Formula Sheet: Work and Energy

Work and Energy Formula Sheet

Work

Definition of Work

• Work done by constant force: \[W = F \cdot d \cdot \cos\theta\]
  • \(W\) = work (J = joules = N·m)
  • \(F\) = magnitude of force (N)
  • \(d\) = displacement (m)
  • \(\theta\) = angle between force and displacement vectors
  • Note: Work is a scalar quantity
  • Positive work: \(\theta < 90°\)="" (force="" component="" in="" direction="" of="">
  • Negative work: \(\theta > 90°\) (force component opposite to motion)
  • Zero work: \(\theta = 90°\) (force perpendicular to displacement) or \(d = 0\)
• Alternative form: \[W = \vec{F} \cdot \vec{d}\]
  • Vector dot product notation
• Work done by variable force: \[W = \int_{x_1}^{x_2} F(x) \, dx\]
  • Area under force vs. displacement graph

Special Cases

• Work done by gravity: \[W_g = mgh\]
  • \(m\) = mass (kg)
  • \(g\) = gravitational acceleration (9.8 m/s²)
  • \(h\) = vertical displacement (m)
  • Positive when object moves downward
  • Negative when object moves upward
• Work done by spring force: \[W_s = -\frac{1}{2}k(x_f^2 - x_i^2)\]
  • \(k\) = spring constant (N/m)
  • \(x_f\) = final displacement from equilibrium (m)
  • \(x_i\) = initial displacement from equilibrium (m)
• Net work (Work-Energy Theorem): \[W_{net} = \Delta KE = KE_f - KE_i\]
  • Net work equals change in kinetic energy

Energy

Kinetic Energy

• Translational kinetic energy: \[KE = \frac{1}{2}mv^2\]
  • \(KE\) = kinetic energy (J)
  • \(m\) = mass (kg)
  • \(v\) = speed (m/s)
  • Note: Energy is always positive; depends on speed squared
• Rotational kinetic energy: \[KE_{rot} = \frac{1}{2}I\omega^2\]
  • \(I\) = moment of inertia (kg·m²)
  • \(\omega\) = angular velocity (rad/s)

Potential Energy

• Gravitational potential energy (near Earth's surface): \[PE_g = mgh\]
  • \(PE_g\) = gravitational potential energy (J)
  • \(m\) = mass (kg)
  • \(g\) = gravitational acceleration (9.8 m/s²)
  • \(h\) = height above reference point (m)
  • Note: Reference point (h = 0) can be chosen arbitrarily
• Elastic potential energy (spring): \[PE_s = \frac{1}{2}kx^2\]
  • \(PE_s\) = elastic potential energy (J)
  • \(k\) = spring constant (N/m)
  • \(x\) = displacement from equilibrium position (m)
  • Note: Always positive; equilibrium position is reference (PE = 0)
• Gravitational potential energy (general): \[PE_g = -\frac{GMm}{r}\]
  • \(G\) = gravitational constant (6.67 × 10^-11 N·m²/kg²)
  • \(M\) = mass of large object (kg)
  • \(m\) = mass of small object (kg)
  • \(r\) = distance between centers (m)
  • Note: Negative value; zero at infinite separation
• Electric potential energy: \[PE_e = \frac{kq_1q_2}{r}\]
  • \(k\) = Coulomb's constant (8.99 × 10⁹ N·m²/C²)
  • \(q_1, q_2\) = charges (C)
  • \(r\) = distance between charges (m)
  • Positive for like charges (repulsive)
  • Negative for opposite charges (attractive)
• Alternative form for electric potential energy: \[PE_e = qV\]
  • \(q\) = charge (C)
  • \(V\) = electric potential (V = volts)

Total Mechanical Energy

• Total mechanical energy: \[E = KE + PE\]
  • \(E\) = total mechanical energy (J)
  • Sum of all kinetic and potential energies in system

Conservation of Energy

Conservative Forces

• Definition: Forces for which work done is path-independent
  • Examples: gravity, elastic (spring) force, electric force
  • Work done over closed path = 0
  • Associated with potential energy
• Conservation of mechanical energy (conservative forces only): \[E_i = E_f\] \[KE_i + PE_i = KE_f + PE_f\]
  • Total mechanical energy remains constant
  • Condition: No non-conservative forces do work
• Expanded form: \[\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f\]
  • For gravitational systems without friction

Non-Conservative Forces

• Definition: Forces for which work done is path-dependent
  • Examples: friction, air resistance, tension, normal force
  • Generally dissipate mechanical energy as heat
• Work done by friction: \[W_f = -f_k \cdot d = -\mu_k N \cdot d\]
  • \(f_k\) = kinetic friction force (N)
  • \(\mu_k\) = coefficient of kinetic friction (unitless)
  • \(N\) = normal force (N)
  • \(d\) = distance traveled (m)
  • Note: Always negative (opposes motion)
• Energy conservation with non-conservative forces: \[E_i + W_{nc} = E_f\] \[KE_i + PE_i + W_{nc} = KE_f + PE_f\]
  • \(W_{nc}\) = work done by non-conservative forces (J)
• Energy dissipated by friction: \[E_{dissipated} = |W_f| = \mu_k N \cdot d\]
  • Mechanical energy lost (converted to heat)

Universal Energy Conservation

• Total energy of isolated system: \[E_{total} = constant\]
  • Energy cannot be created or destroyed, only transformed
  • Includes all forms: mechanical, thermal, chemical, nuclear, etc.

Power

Definition of Power

• Average power: \[P_{avg} = \frac{W}{t}\]
  • \(P\) = power (W = watts = J/s)
  • \(W\) = work done (J)
  • \(t\) = time interval (s)
• Instantaneous power: \[P = \frac{dW}{dt}\]
  • Rate of doing work at specific instant
• Power in terms of force and velocity: \[P = F \cdot v \cdot \cos\theta\]
  • \(F\) = magnitude of force (N)
  • \(v\) = velocity (m/s)
  • \(\theta\) = angle between force and velocity vectors
• Alternative form: \[P = \vec{F} \cdot \vec{v}\]
  • Vector dot product notation
• When force parallel to velocity: \[P = Fv\]

Power and Energy

• Energy delivered: \[E = P \cdot t\]
  • For constant power
  • \(E\) = energy (J)
  • \(P\) = power (W)
  • \(t\) = time (s)
• Power in terms of energy change: \[P = \frac{\Delta E}{t}\]
  • Rate of energy transfer or transformation

Common Power Units

• Watt (W): 1 W = 1 J/s
• Kilowatt (kW): 1 kW = 1000 W
• Horsepower (hp): 1 hp ≈ 746 W
• Kilowatt-hour (kWh): 1 kWh = 3.6 × 10⁶ J
  • Note: kWh is unit of energy, not power

Efficiency

• Efficiency: \[\eta = \frac{W_{out}}{W_{in}} = \frac{E_{out}}{E_{in}} = \frac{P_{out}}{P_{in}}\]
  • \(\eta\) = efficiency (unitless, often expressed as percentage)
  • \(W_{out}\) = useful work output (J)
  • \(W_{in}\) = total work input (J)
  • Range: 0 ≤ \(\eta\) ≤ 1 (or 0% to 100%)
  • Note: Always less than 1 for real machines due to energy dissipation
• Percentage efficiency: \[\eta \% = \frac{W_{out}}{W_{in}} \times 100\%\]
• Energy loss: \[E_{lost} = E_{in} - E_{out}\]
  • Usually dissipated as heat due to friction

Special Systems and Applications

Inclined Planes

• Gravitational PE change on incline: \[\Delta PE = mgh = mg(L\sin\theta)\]
  • \(L\) = length along incline (m)
  • \(\theta\) = angle of incline
  • \(h\) = vertical height (m)
• Work against friction on incline: \[W_f = \mu_k mg\cos\theta \cdot L\]
  • \(\mu_k\) = coefficient of kinetic friction
  • \(L\) = distance traveled along incline

Pendulum

• Energy at highest point (amplitude): \[E = mgh = mgL(1 - \cos\theta_{max})\]
  • \(L\) = length of pendulum (m)
  • \(\theta_{max}\) = maximum angular displacement
  • All potential energy, no kinetic energy
• Energy at lowest point: \[E = \frac{1}{2}mv_{max}^2\]
  • All kinetic energy, minimum potential energy
• Conservation for pendulum: \[mgL(1 - \cos\theta_{max}) = \frac{1}{2}mv_{max}^2\]

Projectile Motion

• Energy conservation in projectile motion: \[\frac{1}{2}mv_0^2 = \frac{1}{2}mv^2 + mgh\]
  • \(v_0\) = initial speed (m/s)
  • \(v\) = speed at height h (m/s)
  • Note: Ignores air resistance
• Maximum height from energy: \[h_{max} = \frac{v_{0y}^2}{2g}\]
  • \(v_{0y}\) = initial vertical velocity component (m/s)

Circular Motion

• Minimum speed at top of vertical circle: \[v_{top} = \sqrt{gr}\]
  • \(r\) = radius of circle (m)
  • Required to maintain circular path with tension ≥ 0
• Energy conservation in vertical circle: \[\frac{1}{2}mv_{bottom}^2 = \frac{1}{2}mv_{top}^2 + mg(2r)\]
  • Comparing bottom and top of circle

Collisions (Energy Perspective)

• Elastic collision:
  • Both momentum and kinetic energy conserved
  • \(KE_i = KE_f\)
  • \(\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2\)
• Inelastic collision:
  • Momentum conserved, kinetic energy NOT conserved
  • \(KE_f <>
  • Energy lost to heat, sound, deformation
• Energy lost in inelastic collision: \[\Delta KE = KE_i - KE_f\]
  • Always positive (energy lost from mechanical system)
• Perfectly inelastic collision (objects stick together): \[\Delta KE = \frac{1}{2}\frac{m_1m_2}{m_1 + m_2}(v_{1i} - v_{2i})^2\]
  • Maximum kinetic energy loss for given initial velocities

Important Relationships and Concepts

Work-Energy Theorem Applications

• Net work equals change in kinetic energy: \[W_{net} = \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2)\]
  • True regardless of path taken
  • Accounts for all forces acting on object
• Relating acceleration to work: \[W_{net} = ma \cdot d\]
  • For constant acceleration along straight line
  • Combines with \(v_f^2 = v_i^2 + 2ad\)

Energy Transformations

• Common transformations:
  • Gravitational PE ↔ KE (falling/rising objects)
  • Elastic PE ↔ KE (spring systems)
  • Electric PE ↔ KE (charged particles in fields)
  • Mechanical energy → Thermal energy (friction)
  • Chemical energy → Mechanical energy (muscles, engines)

Sign Conventions

• Work:
  • Positive: force has component in direction of displacement
  • Negative: force has component opposite to displacement
  • Zero: force perpendicular to displacement or no displacement
• Potential energy changes:
  • ΔPE > 0: energy stored (work done against conservative force)
  • ΔPE < 0:="" energy="" released="" (conservative="" force="" does="" positive="">

Key Distinctions

• Energy vs. Work:
  • Energy: capacity to do work (state function)
  • Work: energy transfer via force over displacement (process function)
• Conservative vs. Non-conservative forces:
  • Conservative: path-independent, reversible, have PE associated
  • Non-conservative: path-dependent, irreversible, no PE function
• Power vs. Energy:
  • Power: rate of energy transfer (J/s)
  • Energy: total amount transferred (J)

Problem-Solving Strategies

Choosing Energy Methods

• Use energy methods when:
  • Path is complex or unknown
  • Finding final velocity from initial conditions
  • Force varies with position
  • Time information not needed
• Use force/kinematics methods when:
  • Acceleration or force information needed
  • Time-dependent information required
  • Analyzing specific points along path

General Approach

• Step 1: Identify system and choose reference points (PE = 0)
• Step 2: Determine initial and final states
• Step 3: Calculate all energies at each state
• Step 4: Apply conservation or work-energy theorem
• Step 5: Account for non-conservative forces if present
• Step 6: Solve for unknown quantity