Simple Nature - Light and Matter

bra symbols.

1 J = 1

kg·m^2

s^2

= 1

kg·m^2

s^2

×

1000 g

1 kg

×

(

100 cm

1 m

) 2

= 10^7

g·cm^2

s^2

= 10^7 erg

Cabin air in a jet airplane example 3

.A jet airplane typically cruises at a velocity of 270 m/s. Outside

air is continuously pumped into the cabin, but must be cooled off

first, both because (1) it heats up due to friction as it enters the en-

gines, and (2) it is heated as a side-effect of being compressed to

cabin pressure. Calculate the increase in temperature due to the

first effect. The specific heat of dry air is about 1.0× 103 J/kg·◦C.

.This is easiest to understand in the frame of reference of the

plane, in which the air rushing into the engine is stopped, and its

kinetic energy converted into heat.^6 Conservation of energy tells

us

0 =∆E

=∆K+∆Eheat.

In the plane’s frame of reference, the air’s initial velocity isvi=270

m/s, and its final velocity is zero, so the change in its kinetic en-

ergy is negative,

∆K=Kf−Ki

= 0−(1/2)mvi^2

=−(1/2)mvi^2.

Assuming that the specific heat of air is roughly independent of

temperature (which is why the number was stated with the word

“about”), we can substitute into 0 =∆K+∆Eheat, giving

0 =−

1

2

mvi^2 +mc∆T

1

2

vi^2 =c∆T.

Note how the mass cancels out. This is a big advantage of solving

problems algebraically first, and waiting until the end to plug in

(^6) It’s not at all obvious that the solution would work out in the earth’s frame of
reference, although Galilean relativity states that it doesn’t matter which frame
we use. Chapter 3 discusses the relationship between conservation of energy and
Galilean relativity.
Section 2.1 Energy 79