Chapter 5: Work, Energy and Power | Summary

Work done by a constant force F on an object undergoing displacement d is defined as W = F·d = Fd cos θ, where θ is the angle between the force and displacement vectors. Work is a scalar quantity with SI unit joule (J) and dimensions [ML²T⁻²]. It can be positive, negative, or zero depending on the angle between force and displacement.

No work is done when: (i) displacement is zero (e.g., pushing a rigid wall), (ii) force is zero, or (iii) force and displacement are mutually perpendicular (e.g., gravitational force on an object moving horizontally). Friction typically does negative work since it opposes displacement. Alternative units of energy include erg (10⁻⁷ J), electron volt (1.6×10⁻¹⁹ J), calorie (4.186 J), and kilowatt hour (3.6×10⁶ J).

The kinetic energy of an object of mass m moving with velocity v is K = (1/2)mv². It is always a positive scalar quantity. Kinetic energy represents the work an object can perform by virtue of its motion. Examples include the kinetic energy of flowing streams used to grind corn, and wind energy used in sailing ships.

Kinetic energy depends on both mass and the square of speed. A bullet of mass 5×10⁻² kg moving at 200 m/s has kinetic energy of 10³ J, while a car of 2000 kg at 25 m/s has 6.3×10⁵ J. The work-energy theorem connects changes in kinetic energy to net work done, and is valid for all types of forces acting on a body.

When a force varies with position, the work done cannot be computed directly using W = Fd cos θ. Instead, for small displacements Δx, the work element is ΔW ≈ F(x)Δx. Summing over all such small displacements and taking the limit as Δx → 0, the total work is expressed as the definite integral W = ∫F(x)dx from xi to xf. Graphically, this equals the area under the F-x curve between initial and final positions.

The work-energy theorem for a variable force follows from integrating Newton's second law: dK = F dx, which upon integration gives Kf − Ki = ∫F dx = W. This confirms that the work-energy theorem holds for variable forces as well. The theorem provides energy information but not temporal details, unlike Newton's second law which gives instantaneous acceleration.

Potential energy is the energy stored in a body by virtue of its position or configuration. It is applicable only to conservative forces, where the work done against the force gets stored and can be fully recovered as kinetic energy. For example, lifting a ball to height h stores gravitational potential energy V(h) = mgh, which converts entirely to kinetic energy when the ball is released.

For a conservative force in one dimension, the potential energy function V(x) is defined such that F(x) = −dV/dx. This means the force equals the negative gradient of potential energy. The change in potential energy ΔV equals the negative of work done by the conservative force: ΔV = −F(x)Δx. Non-conservative forces like friction do not have an associated potential energy.

The total mechanical energy E = K + V of a system remains constant if only conservative forces act on it. This follows from the work-energy theorem combined with the definition of potential energy: ΔK + ΔV = 0, or Δ(K + V) = 0. The individual values of K and V may change from point to point, but their sum remains constant throughout the motion.

A conservative force can be defined in three equivalent ways: it is derivable from a potential energy function; its work depends only on end points and not the path; and its work over any closed path is zero. When non-conservative forces like friction are also present, the mechanical energy is not conserved, and the change in mechanical energy equals the work done by non-conservative forces: Ef − Ei = Wnc.

The spring force follows Hooke's Law: Fs = −kx, where k is the spring constant (unit: N m⁻¹) and x is the displacement from the equilibrium position. The negative sign shows the restoring nature of the force. The work done by the spring force when the block moves from xi to xf is Ws = (kxi²/2) − (kxf²/2), depending only on end positions, confirming the spring force is conservative.

The potential energy of a spring stretched or compressed by x is V(x) = kx²/2. At the equilibrium position (x = 0), V = 0 and kinetic energy is maximum. At maximum displacement xm, all energy is potential. The total mechanical energy E = (1/2)kxm² remains constant, and the kinetic and potential energies vary as complementary parabolic functions of displacement as shown in Fig. 5.8.

Power is defined as the time rate of doing work or transferring energy. Average power Pav = W/t, and instantaneous power P = dW/dt = F·v, where v is the instantaneous velocity. Power is a scalar quantity with dimensions [ML²T⁻³] and SI unit watt (W), equal to 1 joule per second. The unit is named after James Watt, inventor of the steam engine.

Another unit of power is the horsepower: 1 hp = 746 W, still used for automobiles and engines. Energy consumption in electrical appliances is measured in kilowatt hours (kWh): 1 kWh = 3.6×10⁶ J. A 100 W bulb burning for 10 hours uses 1 kWh of energy. Note that kWh is a unit of energy, not power.

In all collisions, total linear momentum is conserved due to Newton's third law: the mutual impulsive forces are equal and opposite, so the total change in momentum is zero. However, kinetic energy need not be conserved. An elastic collision conserves both momentum and kinetic energy. A completely inelastic collision is one where the bodies move together after collision, resulting in maximum kinetic energy loss. An inelastic collision lies in between, with partial recovery.

For a one-dimensional elastic collision between m1 (moving) and m2 (at rest): v1f = (m1−m2)/(m1+m2)·v1i and v2f = 2m1/(m1+m2)·v1i. When masses are equal, m1 comes to rest and m2 moves with the initial speed of m1. When m2 >> m1, the lighter mass reverses velocity while the heavier mass remains nearly undisturbed. For two-dimensional collisions, momentum conservation gives two component equations, and an elastic collision adds a third energy equation.

Chapter 5 develops a thorough understanding of work, energy, and power as precisely defined physical quantities. Work is done only when a force causes displacement with a non-perpendicular component. The work-energy theorem elegantly links mechanics to energy, showing that the net work on a body equals its change in kinetic energy. Conservative forces allow the definition of potential energy, and the principle of conservation of mechanical energy emerges naturally. The spring force illustrates how kinetic and potential energy interconvert while total mechanical energy is preserved. Power quantifies the rate of energy transfer, and collisions demonstrate the universal conservation of momentum alongside the conditional conservation of kinetic energy.

For CBSE board exams, this chapter is highly important as it introduces energy methods that offer alternatives to Newton's force-based approach. Questions frequently test the work-energy theorem, conservation of mechanical energy in spring-block and pendulum systems, calculation of power, and analysis of elastic versus inelastic collisions. Understanding the conditions under which work is zero, the distinction between conservative and non-conservative forces, and the special cases of one-dimensional elastic collisions are particularly testable. Numericals involving spring compression with and without friction (Examples 5.8 and 5.9) are model problems for board-level problem-solving.