For a variable force, work done is the integral of force over displacement, representing the area under the force-displacement curve: $W = \int_{x_i}^{x_f} F(x) \, dx$

Mechanical Power

Power $P$ is the time rate of doing work: $P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v} = F v \cos\theta$

• SI Unit: Watt ($\text{W} = \text{J/s} = \text{kg}\cdot\text{m}^2/\text{s}^3$)
• Practical units: Horsepower ($1\text{ hp} = 746\text{ W}$), Kilowatt-hour ($1\text{ kWh} = 3.6 \times 10^6\text{ J}$).

Work-Energy Theorem & Potential Energy

• Work-Energy Theorem: Net work done on an object equals its change in kinetic energy: $W_{net} = \Delta K = K_f - K_i = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$
• Gravitational Potential Energy: $U = m g h$
• Elastic Potential Energy (Hooke's Law spring): $U = \frac{1}{2} k x^2$, where $k$ is spring constant.

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2. Linear Momentum, Collisions & Rocket Propulsion

Conservation of Linear Momentum

Linear momentum is $\mathbf{p} = m \mathbf{v}$. In an isolated system (where net external force $\sum \mathbf{F}_{ext} = 0$), total linear momentum is strictly conserved: $\sum \mathbf{p}_i = \sum \mathbf{p}_f$

Elastic vs Inelastic Collisions in 1D

• Elastic Collision: Both total linear momentum and total kinetic energy are conserved.
• Inelastic Collision: Linear momentum is conserved, but kinetic energy is not conserved (converted to heat, sound, or deformation energy).
• Perfectly Inelastic Collision: Colliding bodies stick together post-collision, moving with a common velocity $v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$.

For a 1D Head-on Elastic Collision:

1. Relative velocity of approach equals negative relative velocity of separation: $v_{1i} - v_{2i} = -(v_{1f} - v_{2f})$
2. Final velocity solutions: $v_{1f} = \left(\frac{m_1 - m_2}{m_1 + m_2}\right) v_{1i} + \left(\frac{2 m_2}{m_1 + m_2}\right) v_{2i}$ $v_{2f} = \left(\frac{2 m_1}{m_1 + m_2}\right) v_{1i} + \left(\frac{m_2 - m_1}{m_1 + m_2}\right) v_{2i}$

Special Case ($m_1 = m_2$): Bodies completely exchange their initial velocities ($v_{1f} = v_{2i}$ and $v_{2f} = v_{1i}$).

Rocket Propulsion Dynamics

A rocket ejects exhaust gas at high speed $v_e$ relative to the rocket, producing forward thrust $F_{thrust}$: $F_{thrust} = v_e \left(\frac{\Delta m}{\Delta t}\right)$ Rocket acceleration at mass $m$ under gravity $g$: $a = \frac{v_e}{m} \left(\frac{\Delta m}{\Delta t}\right) - g$

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3. Kinetic Theory of Gases & Ideal Gas Law

Kinetic Theory Postulates & Ideal Gas Equation

Ideal gas equation: $P V = n R T = N k_B T$, where $R = 8.314\text{ J/(mol}\cdot\text{K)}$ is universal gas constant and $k_B = 1.38 \times 10^{-23}\text{ J/K}$ is Boltzmann's constant.

From kinetic theory, gas pressure exerted on container walls is: $P = \frac{1}{3} \rho \langle v^2 \rangle = \frac{2}{3} N_0 \langle K \rangle$ where $N_0 = N/V$ is number density and $\langle K \rangle = \frac{1}{2} m \langle v^2 \rangle$ is mean kinetic energy.

Temperature & Molecular Speeds

Absolute temperature $T$ directly measures mean translational kinetic energy per molecule: $\langle K \rangle = \frac{3}{2} k_B T$ Root-mean-square speed ($v_{rms}$): $v_{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3 R T}{M}}$ where $M$ is molar mass.

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4. Thermodynamics: Laws & Thermal Processes

First Law of Thermodynamics

The First Law expresses energy conservation for thermodynamic systems: $\Delta Q = \Delta U + W$

• $\Delta Q$: Heat added to system ($+ \Delta Q$) or removed ($- \Delta Q$).
• $\Delta U$: Change in internal energy (state function, depends only on $T$ for ideal gas: $U = \frac{3}{2} n R T$).
• $W$: Work done by system ($+W$ during expansion) or on system ($-W$ during compression).

Four Fundamental Thermodynamic Processes

For an adiabatic process, $P V^\gamma = \text{constant}$, where $\gamma = C_p / C_v$.

Molar Heat Capacities & Carnot Engine

• Molar heat capacity relation: $C_p - C_v = R$
• Ratio of specific heats: $\gamma = \frac{C_p}{C_v}$ ($1.67$ for monoatomic gas, $1.40$ for diatomic gas).
• Thermal efficiency of heat engine: $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$
• Maximum theoretical Carnot efficiency: $\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$ (temperatures strictly in Kelvin).

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5. Worked AMC Numerical Examples

Example 1: 1D Head-on Elastic Collision Velocity Exchange

Question: A $2\text{ kg}$ sphere moving at $+10\text{ m/s}$ collides head-on elastically with an identical $2\text{ kg}$ sphere initially at rest. Find the final velocity of each sphere.

Solution:

1. Identify given values: $m_1 = m_2 = 2\text{ kg}$, $v_{1i} = 10\text{ m/s}$, $v_{2i} = 0\text{ m/s}$.
2. Use elastic collision formula for equal masses: $v_{1f} = \left(\frac{m_1 - m_2}{m_1 + m_2}\right) v_{1i} + \left(\frac{2m_2}{m_1 + m_2}\right) v_{2i} = 0 + 0 = 0\text{ m/s}$ $v_{2f} = \left(\frac{2m_1}{m_1 + m_2}\right) v_{1i} + \left(\frac{m_2 - m_1}{m_1 + m_2}\right) v_{2i} = \frac{4}{4}(10) + 0 = 10\text{ m/s}$
3. The incoming sphere stops completely ($v_{1f} = 0$), transferring all momentum to the second sphere ($v_{2f} = 10\text{ m/s}$).

Example 2: Isothermal Gas Expansion

Question: An ideal gas expands isothermally at $300\text{ K}$, performing $500\text{ J}$ of work against external pressure. Calculate heat added $\Delta Q$ and internal energy change $\Delta U$.

Solution:

1. In an isothermal process, temperature is constant: $\Delta T = 0$.
2. Since internal energy of an ideal gas depends solely on temperature, $\Delta U = 0\text{ J}$.
3. Apply First Law: $\Delta Q = \Delta U + W = 0 + 500 = 500\text{ J}$.

Example 3: Carnot Engine Efficiency

Question: A Carnot steam turbine operates between $600\text{ K}$ and $300\text{ K}$. Calculate its thermal efficiency and work output per $1000\text{ J}$ of heat absorbed.

Solution:

1. Efficiency $\eta = 1 - \frac{T_C}{T_H} = 1 - \frac{300}{600} = 0.50 = 50%$.
2. Work output: $W = \eta \times Q_H = 0.50 \times 1000\text{ J} = 500\text{ J}$.