Work and Energy in Physics

• Work and energy are fundamental concepts in physics, closely related and essential for understanding motion and forces.

Work

• Work occurs when a force acts on an object, resulting in its displacement.
• Work is a scalar quantity possessing magnitude but lacking direction.
• The magnitude of work is the product of the force applied and the distance over which it causes displacement.
• If force and displacement aren't parallel, use the component of force along the displacement direction.
• The formula for work done by a constant force is W = Fd cosθ, with W denoting work, F as the force magnitude, d as the displacement magnitude, and θ as the angle between force and displacement vectors.
• The SI unit of work is the joule (J), where 1 J = 1 N·m.
• Work can be positive, negative, or zero; positive work signifies force aiding motion, negative means force opposing, and zero indicates no displacement, no force, or force perpendicular to displacement.
• For variable forces, work is W = ∫F(x) dx from x1 to x2.
• Work done by a variable force equals the area under the force vs. displacement curve.

Energy

• Energy signifies the capacity to perform work.
• It is a scalar quantity and it is measured in joules (J).
• Energy manifests in various forms: kinetic, potential, thermal, electromagnetic, and nuclear.

Kinetic Energy

• Kinetic energy (KE) is an object's energy due to its motion.
• It relies on mass (m) and velocity (v).
• KE = (1/2)mv².
• Kinetic energy is always positive and remains a scalar quantity.
• The work-energy theorem states Wnet = ΔKE = KEf - KEi, where net work equals the change in kinetic energy.

Potential Energy

• Potential energy (PE) is stored energy based on position or condition.
• It can be converted into kinetic or other energy forms.
• Gravitational potential energy (GPE) is energy from an object's height above a reference.
• GPE = mgh, with m as mass, g as gravitational acceleration, and h as height above the chosen reference, often the ground.
• Elastic potential energy (EPE) is energy stored in a spring when stretched or compressed.
• EPE = (1/2)kx², with k as the spring constant and x as displacement from equilibrium.

Conservative and Non-Conservative Forces

• A conservative force's work is path-independent between two points.
• Work by conservative forces is recoverable as kinetic energy.
• Gravitational and elastic (spring) forces exemplify conservative forces.
• Potential energy is definable only for conservative forces.
• A non-conservative force's work depends on the path taken between two points.
• Work done by non-conservative forces isn't fully recoverable as kinetic energy; some dissipates as heat.
• Friction, air resistance, and rope tension are examples of non-conservative forces.

Conservation of Energy

• The conservation of energy principle: the total energy in an isolated system remains constant.
• Energy transforms between forms but isn't created or destroyed.
• With only conservative forces, total mechanical energy (E) is conserved: E = KE + PE = constant.
• With non-conservative forces, their work equals mechanical energy change: Wnc = ΔE = (KEf + PEf) - (KEi + KEi).

Power

• Power is the rate of work done or energy transferred.
• It is a scalar quantity measured in watts (W).
• 1 watt (W) = 1 joule per second (J/s).
• Power is calculated as P = W/t, with P as power, W as work, and t as time.
• Power is also expressed as P = Fv cosθ, with F as force, v as velocity, and θ as the angle between force and velocity vectors.
• Average power is the total work divided by the total time.
• Instantaneous power is the power at a specific moment, representing the limit of average power as the time interval approaches zero.

Work-Energy Theorem

• The work-energy theorem links net work and kinetic energy change directly.
• It is a scalar quantity, possessing magnitude but lacking direction, is useful for solving motion and force problems independently of kinematic equations.
• Wnet = ΔKE = (1/2)mvf² - (1/2)mvi², with vf as final and vi as initial velocities.

Problem-Solving Strategies

• Identify all forces and classify them as conservative or non-conservative.
• Define a reference point for potential energy.
• Use the work-energy theorem or energy conservation to relate initial and final system states.
• Account for work by non-conservative forces like friction.
• Use power calculations to find the rates of work or energy transfer.