NDA Physics · Formula sheet

Work, Energy and Power formulas

11 formulas, 3 reference tables and 19 common traps for NDA Physics Work, Energy and Power, grouped by subtopic.

Work — Force Times Displacement Times Cosine

What work means in physics — W = F d cos θ

Work done by a constant force

W = F\,d\cos\theta

$W$ work done (joules, J)
$F$ magnitude of the applied force (N)
$d$ magnitude of the displacement (m)
$\theta$ angle between the force and the displacement

The sign of work — positive, zero, or negative by the angle

Sign cases of W = F d cos θ

\cos 0^\circ = 1,\quad \cos 90^\circ = 0,\quad \cos 180^\circ = -1

$\theta = 0^\circ$ force along motion — positive work
$\theta = 90^\circ$ force perpendicular — zero work
$\theta = 180^\circ$ force opposes motion — negative work

Work done by gravity depends only on the height change

Work done by gravity over a height change h

W_\text{gravity} = \pm\,mgh

$m$ mass of the body (kg)
$g$ acceleration due to gravity ( $\approx 9.8$ m/s²)
$h$ vertical height change only (m)

Common traps

Work needs MOVEMENT in the force's direction — holding a weight is zero work

If the displacement is zero, the work is zero no matter how large the force. Holding a heavy load still, or pushing a wall that does not move, does no physics work even though you feel tired.

Perpendicular force does ZERO work — not maximum

Carrying a weight at constant height, or the centripetal force on an orbiting body, does no work because the force is at

90^\circ

to the motion. Students often confuse "large force" with "large work" — but with

\cos 90^\circ = 0

the work is exactly zero.

Negative work means the force OPPOSES motion

When force and displacement are anti-parallel (

\theta = 180^\circ

\cos\theta = -1

and the work is negative — the force is taking energy away (friction, air resistance, a brake). It does not mean "no work".

Gravity's work does NOT depend on the path

Because gravity is conservative, the work it does between two points is fixed by the height difference alone. A statement that "work done by gravity depends on the path followed" is the wrong one the NDA tests.

Energy — Kinetic, Potential, and Conservation

Learn this subtopic in the notes

Kinetic energy — energy of motion (½mv²)

Kinetic energy

KE = \tfrac{1}{2}mv^2

$KE$ kinetic energy (J)
$m$ mass of the body (kg)
$v$ speed of the body (m/s)

Potential energy — energy of position (mgh)

Gravitational potential energy

PE = mgh

$PE$ gravitational potential energy (J)
$m$ mass of the body (kg)
$g$ acceleration due to gravity ( $\approx 9.8$ m/s²)
$h$ height above the reference level (m)

Conservation of energy — PE converts to KE as a body falls

Energy conservation for a freely falling body

mgh = \tfrac{1}{2}mv^2 \;\Rightarrow\; v = \sqrt{2gh}

mghpotential energy at the top (J)
$\tfrac{1}{2}mv^2$ kinetic energy at the bottom (J)
$v$ landing speed (m/s)

Kinetic energy and its change depend on the reference frame

Frame-dependent change in kinetic energy

\Delta K_{S'} = mgh + mu\sqrt{2gh} \;>\; \Delta K_{S} = mgh

$\Delta K_S$ KE change in the rest frame = mgh
$\Delta K_{S'}$ KE change in a frame moving with speed u
$u$ relative speed of the two frames (m/s)

Conservative forces and energy transformations

Item	Classification / sequence	Note
Gravitational force	Conservative	work depends only on height change
Spring (elastic) force	Conservative	energy fully recovered on release
Electrostatic force	Conservative	path-independent work
Frictional force	Non-conservative	dissipates energy as heat — the bank's answer "Which is NOT a conservative force?" — the answer is friction.
Air resistance / drag	Non-conservative	removes mechanical energy as heat
Apple falling to ground	GPE → KE → Sound → Heat	PE turns to motion, then a thud, then heat on impact The correct transfer sequence: gravitational PE → kinetic → sound → heat.

Friction is the standard "not conservative" answer; the falling-apple sequence runs gravitational PE → KE → sound → heat.

Common traps

Kinetic energy grows with the SQUARE of speed

Doubling the speed multiplies kinetic energy by 4, tripling it by 9. A body at twice the speed of another (same mass) carries four times the kinetic energy — not twice.

Convert grams to kilograms before substituting

The mass in

\tfrac{1}{2}mv^2

must be in kilograms. A mass given as 2000 g is 2 kg; forgetting to convert gives an answer 1000 times too large.

Potential energy is about POSITION or SHAPE — not motion

A body raised to a height, a compressed spring, or a stretched bow all store potential energy because of their configuration. Energy due to motion is kinetic energy — keep the two definitions distinct.

Energy is conserved for an ISOLATED system

The total energy stays constant only when no energy crosses the system boundary — that is the isolated case. For real bodies, friction and air resistance carry mechanical energy away as heat and sound, so MECHANICAL energy alone is not conserved, but total energy still is.

At the bottom of a free fall, KE equals the starting PE

For a body dropped from rest, the kinetic energy on landing equals the potential energy it had at the top:

\tfrac{1}{2}mv^2 = mgh

. Set them equal — do not add them.

Friction is the standard NON-conservative force

Gravity, spring force, and electrostatic force are conservative (recoverable, path-independent). Friction and air resistance are non-conservative — they turn mechanical energy into heat and cannot give it back.

The falling-apple sequence ends in HEAT, not sound

The order is gravitational PE → kinetic → sound → heat. KE turns to sound (the thud) AND heat on impact — distractors reorder these or put heat before kinetic energy.

Even the CHANGE in kinetic energy is frame-dependent

It is tempting to think

\Delta K

is the same for all observers since both see the same fall. It is not — the moving frame adds a cross term

mu\sqrt{2gh}

, because work depends on the displacement seen in that frame.

Work-Energy Theorem and Power

Learn this subtopic in the notes

Work-energy theorem — net work equals change in kinetic energy

Work-energy theorem

W_\text{net} = \Delta KE = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2

$W_\text{net}$ net work done by all forces (J)
$v_i$ initial speed (m/s)
$v_f$ final speed (m/s)

Power — the rate of doing work (P = W/t = Fv)

Power

P = \dfrac{W}{t} = Fv

$P$ power (watts, W)
$W$ work done (J)
$t$ time taken (s)
$F$ applied force (N)
$v$ speed (m/s)

Potential energy from a force — U = − ∫ F dx

Potential energy from a conservative force

U(x) = -\int F(x)\,dx \quad\Longleftrightarrow\quad F(x) = -\dfrac{dU}{dx}

$U(x)$ potential energy as a function of position (J)
$F(x)$ conservative force along x (N)

Units of work, energy, and power

Quantity / unit	Definition	In SI base
Joule (J)	1 N acting through 1 m	work / energy unit
1 joule of work	force of 4 N over 0.25 m	$4 \times 0.25 = 1$ J
Watt (W)	1 joule per second	power unit, J/s
Kilowatt-hour (kWh)	energy of a 1 kW device in 1 hour	$3.6 \times 10^{6}$ J 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J — the commercial unit of electrical energy.
Kilowatt (kW)	1000 watts	power unit

The two recall favourites: 1 J = 4 N over 0.25 m, and 1 kWh = 3.6 × 10⁶ J.

Common traps

The theorem uses NET work — not the work of one force

W_\text{net} = \Delta KE

sums the work of every force acting. If only friction acts on a sliding block, then friction's work equals the KE change; but when several forces act, add them all before equating to

\Delta KE

Power is a RATE — do not confuse it with energy

Power (watt) is energy per unit time; energy (joule) is the total amount. Two machines that do the same work have the same energy output but different power if they take different times. "How much" is energy; "how fast" is power.

P = Fv uses the speed at that instant

When a force moves a body, the instantaneous power is

P = Fv

. At higher speed the same force delivers more power — which is why a constant-power engine cannot keep accelerating at the same rate.

1 kWh is 3.6 × 10⁶ J — not 1000 or 3600

A kilowatt-hour combines 1000 W with 3600 s:

1000 \times 3600 = 3.6 \times 10^{6}

J. Multiplying only one of the two factors is the standard wrong answer.

Do not forget the MINUS sign when integrating

U = -\int F\,dx

— the negative sign is essential. Dropping it (writing

U = +\int F\,dx

) flips the sign of the whole potential energy and gives the wrong option.

Simple Machines — Levers and Mechanical Advantage

Learn this subtopic in the notes

The lever and mechanical advantage

Mechanical advantage of a lever

MA = \dfrac{\text{load}}{\text{effort}} = \dfrac{\text{effort arm}}{\text{load arm}}

$MA$ mechanical advantage (no units)
$\text{effort arm}$ distance from fulcrum to effort (m)
$\text{load arm}$ distance from fulcrum to load (m)

The three orders of levers

Order	What is in the middle	Examples
First class	Fulcrum in the middle (E–F–L)	seesaw, scissors, crowbar, beam balance
Second class	Load in the middle (F–L–E)	wheelbarrow, bottle opener, nutcracker The bank's favourite. Second class = load in the middle; example = bottle opener / wheelbarrow.
Third class	Effort in the middle (F–E–L)	forceps, tongs, fishing rod, human forearm

Tell them apart by what sits in the middle: fulcrum (1st), load (2nd), effort (3rd). Second-class levers always have mechanical advantage greater than 1.

Common traps

Mechanical advantage multiplies FORCE, not work

A lever lets a small effort move a big load, but it does not create energy: you move the effort end through a larger distance. The work in roughly equals the work out — only the force is multiplied.

Second class = LOAD in the middle (not fulcrum)

Tell the orders apart by the MIDDLE element: fulcrum (first), load (second), effort (third). A second-class lever has the load between the fulcrum and the effort — a bottle opener or wheelbarrow, not a seesaw.

Work, Energy and Power formulas

Work — Force Times Displacement Times Cosine

What work means in physics — W = F d cos θ

The sign of work — positive, zero, or negative by the angle

Work done by gravity depends only on the height change

Common traps

Energy — Kinetic, Potential, and Conservation

Kinetic energy — energy of motion (½mv²)

Potential energy — energy of position (mgh)

Conservation of energy — PE converts to KE as a body falls

Kinetic energy and its change depend on the reference frame

Conservative forces and energy transformations

Common traps

Work-Energy Theorem and Power

Work-energy theorem — net work equals change in kinetic energy

Power — the rate of doing work (P = W/t = Fv)

Potential energy from a force — U = − ∫ F dx

Units of work, energy, and power

Common traps

Simple Machines — Levers and Mechanical Advantage

The lever and mechanical advantage

The three orders of levers

Common traps

More NDA Physics formula sheets