Work as a scalar product
The scalar (dot) product of two vectors is A · B = AB cos θ, where θ is the angle between them. It is a scalar. In components, A · B = AₓBₓ + A_yB_y + A_zB_z. Note that A · B = 0 when the vectors are perpendicular.
The work done by a constant force F producing a displacement d is
W = F · d = Fd cos θ
0° ≤ θ < 90°: W is positive (a force pulling a trolley forward).
θ = 90°: W = 0 (gravity on a body moving horizontally; the normal force).
90° < θ ≤ 180°: W is negative (friction on a sliding block).
The unit is the joule (1 J = 1 N m). In atomic physics, the electron volt is used: 1 eV = 1.6 × 10⁻¹⁹ J.
Work done by a variable force
If the force changes with position, the work is the area under the force-displacement graph:
W = ∫ F(x) dx from xᵢ to x_f.
The work-energy theorem
Kinetic energy K = ½mv². The work-energy theorem states that
The work done by the net force on a body equals the change in its kinetic energy: W_net = K_f − K_i.
Example. A 0.05 kg bullet moving at 200 m/s passes through a wooden block and emerges at 100 m/s. The work done by the block on it is ½ × 0.05 × (100² − 200²) = −750 J. If the block is 0.1 m thick, the average resistive force is 750 ⁄ 0.1 = 7500 N.
Conservative forces and potential energy
A force is conservative if the work it does depends only on the start and end points, not on the path, and is zero around any closed path. Gravity and spring forces are conservative; friction is not.
For conservative forces we define potential energy V, with F = −dV/dx.
Gravitational PE near the Earth's surface: V = mgh.
If only conservative forces act, the total mechanical energy K + V is conserved.
Example. A ball dropped from 20 m (g = 10 m/s²) hits the ground at v = √(2gh) = √400 = 20 m/s, regardless of its mass.
The spring
A spring obeys Hooke's law: F = −kx, where x is the extension or compression and k the spring constant (N/m). The minus sign means the force opposes the displacement.
The potential energy stored in a spring:
V = ½kx²
Example. A block of mass 1 kg moving at 2 m/s hits a spring of k = 400 N/m on a smooth surface. Its kinetic energy, ½ × 1 × 4 = 2 J, becomes spring energy: ½ × 400 × x² = 2, so the maximum compression x = 0.1 m.
The law of conservation of energy
Energy can change form but the total energy of an isolated system is constant. When friction acts, mechanical energy decreases, but an equal amount of heat is produced. Einstein showed mass itself is a form of energy: E = mc².
Power
Power is the rate of doing work: average power P = W/t, and instantaneous power P = F · v.
The unit is the watt (W). 1 horsepower (hp) = 746 W. The commercial unit is the kilowatt hour: 1 kWh = 3.6 × 10⁶ J.
Example. A lift of mass 1000 kg moves up at a constant 2 m/s against a friction force of 2000 N. The motor must supply F = mg + f = 10,000 + 2,000 = 12,000 N, and its power is 12,000 × 2 = 24 kW.
Collisions
In all collisions where no external force acts, linear momentum is conserved. Kinetic energy may or may not be.
Type | Momentum | Kinetic energy |
|---|---|---|
Elastic | conserved | conserved |
Inelastic | conserved | not conserved (some becomes heat, sound, deformation) |
Perfectly inelastic | conserved | maximum loss; the bodies stick together |
Elastic collision in one dimension, mass m₁ (velocity u₁) hitting m₂ at rest:
v₁ = (m₁ − m₂)u₁ ⁄ (m₁ + m₂) and v₂ = 2m₁u₁ ⁄ (m₁ + m₂)
If m₁ = m₂, the bodies exchange velocities: the first stops and the second moves off with u₁. That is why a moving billiard ball can stop dead after a head-on hit on an identical ball.
A light ball hitting a very heavy one bounces back with almost the same speed.
Perfectly inelastic example. A 2 kg block at 6 m/s strikes a 1 kg block at rest and they stick. Final velocity = (2 × 6) ⁄ 3 = 4 m/s. KE before = 36 J, after = ½ × 3 × 16 = 24 J, so 12 J is lost.
Summary
W = F · d = Fd cos θ; for a variable force, the area under the F-x graph.
W_net = ΔK (work-energy theorem).
Conservative forces have a potential energy; spring PE = ½kx².
P = F · v; momentum is conserved in all collisions, KE only in elastic ones.