xSkill: notes, lectures and practice for CBSE and CSE

1.1 Work and the work-energy theorem Work is the scalar product of the force vector with the displacement vector, so only the part of the force that lies along the motion contributes. A force at right angles to the motion contributes nothing, which is why the centripetal force and the magnetic force leave the speed unchanged. The work done by the net force equals the change in kinetic energy, and this statement is the work-energy theorem. For a varying force, the work is the area under the force-displacement graph, and for a force at an angle the product carries a cosine. Work is a scalar and can be positive, negative or zero, and its SI unit is the joule.

W = F \cdot d = F d cos θ, W = \int_{x_{i}}^{x_{f}} F d x

Work by a constant force and by a one dimensional variable force

W_{net} = Δ K = \frac{1}{2} m v^{2} - \frac{1}{2} m u^{2}

The work-energy theorem

1.2 Potential energy A force is conservative when the work it does depends only on the endpoints and not on the path. Gravity and the spring force are conservative, and only for such forces can a potential energy be defined. The change in potential energy is the negative of the work that the conservative force does, so a body of mass m at height h carries mgh and a spring carries half k times the square of its deformation. Mechanical energy is the sum of kinetic and potential energy. When only conservative forces act, the total mechanical energy stays constant, and the kinetic and potential parts trade with each other. When friction acts, the mechanical energy falls by the work done against friction, which appears as heat.

U_{g} = m g h, U_{s} = \frac{1}{2} k x^{2}, E = K + U = constant

Gravitational and spring potential energy, and conservation of mechanical energy

W_{friction} = Δ E, Δ E = E_{final} - E_{initial}

The change in mechanical energy when a non conservative force acts

1.3 Power and collisions Power is the rate at which work is done, equal to the work divided by the time for an average and to the scalar product of force and velocity for an instantaneous value. Its unit is the watt, one joule per second. A machine that lifts a load quickly needs more power than one that lifts the same load slowly, even though the work is the same. In every collision the total momentum stays conserved when no external force acts. An elastic collision also conserves kinetic energy, while an inelastic collision loses part of it to heat and sound. For a perfectly inelastic collision the two bodies stick together and move with the common velocity given by momentum conservation. The coefficient of restitution measures the ratio of the relative separation speed to the relative approach speed, and it is one for an elastic collision and zero for a perfectly inelastic one.

P = \frac{W}{t}, P = F \cdot v, e = \frac{v _{2} - v _{1}}{u _{1} - u _{2}}

Average and instantaneous power, and the coefficient of restitution

1.4 Units and habits Work and energy are in joule, power in watt, force in newton, displacement in metre and time in second. A kilowatt hour is 3.6 million joule, which is the unit used on electricity bills. Before substituting, convert a power in horsepower to watt and a mass in gram to kilogram. The standard method is to choose the two states, write the energy or momentum equation between them, and then solve. For a single body with only conservative forces use energy; for a collision use momentum; for a machine use power. The three tools cover almost every numerical in the chapter.

Exam

Friction always opposes relative sliding, so the work done by friction is usually negative. In an energy equation the frictional loss is written as the work done against friction, a positive quantity subtracted from the initial energy.