Work-energy principle, power and efficiency

(a) Principle of work and energy

Statement: the work done by all forces acting on a particle as it moves from position 1 to position 2 equals the change in its kinetic energy.

$$U_{1\to2} = T_2 - T_1 = \tfrac12 m v_2^2 - \tfrac12 m v_1^2$$

Proof: Let the resultant force $\mathbf{F}$ act on a particle of mass $m$ moving along a path with tangential acceleration $a_t$. Newton's second law along the tangent gives $F_t = m a_t = m\,\dfrac{dv}{dt}$. Using $a_t = v\,\dfrac{dv}{ds}$:

$$F_t\,ds = m v\,dv$$

Integrating from position 1 ($s_1$, $v_1$) to position 2 ($s_2$, $v_2$):

$$\int_{s_1}^{s_2} F_t\,ds = m\int_{v_1}^{v_2} v\,dv = \tfrac12 m v_2^2 - \tfrac12 m v_1^2$$

The left side is the work $U_{1\to2}$ of the resultant force, so $T_1 + U_{1\to2} = T_2$. The normal component of the force does no work, because it is perpendicular to the displacement.

(b) Power and efficiency

Power is the rate of doing work:

$$P = \frac{dU}{dt} = \mathbf{F}\cdot\mathbf{v} = Fv\cos\alpha$$

where $\alpha$ is the angle between the force and the velocity. Unit: watt (W = J/s); 1 hp = 746 W. For rotation, $P = M\omega$ (torque times angular velocity).

Mechanical efficiency is the ratio of useful output power (or work) to input power (or work):

$$\eta = \frac{\text{output power}}{\text{input power}} = \frac{P_{out}}{P_{in}} < 1$$

The difference between input and output is lost mainly to friction and appears as heat.