NEET Physics — Chapter 6

Work, Energy & Power

Work-energy theorem, conservative forces, elastic and inelastic collisions, power, and vertical circular motion — complete NEET notes with formulas and exam traps.

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1. Work — Definition and Formula

Work is done when a force causes displacement in the direction of the force (or its component). The SI unit of work is the Joule (J).

W = ec{F} cdot ec{d} = Fdcos heta

where $heta$ is the angle between force $ec{F}$ and displacement $ec{d}$ .

Special cases:

$heta = 0°$ : $W = Fd$ (maximum positive work — force and displacement in same direction)
$heta = 90°$ : $W = 0$ (no work done — e.g., centripetal force, normal force on horizontal surface)
$heta = 180°$ : $W = -Fd$ (negative work — force opposes displacement, e.g., friction)

Work by a variable force: When force varies with position, use integration:

W = int_{x_i}^{x_f} F(x), dx

This equals the area under the F–x graph between the initial and final positions.

Work done by a spring: For a spring with spring constant $k$ , compressed or stretched by $x$ from natural length:

W_{spring} = - rac{1}{2}kx^2 quad ext{(work done BY spring on object)}

Work done ON the spring (by external force) = $+ rac{1}{2}kx^2$

NEET tip: Work is a scalar quantity even though force and displacement are vectors. Zero displacement → zero work, regardless of force. A satellite in circular orbit: centripetal force does zero work.

2. Kinetic Energy and Work-Energy Theorem

Kinetic Energy (KE) is the energy possessed by a body due to its motion:

KE = rac{1}{2}mv^2

Also expressible as: $KE = rac{p^2}{2m}$ where $p$ is linear momentum.

Work-Energy Theorem: The net work done on an object equals the change in its kinetic energy:

W_{net} = Delta KE = rac{1}{2}mv_f^2 - rac{1}{2}mv_i^2

This theorem holds even when forces are not constant — it's a scalar energy equation, much easier than Newton's 2nd law for finding speeds.

Relation between KE and momentum:

KE = rac{p^2}{2m} implies p = sqrt{2mKE}

If two bodies have equal KE: $rac{p_1}{p_2} = sqrt{ rac{m_1}{m_2}}$ (heavier body has more momentum)

If two bodies have equal momentum: $rac{KE_1}{KE_2} = rac{m_2}{m_1}$ (lighter body has more KE)

Pro tip: Dimensional formula of energy:

[ML^2T^{-2}]

. KE is always non-negative (

KE geq 0

). A body can have zero KE only if its velocity is zero. Use the work-energy theorem whenever you need to find speed after traveling some distance under a known net force.

3. Potential Energy — Gravitational and Spring

Potential Energy (PE) is energy stored in a system due to position or configuration. It is defined only for conservative forces.

Gravitational PE (near Earth's surface, taking ground as reference):

U_g = mgh

where $h$ is height above chosen reference level. PE can be negative if below reference.

Elastic PE (spring compressed/stretched by $x$ ):

U_s = rac{1}{2}kx^2

Always positive (stored regardless of direction of deformation).

Relation between PE and conservative force:

F = - rac{dU}{dx}

Force acts in the direction of decreasing PE. Equilibrium where $dU/dx = 0$ .

Stable, unstable, and neutral equilibrium:

Stable: $d^2U/dx^2 > 0$ — PE is minimum; body returns after small displacement
Unstable: $d^2U/dx^2 < 0$ — PE is maximum; body moves away after displacement
Neutral: $d^2U/dx^2 = 0$ — PE constant; body stays in new position

NEET tip: The choice of reference (where

U = 0

) is arbitrary — only differences in PE matter physically. For spring problems, the elastic PE

rac{1}{2}kx^2

is always measured from the natural (unstretched) length, not from any external reference.

4. Conservation of Mechanical Energy

For a system with only conservative forces acting:

KE + PE = ext{constant} quadimpliesquad rac{1}{2}mv_1^2 + U_1 = rac{1}{2}mv_2^2 + U_2

This is the Law of Conservation of Mechanical Energy. When non-conservative forces (friction, air drag) act, mechanical energy is not conserved — it converts to heat:

W_{non-conservative} = Delta KE + Delta PE = Delta E_{mechanical}

$W_{friction}$ is negative, so mechanical energy decreases by $|W_{friction}|$ .

Common Applications:

Scenario	Result
Ball dropped from height $h$	$v = sqrt{2gh}$ at bottom
Ball projected up with speed $v_0$	Max height $= v_0^2/2g$
Spring-mass system (spring releases from $x$ )	$v_{max} = xsqrt{k/m}$ at natural length
Block slides down frictionless incline (height $h$ )	$v = sqrt{2gh}$ (independent of incline angle)

NEET tip: When a problem says "smooth" or "frictionless," apply conservation of mechanical energy directly. When friction is present, use:

rac{1}{2}mv_f^2 = rac{1}{2}mv_i^2 + mgh - mu mgcos heta cdot d

. Always define your reference level for PE at the start.

5. Power

Power is the rate of doing work (or rate of energy transfer):

P = rac{W}{t} = rac{dW}{dt}

For instantaneous power: $P = ec{F} cdot ec{v} = Fvcos heta$

SI unit: Watt (W) = J/s. Also: 1 horsepower (hp) = 746 W

Average power: $ar{P} = W_{total} / t_{total}$

Efficiency ( $eta$ ):

eta = rac{P_{output}}{P_{input}} imes 100% = rac{W_{useful}}{W_{input}} imes 100%

Vehicle on a level road at constant speed: Engine power equals power dissipated by friction/resistance:

P = Fv quad ext{where } F = ext{friction/drag force}

At maximum speed ( $v_{max}$ ), driving force = resistance force, acceleration = 0.

Pro tip: Dimensional formula of power:

[ML^2T^{-3}]

. For a body moving with velocity

v

under force

F

at angle

heta

P = Fvcos heta

. Note: kilowatt-hour (kWh) is a unit of energy, not power. 1 kWh = 3.6 × 10⁶ J.

6. Collisions — Elastic, Inelastic, and Perfectly Inelastic

A collision is an event where two bodies exert large impulsive forces on each other for a short time. Momentum is always conserved in collisions (if no external force). Energy may or may not be conserved.

Type	Momentum	KE	e (COR)
Perfectly elastic	Conserved	Conserved	e = 1
Inelastic	Conserved	Not conserved	0 < e < 1
Perfectly inelastic	Conserved	Max loss	e = 0

Coefficient of Restitution (e):

e = rac{ ext{relative speed of separation}}{ ext{relative speed of approach}} = rac{v_2 - v_1}{u_1 - u_2}

Elastic collision (1D) — final velocities:

v_1 = rac{(m_1 - m_2)u_1 + 2m_2 u_2}{m_1 + m_2}, quad v_2 = rac{(m_2 - m_1)u_2 + 2m_1 u_1}{m_1 + m_2}

Special cases:

$m_1 = m_2$ : velocities exchange — $v_1 = u_2$ , $v_2 = u_1$
$m_1 gg m_2$ (heavy hits light, $u_2 = 0$ ): $v_1 approx u_1$ , $v_2 approx 2u_1$
$m_1 ll m_2$ (light hits fixed wall): $v_1 approx -u_1$ (bounces back with same speed)

Perfectly inelastic collision: Bodies stick together; velocity of combined mass:

v = rac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

Delta KE_{lost} = rac{m_1 m_2}{2(m_1+m_2)}(u_1 - u_2)^2

NEET caution: KE is NOT conserved in an inelastic collision — the lost KE converts to heat, sound, and deformation energy. In a perfectly inelastic collision, KE loss is maximum (not total KE loss — the system still moves, so some KE remains).

7. Vertical Circular Motion — Energy Approach

A body moving in a vertical circle under gravity is a classic application of energy conservation. The tension varies with position.

At the bottom (speed $v_b$ ):

T_{bottom} - mg = rac{mv_b^2}{r} implies T_{bottom} = mg + rac{mv_b^2}{r}

At the top (speed $v_t$ ):

T_{top} + mg = rac{mv_t^2}{r} implies T_{top} = rac{mv_t^2}{r} - mg

Minimum speed at top (for the string/track to just maintain contact, $T_{top} geq 0$ ):

v_{t,min} = sqrt{gr}

Using energy conservation from bottom to top: $v_{b,min} = sqrt{5gr}$

Relation between speeds at bottom and top (energy conservation, height = 2r):

v_b^2 = v_t^2 + 4gr

Tension difference: $T_{bottom} - T_{top} = 6mg$ (independent of speed and radius!)

NEET tip: The condition for completing the vertical circle (

v_{min} = sqrt{5gr}

at bottom) is extremely commonly tested. If

v_b < sqrt{5gr}

, the string goes slack before the top. For a ball on the inside of a loop (roller coaster track), the normal force replaces tension — same equations apply.

8. NEET Traps & Formula Summary

Trap 1 — Zero work ≠ zero force: Centripetal force, normal force (horizontal motion), and gravity (horizontal motion) all do zero work because force ⊥ displacement.

Trap 2 — Spring PE formula: The elastic PE

rac{1}{2}kx^2

uses

x

measured from the natural (unstretched) length, not from the equilibrium position. In SHM with a hanging spring, equilibrium is displaced — be careful.

Trap 3 — KE is always ≥ 0: You cannot have negative KE. If energy conservation gives a negative KE, the body cannot reach that point — it stops earlier.

Trap 4 — Momentum conserved, not KE in inelastic: Even if they "stick together," momentum is still conserved. Never apply KE conservation in an inelastic collision.

Quick Formula Sheet:

Work	$W = Fdcos heta$
Work-Energy Theorem	$W_{net} = Delta KE$
Kinetic Energy	$KE = rac{1}{2}mv^2 = rac{p^2}{2m}$
Gravitational PE	$U = mgh$
Spring PE	$U = rac{1}{2}kx^2$
Power	$P = Fvcos heta$
Vertical circle min speed (top)	$v_{min} = sqrt{gr}$
Vertical circle min speed (bottom)	$v_{min} = sqrt{5gr}$
COR	$e = rac{v_2 - v_1}{u_1 - u_2}$
Perfectly inelastic velocity	$v = rac{m_1u_1 + m_2u_2}{m_1+m_2}$

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