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Work, Energy and Power

Energy is the ability to do work.

Work done by a Force

$$W = F↖{→}scosθ$$

where $W$ (scalar) done by a constant force F applied to a mass at angle θ to the resultant displacement $s$.

The work done by a varying force is equal to the area under the graph of force versus displacement.

The work-kinetic energy relation: the work done = change in kinetic energy:

$$W = ΔE_K$$

Where there are external forces acting on a system, the work done = change in mechanical energy:

$$W = ΔE$$

Kinetic Energy

$$v_f^2 = v_0^2 + 2ax$$ $$v_f^2 = v_0^2 + 2{F}/{m}x$$ $$Fx = 1/2mv_f^2 - 1/2mv_0^2$$ ∴ $$W = ΔE_K$$
$$E_K = 1/2mv^2$$

where $E_K$ is the kinetic energy of a mass $m$ moving at constant velocity $v$.

$$E_K = {p^2}/{2m}$$

where $E_K$ is the kinetic energy of a mass $m$ moving with constant momentum $p$.

Kinetic is a scalar (not a vector, so has no direction component) value, and cannot be negative.

Potential Energy

$$E_P = mgh$$

where $E_P$ is the potential energy of a mass $m$ at height $h$ in a gravitational field $g$.

$$E_e = 1/2kx^2$$

where $E_e$ is the elastic potential energy of a mass compressing or extending a spring with force constant $k$ by a distance $x$.

The elastic potential energy of a mass moving from extension $x_1$ to $x_2$ is:

$$E_e = 1/2k(x_2^2 - x_1^2)$$

Conservation of Energy

When there is no friction, the total energy of a system is constant:

$$E = E_K + E_P + E_e = 1/2mv^2 + mgh + 1/2kx^2$$

In all collisions, momentum is conserved, but only in elastic collisions kinetic energy is also conserved.

Power

$$P = {ΔW}/{Δt}$$

where the $P$ is the power of a system experiencing a change in work $W$ in time interval $Δt$. Power may also be expressed as:

$$P = Fv$$

Efficiency

A mass is raised up a ramp at an angle of θ to the horizontal. Since the speed of ascent is constant, there is no net force, so the frictional force is equal to the applied force.

What proportion of the energy is used to raise the mass?

The law of conservation of energy dictates that energy is neither gained or lost by a closed system. Energy is put into the system for the purposes of raising the mass, not to heat the ramp with friction.

Therefore, the useful work is work which achieves the aim of the energy input.

Efficiency is the proportion of useful work to the total work done (actual work). Efficiency has the symbol η (eta) and has no units, since it is a ratio of quantities with the same unit.

$$η = {\text"useful work"}/{\text"actual work"}$$

In reality, efficiency is always less than 1.0, since any conversion of energy involves some loss to undesired forms.

A bicycle has the impressive efficiency of over 0.9, while a petrol car has efficiency of 0.1 - 0.15, or 10-15%.

Content © Andrew Bone. All rights reserved. Created : September 2, 2013 Last updated :November 3, 2015

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