Work and Energy

• Work and energy are fundamental in physics for understanding motion, forces, and energy transfer.
• Work occurs when a force displaces an object.
• Energy is the capacity to perform work.

Work

• Work is the energy transferred to or from an object by a force acting upon it.
• Work is a scalar quantity.
• Work done is the product of the force component along the displacement and the displacement's magnitude.
• Only the force component parallel to the displacement performs work.
• Given force F and displacement d, work W = F · d · cos(θ), where θ is the angle between force and displacement vectors.
• Work is positive when force and displacement are in the same direction (0° ≤ θ < 90°).
• Work is negative when force and displacement are in opposite directions (90° < θ ≤ 180°).
• No work is done if the force is perpendicular to the displacement (θ = 90°).
• The SI unit for work is the joule (J), equivalent to 1 N·m.
• For a varying force, work is calculated by integrating the force over displacement: W = ∫ F(x) dx, from x1 to x2.
• The integral equals the area under the force-displacement curve.

Energy

• Energy is the property that must be transferred to an object to perform work or heat it.
• Energy is conserved.
• Energy exists as kinetic, potential, thermal, electromagnetic, and nuclear forms.
• The SI unit of energy is the joule (J), identical to work.

Kinetic Energy

• Kinetic energy (KE) is the energy of an object due to its motion.
• KE is a scalar quantity.
• For mass m moving at velocity v, KE = (1/2) · m · v².
• The work-energy theorem: Wnet = ΔKE = KEf - KEi, where net work equals the change in kinetic energy.
• KEf and KEi represent the final and initial kinetic energies, respectively.

Potential Energy

• Potential energy (PE) is stored energy due to an object's position or condition.
• Gravitational potential energy depends on an object's height above a reference.
• PE due to gravity: PE = m · g · h, where m is mass, g is approximately 9.8 m/s² on Earth, and h is height.
• Elastic potential energy is stored in deformable objects like springs under stretch or compression.
• Spring potential energy: PE = (1/2) · k · x², where k is the spring constant and x is displacement from equilibrium.
• Only conservative forces are associated with potential energies.

Conservative Forces

• Conservative forces do work independent of the path taken between two points.
• Examples: gravitational force, elastic force (spring), and electrostatic force.
• Work by a conservative force: Wc = -ΔPE, the negative change in potential energy.
• Total mechanical energy (E) is the sum of kinetic and potential energies: E = KE + PE.
• In a closed system with only conservative forces, total mechanical energy is conserved: Ei = Ef, or KEi + PEi = KEf + PEf.

Non-Conservative Forces

• Non-conservative force work depends on the path taken between two points.
• Examples: friction, air resistance, tension, and applied forces.
• Work by non-conservative forces dissipates as thermal energy, not stored as potential energy.
• Total mechanical energy is not conserved when non-conservative forces are present.
• Wnc = ΔE = (KEf + PEf) - (KEi + PEi)

Power

• Power is the rate of work done or energy transferred.
• Power is a scalar quantity.
• Average power: Pavg = W / t, where W is work done and t is the time interval.
• Instantaneous power: P = dW / dt.
• Power can also be expressed as P = F · v · cos(θ), where v is velocity, since W = F · d · cos(θ).
• The SI unit of power is the watt (W), equivalent to 1 J/s.
• Horsepower (hp) is another power unit: 1 hp ≈ 746 W.

Work-Energy Theorem

• Connects work done on an object to its change in kinetic energy.
• Net work equals the change in kinetic energy: Wnet = ΔKE.
• Valid for constant and varying forces, simplifying motion analysis.
• Change in speed can be determined by calculating net work, without detailed force analysis.

Conservation of Energy

• Energy cannot be created or destroyed, only transformed or transferred.
• Total energy in a closed system remains constant.
• This principle is a fundamental analysis tool for physical systems.
• Account for all energy forms, including kinetic, potential, and thermal, when considering energy transformations.
• ΔE = ΔKE + ΔPE + ΔU = 0, where ΔU is the change in internal energy.

Efficiency

• Efficiency measures the conversion of input energy/work into useful output energy/work.
• Defined as: η = (Useful Energy Output / Total Energy Input) × 100%.
• Always less than 100% because of energy losses from friction, heat, and sound.
• Indicates how well machines and engines convert input energy into useful work/power.
• Improving conserves resources and reduces waste.