Work, Energy & Power

Work Done by a Constant Force

In everyday speech, work means any kind of mental or physical exertion. In physics, it has a precise mechanical meaning.

Work done is defined as force multiplied by displacement in the direction of the force:

Where:

• $W$ is work done in joules ($\text{J}$)
• $F$ is the constant applied force in newtons ($\text{N}$)
• $s$ is the displacement in metres ($\text{m}$)

Work is a scalar quantity. It has magnitude but no direction in space.

When the applied force acts at an angle $\theta$ to the direction of travel, only the component along the displacement does work:

For example, if you pull a sledge across a horizontal path with a force of $50\text{ N}$ at an angle of $30^\circ$ above the ground for $4.0\text{ m}$:

The vertical component $F\sin\theta$ pulls upward against gravity, but because there is no vertical displacement, it does no work.

Two special angles appear frequently in exam problems:

• Perpendicular force ($\theta = 90^\circ$): Because $\cos 90^\circ = 0$, no work is done. A satellite in a circular orbit experiences a centripetal force directed toward the Earth, but this force acts at right angles to the satellite's velocity, so gravity does zero work on it.
• Opposing force ($\theta = 180^\circ$): Because $\cos 180^\circ = -1$, the work done is negative ($W = -Fs$). Friction opposes motion and does negative work on an object, removing mechanical energy from it and dissipating it as heat.

The SI unit of work and energy is the joule ($\text{J}$). One joule is the work done when a force of 1 newton moves an object through a displacement of 1 metre in the direction of the force ($1\text{ J} = 1\text{ N m} = 1\text{ kg m}^2\text{ s}^{-2}$).

Forms of Mechanical Energy

Energy is defined as the ability to do work. Like work, it is a scalar quantity measured in joules ($\text{J}$).

In mechanics, you will encounter the symbol $E_p$ for two different quantities: gravitational potential energy ($mgh$) and elastic potential energy ($\frac{1}{2}ks^2$). Be clear from the context which form of stored energy you are using.

Kinetic Energy

Kinetic energy ($E_k$) is the energy an object possesses due to its motion:

Where $m$ is mass in kilograms and $v$ is speed in metres per second. Because velocity is squared, doubling an object's speed quadruples its kinetic energy. For the same braking force, doubling the speed makes the braking distance four times longer, because the brakes must remove four times as much kinetic energy ($W = Fs$, so $s \propto E_k$).

The work-energy theorem connects applied force directly to speed changes: the net work done on a body equals its change in kinetic energy:

Gravitational Potential Energy

Gravitational potential energy ($E_p$) is the energy an object possesses by virtue of its position in a gravitational field. When you lift an object of mass $m$ vertically through a height $h$ at constant speed, you apply an upward force equal to its weight ($mg$):

Here $g$ is the acceleration due to gravity ($9.8\text{ m s}^{-2}$). The height $h$ must always be the vertical displacement, regardless of whether the object travelled up a ramp, along a curve, or straight up.

Hooke's Law and Elastic Potential Energy

When you stretch or compress a spring or elastic material, internal forces act to restore it to its original shape. Within the elastic limit, elastic bodies obey Hooke's Law:

Where:

• $F$ is the restoring force in newtons ($\text{N}$)
• $k$ is the spring constant in newtons per metre ($\text{N m}^{-1}$), representing the force needed per metre of extension
• $s$ is the extension or compression from equilibrium in metres ($\text{m}$)

The negative sign shows that the restoring force acts opposite to the direction of displacement. The external force applied to stretch the spring has the same magnitude: $F_{\text{applied}} = ks$.

The elastic limit is the point beyond which an object ceases to obey Hooke's Law and will not return to its original length when the deforming force is removed.

Derivation of Elastic Potential Energy (Higher Level)

Because the applied force increases steadily from zero to $ks$, the force is not constant. The average force needed to produce an extension $s$ is:

Work done equals average force multiplied by displacement:

Graphically, on a plot of applied force $F$ (vertical axis) against extension $s$ (horizontal axis), the graph is a straight line through the origin. The work done equals the triangular area under this line:

This work is stored as elastic potential energy ($E_p$):

Investigation: Verifying Hooke's Law

The specification asks you to carry out this investigation using primary data and to analyse secondary data, so be ready to describe the method or to work from a table of results.

• Apparatus: Retort stand, clamp, spiral spring, weight hanger with slotted masses, and a vertical metre stick.
• Method: Suspend the spring and clamp the metre stick beside it. Note the initial position of the bottom of the spring on the scale with no load ($s_0$). Add slotted weights in equal steps (such as $0.5\text{ N}$ or $1.0\text{ N}$), recording each added load $F = mg$ and the new scale reading $s_1$. Calculate extension as $s = s_1 - s_0$. Unload the spring step by step to confirm the spring returns to its original length.
• Primary and secondary data analysis: Plot applied force $F$ on the vertical axis against extension $s$ on the horizontal axis. A straight line passing through the origin verifies Hooke's Law ($F \propto s$). The slope of this line equals the spring constant $k$ in $\text{N m}^{-1}$. If inspecting a secondary data table, any point lying well away from the trend line represents an anomalous result; identify it, explain why it does not fit, and exclude it when drawing the line of best fit.
• Sources of error: Parallax error when reading the metre stick (minimised by reading the scale with your eye level with the bottom of the spring, at right angles to the metre stick); exceeding the elastic limit by adding too much load.

Conservation of Energy and Real-World Applications

The Principle of Conservation of Energy states that energy cannot be created or destroyed; it can only be transferred from one body to another or converted from one form to another. In any isolated system, the total energy remains constant.

If there is no friction or air resistance, no mechanical energy converts into heat, so kinetic energy plus potential energy stays the same throughout the motion:

For a body dropped from rest through a vertical height $h$ in a vacuum, its initial gravitational potential energy converts entirely into kinetic energy:

When friction acts, mechanical energy is converted into non-mechanical forms like thermal energy (heat) and sound:

Where $W_{\text{against friction}} = F_{\text{friction}}s$.

In modern engineering, regenerative braking in electric and hybrid cars uses an electric generator to convert the vehicle's kinetic energy into chemical energy in a battery when slowing down, rather than dissipating it all as waste heat in brake pads. This improves vehicle efficiency and sustainability.

Investigation: Conservation of Energy

The specification asks you to carry out this investigation using primary data and to analyse secondary data, so be ready to describe the method or to work from a table of results.

• Apparatus: Linear air track, blower, glider of mass $m$ with an interrupt card of width $d$, two light gates connected to a digital timer, a block to elevate one end, and a metre stick.
• Method: Raise one end of the air track on a block. Set up two light gates along the track separated by a measured distance. Measure the vertical height difference $h$ between the card positions at the two light gates using a metre stick. Turn on the air supply. Release the glider from rest above the first light gate so it travels through both gates. The timer records the time intervals $t_1$ and $t_2$ for the card of width $d$ to pass through gate 1 and gate 2.
• Primary and secondary data analysis: Calculate speeds using $v_1 = \frac{d}{t_1}$ and $v_2 = \frac{d}{t_2}$. Calculate the loss in gravitational potential energy, $\Delta E_p = mgh$, and the gain in kinetic energy, $\Delta E_k = \frac{1}{2}m(v_2^2 - v_1^2)$. If they agree within the limits of experimental error, the principle is supported. When working with secondary data tables, look to see if the gain in $E_k$ is consistently slightly smaller than the loss in $E_p$; this systematic shortfall is due to air resistance or residual friction.
• Sources of error: Friction and air resistance (minimised by keeping the track clean and the air holes clear); uncertainty in measuring the small vertical height $h$ (minimised by measuring over a larger distance along the track or using trigonometry to calculate $h = d_{\text{track}} \sin\theta$, where $d_{\text{track}}$ is the distance along the track between the two gates and $\theta$ is the angle of the track to the horizontal).

Power and Efficiency

Power is defined as the rate at which work is done, or the rate at which energy is transferred:

Where $P$ is power in watts ($\text{W}$), $W$ is work done in joules ($\text{J}$), and $t$ is time taken in seconds ($\text{s}$).

The SI unit of power is the watt ($\text{W}$). One watt is an energy transfer rate of one joule per second ($1\text{ W} = 1\text{ J s}^{-1}$).

Rearranging the definition gives the total energy transferred over time:

Always ensure time is converted to seconds before calculating energy in joules.

Power at Constant Velocity

When an engine or applied force $F$ drives an object at a steady speed $v$ along the line of action of the force:

Because velocity $v = \frac{s}{t}$ under uniform motion:

Percentage Efficiency

Because resistive forces like friction and electrical resistance always convert some energy into unwanted heat and sound, no practical machine is $100\%$ efficient.

Percentage efficiency is calculated using:

Where $P_o$ is useful power output and $P_i$ is total power input. When calculating efficiency from energies, use the input and output energy for the same length of time. The easiest way is to compare power in with power out.

Key terms

Work done

Force multiplied by displacement in the direction of the force ($W = Fs$).

Joule

The SI unit of work and energy, defined as the work done when a force of 1 newton acts through a displacement of 1 metre in the direction of the force ($1\text{ J} = 1\text{ N m}$).

Energy

The ability to do work, measured as a scalar in joules.

Kinetic energy

The energy an object possesses due to its motion, given by $E_k = \frac{1}{2}mv^2$.

Gravitational potential energy

The energy an object possesses by virtue of its position in a gravitational field, given near Earth's surface by $E_p = mgh$.

Hooke's Law

The law stating that the restoring force exerted by an elastic object is directly proportional to its displacement from equilibrium, provided its elastic limit is not exceeded ($F = -ks$).

Spring constant

The force needed per metre of extension or compression of an elastic material ($k = -F/s$), measured in $\text{N m}^{-1}$.

Elastic limit

The maximum extension an elastic object can experience such that it still returns to its original length when the deforming force is removed.

Elastic potential energy

The energy stored in an elastic body when deformed by an extension or compression $s$, given by $E_p = \frac{1}{2}ks^2$.

Principle of Conservation of Energy

The fundamental principle stating that energy cannot be created or destroyed; it can only be transferred from one body to another or converted from one form to another.

Power

The rate at which work is done, or the rate at which energy is transferred ($P = W/t$).

Watt

The SI unit of power, equivalent to an energy transfer rate of 1 joule per second ($1\text{ W} = 1\text{ J s}^{-1}$).

Percentage efficiency

The ratio of useful power output to total power input expressed as a percentage: $\text{Efficiency} = \frac{P_o}{P_i} \times 100\%$.