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1. In class, you saw the experiment of burning some paper by compressing air in a glass
cylinder. Instead on pushing on the piston as in class (left-hand figure), we rotate the
cylinder by 180 (upside down) and drop it on the floor so that the piston strikes the floor
(right-hand figure).
(a) From what minimum height we have to drop it to burn the paper? The apparatus
weights 0.35 kg; the inside diameter of glass tube is 0.797 cm; the paper will ignite at
240 C; the initial position of the piston is 18.308 cm above the bottom of the tube. Ignore
the friction and drag forces.
(b) What assumptions did you make? Write a brief sentence justifying each assumption.
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2. A cylinder containing 0.5kg of air and a moveable piston is submerged in an ice-water
bath (shown in the sketch below). The air in the cylinder is initially in equilibrium with the
bath temperature. The piston is very quickly (adiabatically) pushed down from position
1 to position 2 compressing the air to 2/3 of its original volume. The piston is held at
position 2 until the air is again in equilibrium with the bath temperature. Finally, the
piston is very slowly raised back to position 1.
(a) Draw approximate p-V and T-S diagrams for the cycle indicating those parts of the
cycle in which heat flows into or out of the air cylinder.
(b) On the basis of either of your diagrams, explain whether the amount of ice in the
ice-water mixture is increased or decreased over a complete cycle.
(c) Calculate the per cycle change in the mass of ice in the mixture.
Useful information: molar mass of air: 28.97g/mol, latent heat of fusion of ice is: 334
kJ/kg
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3. Two identical springs fixed to a rigid support on one side, with identical 500g masses
attached on the other end, sit next to each other as shown in the diagram below. One
spring (A) is compressed from its equilibrium position by an amount 2.5cm by pushing on
it with a force of 5.0N, the other (B) is stretched from its equilibrium position by the same
amount by applying the same force. Both are let go at the same time; consider this t=0.
Displacements to the right are positive, and to the left are negative.
(a) What is the period of oscillation?
(b) What is the maximum acceleration?
(c) Sketch the displacement vs. time for each. Label the graphs with quantitative values
on both axes.
(d) Sketch the acceleration vs. time for each. Label the graphs with quantitative values
on both axes.
(e) What is the phase di↵erence between the oscillation of B and oscillation of A?
A"
Acceleration of B
Acceleration of A
Displacement of B
Displacement of A
B"
k"
m"
A"
m"
B"
k"
k"
m"
k"
m"
Useful Constants
R = 8.31451J/mol K, 1 atm = 1.013 ⇥ 105 Pa, 1 atm litre = 101.3 J
Stefan-Boltzmann constant = 5.6704 ⇥ 10 8 W m 2 K 4 .
1
K 1 , The density of water is 1 gram/cm3 =1000 kg/m3 .
air = 1.4, CV,air = 20.8Jmol
Mechanics
Linear Motion: x = x0 + 12 (vx0 + vx )t, x = x0 + vx0 t + 12 ax t2 , vx = vx0 + ax t,
vx 2 = vx0 2 + 2ax (x x0 )
2
Circular Motion: ac = vr
d
Forces: F~ = m~a, F~ = dt
p~, Friction: F = µN , Spring: F = kx, Damping: F~ = b~v
Bouyant: FB = ⇢V g
R ~r
~ v.
W = ~rif F~ · d~r, W = F~ · ~r, K = 12 mv 2 , U gravity = mgy, U spring = 12 kx2 , P = dW
dt , P = F · ~
Thermodynamics
L
Thermal Expansion: L = ↵L0 T , V = V0 T , Stress and Strain: F
A =Y L
3
1
Ideal Gas Law: P V = nRT , Kav = 2 kT , 2 kT for each degree of freedom.
Thermal Conductivity: H = Qt = kA Tx
Black Body Radiation: P = e AT 4 , max T = 2.8977685 ⇥ 10 3 m · K
Internal Energy: U = nCV T
R
First Law: dQ = dU + dW for an ideal gas dW = P dV ! W = P dV .
Work for isothermal process W = nRT ln(Vf /Vi ).
For adiabatic expansion T V 1 = constant, if the number of moles is constant P V
constant and = Cp /Cv .
Work for adiabatic process
W =
Z
V2
P dV = C
V1
Z
V2
V1
dV
C
=
(V2 1
V
1
V1 1
= C where C is a
)
Q = mc T , Q = mL. CP = CV + R, CV = f2 R where f =degrees of freedom. f = 3 for monatomic and
f = 5 for diatomic.
dS = dQ/T
e = W/QH COPCooling = |QC |/|W | COPHeating = |QH |/|W |
eCarnot = 1 TC /TH
Integrals
Z
xn+1
+ constant
n+1
xn dx =
n 6=
Trig
sin ✓1 + sin ✓2 = 2 cos(
1
✓1
Z
✓2
2
x
1
) sin(
dx = ln x + constant
✓1 + ✓2
)
2
Area and Volume Area of a sphere A = 4⇡r2 . Area of a cylinder A = 2⇡rl. Area of a circle A = ⇡r2 .
Volume of a cylinder V = l⇡r2 . Volume of a sphere V = 43 ⇡r3 .
Oscillations
! = 2⇡f , T = f1 , x = Acos(!t + ), ! 2 =
Damped Oscillations: x = A0 e
bt
m
bt
2m
k
m.
cos(!t + ), where ! =
q
!0 2
b 2
( 2m
) , Q = 2⇡
Energy for damped E = E0 e
Waves
q
v = Tµ , k = 2⇡ , v = f , P = 12 µ! 2 A2 v, p0 = ⇢!vs0
q
v
(1± D )
RT
Pav
v=
= 10dB log10 ( II0 ) Doppler E↵ect f 0 = f0 (1⌥ vvs )
M , I = 4⇡r 2 ,
v
y = A cos(kx ⌥ !t + )
Interference k x +
= 2⇡n or ⇡(2n + 1) n = 0, ±1, ±2, ±3, ±4, . . .
mv
Standing Waves fm = mv
2L m = 1, 2, 3, . . . fm = 4L m = 1, 3, 5, . . .
Beats
E
E.
f = f2
f1