Chapter 7Conservation of Energy (cont’d)

Mechanical energy conservationExamples

Work by non-conservative forces

March 4, 2010

Recollect: Conservation of Energy

• If only conservative forces are present, the total kinetic plus potential energy of a system is conserved.Mechanical Energy = Potential Energy (U) + Kinetic Energy (K)

Conservative forces interchange U K (work done), but E = K + U is a constant.

ΔE = ΔK + ΔU = 0Work-kinetic energy theorem: ΔK = Wthus ΔU = -W, for conservative forces only.

• Both K and U can change, but E = K + U remains constant.• But, if non-conservative forces act, then energy can be

dissipated in other forms (heat, for example)

Example: The simple pendulum

• Suppose we release a mass m from rest a height h1 above its lowest possible elevation. Assuming no friction (air drag):– What is the maximum speed of the mass and where does

this happen?– To what elevation h2 does it rise on the other side?

v

h1 h2

m

Example: The simple pendulum• Kinetic + potential energy is conserved since gravity

is a conservative force (E = K + U is a constant)• Choose y=0 at the bottom of the swing, and set the constant U=0 at y=0

Thus the general energy function of v, y: E = ½mv2 + mgy

v

h1 h2

m

• E = ½mv2 + mgy– Initially, y = h1 and v = 0, so E = mgh1.

– Since E = mgh1 initially, and energy is conserved, E = mgh1 at all times.

Example: The simple pendulum

h1

m

y = 0

y

Example: The simple pendulum

E = ½mv2 + mgyVelocity is maximum where potential energy is lowest, at bottom of the swing.So, at y = 0, ½mv2 = mgh1 v2 = 2gh1 v = (2gh1)1/2

v

h1

m

y = 0

y

Example: The simple pendulum

• To find maximum elevation on other side, note that maximum is reached when v=0. Since E = mgh1, and maximum potential energy on right is mgh2, h2=h1. The ball returns to its original height.

h1 h2

m

y = 0

y

Example: Airtrack and Glider

• A glider of mass M is initially at rest on a horizontal frictionless track. A mass m is attached to it with a massless string hung over a massless pulley as shown. What is the speed v of M after m has fallen a distance d?

d

M

m

v

v

Example: Airtrack and Glider

• Kinetic + potential energy is conserved since all forces are conservative.

• Choose initial configuration to have U=0.ΔK = -ΔU

d

M

m

v

v

1

2m + M( )v2 =mgd ⇒ v =

2mgd

m +M

Problem: Toy car

• A toy car slides on a frictionless track shown below. It starts at rest, drops a height d, moves horizontally at speed v1, rises a height h, and ends up moving horizontally with speed v2.

– Find v1 and v2.

hd

v1

v2

Problem: Toy car

• K+U is conserved, so ΔK = -ΔU• When the elevation decreases a distance D,

ΔU = -mgd, ΔK = ½mv12.

• Solving for the speed:

hd

v1

v2

€

v1 = 2gd

€

v2 = 2g(d − h)

A projectile of mass m is propelled from ground level with an initial kinetic energy of 450 J. At the exact top of its trajectory, its kinetic energy is 250 J. To what height, in meters, above the starting point does the projectile rise? Assume air resistance is negligible.

A projectile of mass m is propelled from ground level with an initial kinetic energy of 450 J. At the exact top of its trajectory, its kinetic energy is 250 J. To what height, in meters, above the starting point does the projectile rise? Assume air resistance is negligible.

K1 +U1 =K2 +U2 , with Ui=mghi

So mg(h2 -h1)=K 1-K 2

h2 -h1=(K 1-K 2 )/(mg)

Question• A box sliding on a horizontal frictionless surface runs into a

fixed spring, compressing it to a distance x1 from its relaxed position while momentarily coming to rest.– If the initial speed of the box were doubled and its mass

were halved, what would be the distance x2 that the spring would compress?

A) x2=x1 B) x2=x1√2 C) x2=2x1

x

Question• A box sliding on a horizontal frictionless surface runs into a

fixed spring, compressing it to a distance x1 from its relaxed position while momentarily coming to rest.– If the initial speed of the box were doubled and its mass

were halved, what would be the distance x2 that the spring would compress?

A) x2=x1 B) x2=x1√2 C) x2=2x1

x

(solution on next slide)

SolutionAgain, use the fact that WNET = ΔK.

Here, WNET = WSPRING = -½kx2

and ΔK = -½mv2 kx2 = mv2, and thus v = x(k/m)1/2, x = v(m/k)1/2

So doubling v and halving m increases x by a factor of √2.

x

Non-conservative forces:

• If the work done does not depend on the path taken, the force is said to be conservative.

• If the work done does depend on the path taken, the force is said to be non-conservative.

• An example of a non-conservative force is friction.– When pushing a box across the floor, the amount

of work that is done by friction depends on the path taken.

• Work done is proportional to the length of the path!

– The mechanical energy is converted to heat.

Spring pulls on mass: with friction

• Suppose spring pulls on block, but now there is a nonzero coefficient of friction μ between the block and the floor.

• The total work done on the block is now the sum of the work done by the spring, Ws (same as before), and the work done by the friction Wf,

(not related to either kinetic energy or potential energy) Work-energy theorem now reads: Wnet = Wcons + Wf = ΔK

stretched position (at rest)

d

relaxed position

vr

m

m

ii

f = μmg

rr

€

Wf =r f ⋅Δr r = −μmgd

Spring pulls on mass: with friction

• Again use Wnet = Ws + Wf = ΔK

stretched position (at rest)

d

relaxed position

vr

m

m

ii

f = μmg

rr

€

12

kd2 −μmgd =12

mvr2

€

vr =km

d2 − 2μgd

€

Wf = μmgd

€

Ws =12

kd2

€

K =12

mvr2

Many forms of energy:

Question

Which statement is true?A. Mechanical energy is always conservedB. Total energy is always conservedC. Potential energy is always conserved

Question

Which statement is true?A. Mechanical energy is always conservedB. Total energy is always conservedC. Potential energy is always conserved

Energy of one form can be converted to another form, but the total energy remains the same.

Conservation of energy is more broadly applicable than Newton’s laws

• Newton’s laws do not apply to systems that are fast-moving close to the speed of light (where Einstein’s theory of relativity applies) and to very small systems

(where quantum mechanics applies), but conservation of energy is always valid.

Conservation laws are the consequence of symmetries: Energy conservation Time translation invariance.