Worksheet: Thermal Expansion of Solids and Liquids

In this worksheet, we will practice relating the linear and volume expansion coefficients of solids and liquids to objects' size changes with temperature.

Q1:

The height of the Washington Monument is measured to be 170.00 m on a day when the temperature is 35.000C.

What will its height be on a day when the temperature falls to 10.000C? Assume that the monument’s coefficient of thermal expansion is 2.5000×10C.

Q2:

You are looking to buy a small piece of land in Hong Kong. The price is “only” $60,000 per square meter. The land title says the dimensions are 20 m by 30 m. By how much would the total price change if you measured the parcel with a steel tape measure on a day when the temperature was 20C above the temperature that the tape measure was designed for? The dimensions of the land do not change. The coefficient of thermal expansion of steel is 12×10C.

Q3:

A volume of mercury is contained in a tall, thin, cylinder. The mercury fills the cylinder to a height of 3.0 cm and is at a temperature of 37C. If the mercury is heated to a temperature of 40C, how much does the height of the column of mercury increase? Use a value of 61×10C for the linear coefficient of expansion of mercury.

  • A 5 . 4 × 1 0 m
  • B 5 . 0 × 1 0 m
  • C 4 . 4 × 1 0 m
  • D 5 . 5 × 1 0 m
  • E 4 . 5 × 1 0 m

Q4:

Most cars have a coolant reservoir to catch radiator fluid that may overflow when the engine is hot. A radiator is made of copper and is filled to its 16.0 L capacity when at 10.0C. What volume of radiator fluid will overflow when the radiator and fluid reach a temperature of 95.0C given that the fluid’s volume coefficient of expansion is 𝛽=400×10C and copper’s volume coefficient of expansion is 51.0×10C?

Q5:

Find the stress exerted on a steel beam along its length if the beam has no lengthwise space to expand into and its temperature increases from 15C to 40C. Use a value of 35×10C for the Young’s modulus of steel and use a value of 12×10C for the coefficient of thermal expansion of steel. Ignore any change in the cross-sectional area of the beam due to heating.

  • A 1 . 2 × 1 0 Pa
  • B 1 . 0 × 1 0 Pa
  • C 1 . 4 × 1 0 Pa
  • D 0 . 8 7 × 1 0 Pa
  • E 0 . 7 3 × 1 0 Pa

Q6:

A 1.0 m long stick made of steel and a 1.0 m long stick made of aluminum have the same length as each other when they are both at a temperature of 0.0C. Both sticks expand when heated. Use a value of 12×10C for the coefficient of thermal expansion of steel and a value of 25×10C for the coefficient of thermal expansion of aluminum.

What is the difference in length of the two sticks when both are at a temperature of 22C?

  • A 2 . 4 × 1 0 m
  • B 1 . 1 × 1 0 m
  • C 3 . 5 × 1 0 m
  • D 1 . 8 × 1 0 m
  • E 2 . 9 × 1 0 m

A steel tape and an aluminum tape are both 30.0 m in length at a temperature of 0.0C. What is the difference in length of the tapes when both are at a temperature of 22C?

  • A 5 . 5 × 1 0 m
  • B 6 . 4 × 1 0 m
  • C 8 . 6 × 1 0 m
  • D 7 . 2 × 1 0 m
  • E 4 . 2 × 1 0 m

Q7:

A glass beaker has a capacity of 397 mL. The beaker is filled to the brim with water that is at a temperature of 15.0C. Determine the volume of water that will overflow the beaker when the water’s temperature increases to 25.0C. Use a value of 207×10C for the coefficient of volumetric expansion of water.

Q8:

An engineer wants to design a structure in which the difference in length between a steel beam and an aluminum beam remains at 0.730 m when they both vary in temperature by the same amount. In determining the relationship between the lengths of the beams, use a value of 25.0×10C for the linear thermal expansion coefficient of aluminum and 12.0×10C for the linear thermal expansion coefficient of steel.

What must the length of the steel beam be?

What must the length of the aluminum beam be?

Q9:

An aluminum pendulum used to drive a grandfather clock has a length 𝑙=1.26 m and a mass 𝑀 at temperature 𝑇=20C. The clock can be modeled as a physical pendulum consisting of a rod oscillating around one of its ends. Determine the percent increase in the period of the clock if the temperature increases by 15C, assuming the length of the rod changes linearly with temperature, where 𝑙=𝑙(1+𝛼Δ𝑇). Use a value of 21×10C for 𝛼, the linear temperature coefficient of expansion of aluminum.

  • A 0 . 0 2 1 %
  • B 0 . 0 1 6 %
  • C 0 . 0 0 5 2 %
  • D 0 . 0 3 7 %
  • E 0 . 0 2 6 %

Q10:

A farmer making grape juice fills a glass bottle to the brim and caps it tightly. After capping, the juice and bottle warm up and expand. The juice’s thermal expansion is greater than the glass’s thermal expansion. The juice’s volume increases 0.20% more than the bottle’s volume increases. Take the bulk modulus of the juice to be 1.8×10 N/m2 and calculate the force exerted per square centimeter by the juice on the bottle.

  • A 6 . 8 × 1 0 N/cm
  • B 5 . 7 × 1 0 N/cm
  • C 3 . 6 × 1 0 N/cm
  • D 7 . 5 × 1 0 N/cm
  • E 2 . 1 × 1 0 N/cm

Q11:

A brass rod that is at a temperature of 20.0C has a length of 1.12 m and a diameter of 0.6700 cm. The rod is fixed at both of its ends and then its temperature is increased, creating a compressive force along the length of the rod. Determine the temperature at which the rod will be subject to a force of 28.0 kN. Use a value of 9.00×10 Pa for the Young’s modulus of brass and use a value of 19.0×10C for the linear thermal expansion coefficient of brass. Assume that any change in the radius of the rod due to thermal expansion is negligible.

Q12:

A physicist makes 2.50×10 cm3 of instant coffee, which he pours into a cup that has a radius of 10.00 cm. As the coffee cools from a temperature of 80.0C to 25.0C, the physicist notices that the vertical height of the coffee decreases by 4.50 mm. Determine what percent of the decrease of the height of the coffee is due to its thermal contraction. Use a value of 207×10C for the coefficient of volumetric expansion of the coffee.

  • A 5 5 . 6 %
  • B 7 . 8 4 %
  • C 8 . 0 4 %
  • D 6 3 . 3 %
  • E 5 . 5 6 %

Q13:

Determine how large of an expansion gap should be left between steel railroad rails if they may reach a maximum temperature of 40.0C greater than when they were laid down. The original length of the rails is 8.0 m. Use a value of 7.2×10C for the coefficient of linear expansion of steel.

Q14:

Determine how much taller the Eiffel Tower becomes at the end of a day when the temperature has increased by 21C. Use a value of 312.0 m as the original height of the tower and 7.2×10C for the coefficient of linear expansion of steel.

Q15:

Global warming will produce rising sea levels partly due to melting ice caps and partly due to the expansion of water as average ocean temperatures rise. To get some idea of the size of this effect, determine the change in length of a column of water 3.50 km high for a temperature increase of 5.00C. Assume the column is not free to expand sideways and neglect the variation of ocean water warming with depth. Use a value of 7.00×10C for the coefficient of linear expansion of water.

Q16:

A mercury thermometer still in use for meteorology has a bulb with a volume of 0.780 cm3 and a tube for the mercury to expand into of inside diameter 0.130 mm. The coefficient of thermal expansion of mercury is 180×10C. The coefficient of thermal expansion of ordinary glass is 27.0×10C.

Neglecting the thermal expansion of the glass, what is the spacing between marks 1C apart?

If the thermometer is made of ordinary glass, what is the spacing?

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