Conservation of Mechanical Energy

(Definition)

Conservation of Mechanical Energy

Kinetic energy describes energy associated with motion. Potential energy describes energy associated with position in a conservative interaction. When only conservative forces do work, these two forms can exchange while their sum remains constant.

Define the mechanical energy

|----------------| |Emech = K + U. | -----------------

(1)

If only conservative forces do work,

|--------------------| |K + U = K + U .| ---i----i----f-----f-

(2)

Equivalently,

|------------------| -Emech-=-constant.-|

(3)

This is the conservation of mechanical energy.

Figure 1. In a conservative system, kinetic and potential energy can convert into one another while the total mechanical energy remains constant.

1 Derivation from the work-energy theorem

The work-energy theorem gives

W = ΔK. net

(4)

Suppose every force doing work is conservative. Then

Wnet = Wc.

(5)

For a conservative force,

Wc = − ΔU.

(6)

Therefore

ΔK = − ΔU.

(7)

Move both changes to the same side:

ΔK + ΔU = 0.

(8)

Since

Δ (K + U ) = ΔK + ΔU,

(9)

we obtain

|----------------| |Δ (K + U) = 0. | -----------------

(10)

Thus

|------------------| K + U = constant.| --------------------

(11)

Between initial and final states,

|--------------------| |K + U = K + U .| ---i----i----f-----f-

(12)

2 Mechanical energy is a sum

Mechanical energy is not a new form of energy separate from kinetic and potential energy. It is the sum

E = K + U. mech

(13)

For a particle of mass m moving near Earth’s surface,

K = 1mv2 2

(14)

and

U = mgy. g

(15)

Thus

|----------------------| | 1 | |Emech = --mv2 + mgy. | ---------2-------------

(16)

For a mass attached to an ideal spring,

1 2 Us = 2kx ,

(17)

|----------------------| Emech = 1mv2 + 1-kx2.| ---------2-------2------

(18)

If both gravity and a spring are relevant,

|----------------------| -Emech-=-K--+-Ug-+-Us.-|

(19)

3 Energy-bar interpretation

A useful conceptual picture is to think of kinetic and potential energy as two accounts whose sum remains fixed.

ΔU < 0,

(20)

then

ΔK > 0.

(21)

ΔU > 0,

(22)

then

ΔK < 0.

(23)

For a conservative system,

ΔK = − ΔU.

(24)

Figure 2. Two states of the same conservative system. The partition between kinetic and potential energy changes, but the total mechanical energy is unchanged.

4 Solving for speed without time

Mechanical-energy conservation is especially powerful when speed is needed after a known change in position.

From

Ki + Ui = Kf + Uf ,

(25)

we can write

1-mv2 + U = 1mv2 + U . 2 i i 2 f f

(26)

Solve for v_f²:

2 2 2- v f = vi + m (Ui − Uf).

(27)

Therefore

|----∘-------------------| | 2 2- | vf = vi + m (Ui − Uf).| --------------------------

(28)

The energy method often avoids finding acceleration and elapsed time.

5 Example 1: falling under uniform gravity

A ball is released from rest from height h above a reference level. Neglect air resistance.

Initially,

Ki = 0

(29)

and

U = mgh. i

(30)

At the reference level,

Uf = 0.

(31)

Conservation of mechanical energy gives

1 2 mgh = 2-mv f.

(32)

Cancel m:

1 2 gh = --vf. 2

(33)

Thus

|-----∘------| |vf = 2gh. | -------------

(34)

This is the same speed obtained from constant-acceleration kinematics, but the energy method never uses the fall time.

6 Example 2: upward launch

A particle is launched vertically upward with speed v₀. Neglect air resistance.

Take the launch point as

U = 0.

(35)

Initially,

E = 1mv2 . mech 2 0

(36)

At the maximum height,

v = 0,

(37)

E = mgh . mech max

(38)

Therefore

1 2 2-mv 0 = mghmax.

(39)

Hence

|--------2--| | v0- | hmax = 2g. | -------------

(40)

7 Example 3: frictionless incline

A block starts from rest at vertical height h and slides down a frictionless incline.

Gravity is conservative and the Normal force does no work because it is perpendicular to the displacement.

Thus

mgh = 1-mv2. 2

(41)

Therefore

|-----------| | ∘ ---- | v-=----2gh.-

(42)

The angle and length of the incline do not appear. Only the vertical height change matters.

Figure 4. On a frictionless track under uniform gravity, the speed at a given height depends only on the vertical change in gravitational potential energy, not on the shape of the path.

8 Spring-mass systems

For a mass attached to an ideal spring on a frictionless horizontal surface,

1- 2 1- 2 Emech = 2mv + 2 kx .

(43)

At maximum extension or compression,

v = 0.

(44)

If the maximum displacement from equilibrium is A, then

1 2 Emech = -kA . 2

(45)

At equilibrium,

x = 0,

(46)

so all the mechanical energy is kinetic:

1 1 --mv2max = --kA2. 2 2

(47)

Thus

|--------∘-----| | k | |vmax = A --.| ------------m---

(48)

Figure 5. In an ideal spring-mass system, energy alternates between elastic potential energy and kinetic energy. The total mechanical energy remains fixed.

9 Multiple conservative forces

If several conservative forces act, each can have its own potential energy.

Suppose gravity and a spring both act. Then

Utotal = Ug + Us.

(49)

Mechanical energy is

Emech = K + Ug + Us.

(50)

If no nonconservative force does work,

|----------------------------------| Ki-+--Ug,i-+-Us,i =-Kf-+-Ug,f +-Us,f.-

(51)

The method is unchanged. Every conservative interaction contributes its potential-energy term.

10 Choice of potential-energy zero

The absolute zero of potential energy is arbitrary.

Suppose

′ U = U + C.

(52)

Then

K_i + U_i′	= K_i + U_i + C,	(53)
K_f + U_f′	= K_f + U_f + C.	(54)

The constant appears on both sides of the energy equation and cancels.

Therefore the conservation law is unaffected by the chosen reference level.

A convenient choice can simplify the algebra substantially.

11 Potential-energy diagrams

Suppose a particle moves in one dimension with conserved total mechanical energy

E.

(55)

Then

E = K + U(x ).

(56)

Therefore

K = E − U (x ).

(57)

Because kinetic energy cannot be negative,

K ≥ 0.

(58)

Hence allowed positions satisfy

|----------| E ≥ U (x).| ------------

(59)

Regions where

U (x ) > E

(60)

are classically forbidden.

12 Turning points

A turning point occurs where the particle momentarily stops before reversing direction.

At a turning point,

v = 0,

(61)

K = 0.

(62)

Thus

|--------------| -E-=--U(xturn).|

(63)

On a potential-energy graph, turning points are the intersections between the horizontal total-energy line and the potential curve.

Figure 3. A particle with total mechanical energy E can occupy only regions where E ≥ U(x). Points where E = U(x) are turning points because the kinetic energy vanishes there.

13 Speed from a potential-energy diagram

From

1 E = --mv2 + U (x), 2

(64)

we obtain

1mv2 = E − U(x). 2

(65)

Therefore

|------------------------| | ∘ -------------- | |v(x) = 2-[E − U (x)].| -----------m-------------

(66)

The speed is greatest where the potential energy is smallest.

At a turning point,

E − U = 0,

(67)

so the speed is zero.

14 Bound and unbound motion

Potential-energy diagrams can also classify motion.

If a particle is trapped between two turning points, the motion is bound.

If the energy is high enough to escape to arbitrarily large distance, the motion is unbound.

For example, with Newtonian gravity and the standard reference

U (∞ ) = 0,

(68)

the sign of the total mechanical energy distinguishes important regimes.

For a two-body gravitational system,

E < 0	corresponds to bound motion,	(69)
E = 0	marks the escape threshold,	(70)
E > 0	corresponds to unbound motion.	(71)

A later celestial-mechanics treatment develops this in greater depth.

15 What changes when nonconservative forces do work?

Mechanical energy is conserved only when the net work done by nonconservative forces is zero.

Start from

Wnet = ΔK.

(72)

Split the net work into conservative and nonconservative parts:

Wc + Wnc = ΔK.

(73)

Use

W = − ΔU. c

(74)

Then

− ΔU + Wnc = ΔK.

(75)

Therefore

|------------------| |Wnc = ΔK + ΔU. | -------------------

(76)

Hence

|---------------| W = ΔE . | --nc-------mech---

(77)

Between two states,

|--------------------------| -Kf-+-Uf--=-Ki-+-Ui-+-Wnc.--

(78)

Thus nonconservative work changes the mechanical energy.

Figure 6. Nonconservative work changes the total mechanical energy. Positive nonconservative work adds mechanical energy, while negative nonconservative work removes it.

16 Example 4: block with kinetic friction

A block of mass m starts from rest at height h, slides down a track, and reaches the bottom after kinetic friction has done work

Wf = − fkL.

(79)

Choose

Uf = 0.

(80)

Initially,

Ki = 0, Ui = mgh.

(81)

The mechanical-energy accounting equation is

K + U = K + U + W . f f i i f

(82)

Therefore

1mv2 = mgh − f L. 2 f k

(83)

Thus

|--------------------| | ∘ ------------ | |v = 2gh − 2fkL-.| --f-------------m----

(84)

Friction reduces the final mechanical energy.

17 Energy conservation versus mechanical-energy conservation

The phrase conservation of energy is broader than conservation of mechanical energy.

Mechanical energy

K + U

(85)

may decrease because friction converts organized mechanical energy into internal energy.

The total energy of an appropriately defined closed system is still conserved.

For example, when a block slides with kinetic friction,

ΔEmech < 0,

(86)

but the lost mechanical energy appears mainly as thermal energy of the surfaces.

Therefore:

|-----------------------------------------------------------------------------| mechanical energy need not be conserved even when total energy is conserved. | -------------------------------------------------------------------------------

(87)

18 Power form of mechanical-energy conservation

If only conservative forces do work,

Emech = K + U

(88)

is constant.

Therefore

|------------| |dE | |---mech= 0.| ---dt---------

(89)

If nonconservative forces are present,

Wnc = ΔEmech.

(90)

Taking the time derivative gives

|--------------| | dEmech- | |Pnc = dt .| ----------------

(91)

Thus nonconservative Power measures the rate at which mechanical energy is added or removed.

19 Common mistakes

Using conservation of mechanical energy when friction or another nonconservative force does net work.
Forgetting to include all relevant conservative potential-energy terms.
Setting kinetic energy equal to zero except at actual turning points.
Confusing a turning point with an equilibrium point.
Forgetting that allowed motion requires E ≥ U(x).
Treating potential energy as necessarily positive.
Assuming the zero of potential energy affects physical predictions.
Mixing heights measured from different gravitational reference levels in the same equation.
Forgetting that the normal force can be nonzero while doing zero work.
Saying energy is destroyed by friction instead of transferred into nonmechanical forms.
Using K + U = constant when there is known nonconservative work.

20 Practice exercises

A 2.0 kg object is dropped from rest through 10 m. Neglect air resistance. Find its speed at the bottom.
A projectile is launched vertically upward at 20 m∕s. Find its maximum height above the launch point.
A block slides frictionlessly from rest down a track through a vertical drop of 4.0 m. Find its speed at the bottom.
A spring with k = 300 N∕m is compressed by 0.15 m and launches a 0.50 kg block on a frictionless surface. Find the block’s speed when the spring reaches equilibrium.
A 1.0 kg particle has total mechanical energy 20 J and potential energy 12 J at some position. Find its speed.
For a potential U(x) and conserved energy E, explain why positions satisfying U(x) > E are classically forbidden.
A particle has E = 8 J and moves in a potential with turning points at x = −2 m and x = 5 m. What is its kinetic energy at either turning point?
A block starts from rest at height h and reaches the bottom with speed smaller than . Give two physically distinct reasons this can occur.
Derive W_nc = ΔE_mech from the work-energy theorem.
A 4.0 kg block slides 6.0 m down a track while friction of magnitude 5.0 N acts. If the block loses 50 J of gravitational potential energy, find its increase in kinetic energy.
A spring-mass oscillator has amplitude A. Find the kinetic energy at x = A∕2 in terms of k and A.
For Newtonian gravity with U(∞) = 0, explain why negative total mechanical energy corresponds to bound motion.
If nonconservative power is P_nc(t), show that
$∫ tf Emech (tf) − Emech (ti) = Pnc(t)dt. ti$ (92)