Mechanical Engineering Principles

THERMAL EXPANSION 227

(b) Initial volume of glass,

V 1 = 100 × 50 × 20 =100 000 mm^3

Final volume of glass,

V 2 =V 1 [1+γ(t 2 −t 1 )]

=V 1 [1+ 3 α(t 2 −t 1 )],

sinceγ= 3 αto a very close approximation

Hence,increase in volume of glass

=V 1 ( 3 α)(t 2 −t 1 )

=(100 000)( 3 × 9 × 10 −^6 )( 60 )

=162 mm^3

Now try the following exercise

Exercise 103 Further questions on the co-

efficients of superficial and

cubic expansion

1. A silver plate has an area of 800 mm^2 at
   15 °C. Determine the increase in the area
   of the plate when the temperature is raised
   to 100°C. Assume the coefficient of linear
   expansion of silver to be 19× 10 −^6 K−^1.
   [2.584 mm^2 ]


2. At 283 K a thermometer contains 440 mm^3
   of alcohol. Determine the temperature at
   which the volume is 480 mm^3 assuming
   that the coefficient of cubic expansion of
   the alcohol is 12× 10 −^4 K−^1. [358.8 K]


3. A zinc sphere has a radius of 30.0 mm
   at a temperature of 20°C. If the temper-
   ature of the sphere is raised to 420°C,
   determine the increase in: (a) the radius,
   (b) the surface area, (c) the volume of the
   sphere. Assume the coefficient of linear
   expansion for zinc to be 31× 10 −^6 K−^1.

[(a) 0.372 mm (b) 280.5 mm^2

(c) 4207 mm^3 ]

4. A block of cast iron has dimensions
   of 50 mm by 30 mm by 10 mm at
   15 °C. Determine the increase in volume
   when the temperature of the block is
   raised to 75°C. Assume the coefficient

of linear expansion of cast iron to be

11 × 10 −^6 K−^1. [29.7 mm^3 ]

5. Two litres of water, initially at 20°C, is
   heated to 40°C. Determine the volume
   of water at 40°C if the coefficient of
   volumetric expansion of water within this
   range is 30× 10 −^5 K−^1. [2.012 litres]


6. Determine the increase in volume, in
   litres, of 3 m^3 of water when heated from
   293 K to boiling point if the coefficient
   of cubic expansion is 2.1 × 10 −^4 K−^1.
   (1 litre≈ 10 −^3 m^3 ). [50.4 litres]


7. Determine the reduction in volume when
   the temperature of 0.5 litre of ethyl alco-
   hol is reduced from 40°Cto− 15 °C.
   Take the coefficient of cubic expansion
   for ethyl alcohol as 1. 1 × 10 −^3 K−^1.
   [0.03025 litres]

Exercise 104 Short answer questions on

thermal expansion

1. When heat is applied to most solids and
   liquids........ occurs.


2. When solids and liquids are cooled they
   usually........


3. State three practical applications where
   the expansion of metals must be allowed
   for.


4. State a practical disadvantage where the
   expansion of metals occurs.


5. State one practical advantage of the
   expansion of liquids.


6. What is meant by the ‘coefficient of
   expansion’.


7. State the symbol and the unit used for
   the coefficient of linear expansion.


8. Define the ‘coefficient of superficial ex-
   pansion’ and state its symbol.


9. Describe how water displays an unex-
   pected effect between 0°Cand4°C.


10. Define the ‘coefficient of cubic expan-
    sion’ and state its symbol.