# IV.5. Recapitulative solved problems - Thermal phenomena.

🔓 **Recapitulative solved problems - Thermal phenomena.**

**1. In order to understand why the railway rails are not welded and when they are installed, a space is left between them, called the joint, to find out how much a railway rail expands at 40 °C, which at 0 °C has a length of 10 m? Give the coefficient of linear thermal expansion of iron α = 0,000012 K ^{-1}.**

*We write the problem data:*

t_{1} = 40 °C

t_{0} = 0 °C

l_{0} = 10 m

α_{Fe} = 0,000012 K^{-1}

*We write the formula that defines the coefficient of linear expansion and which is the constant of proportionality between the relative variation of the bar length and the variation of the bar temperature:*

*If a 10 m bar expands by almost half a centimeter, then if it were welded to another bar, after expansion when it is hot, it would deform and the train would derail from the rails.*

**2. Two bodies, 1 and 2, were placed in thermal contact. Here is the graph of the evolution of the temperature of the two bodies over time:**

Is required:

a) What are the initial temperatures of the two bodies?

b) What is the equilibrium temperature?

c) After how many minutes did the two bodies reach thermal equilibrium?

d) What phenomenon does each body suffer during thermal contact?

**Solution:**

a) Body 1 has the initial temperature of 60 °C, being the warm body and body 2 has the initial temperature of 10 °C, being the cold body (at the moment 0 min).

b) The equilibrium temperature of the two bodies (their final temperature) is 30 °C.

c) The two bodies reached thermal equilibrium after 6 min.

d) During thermal contact, body 1 (cold) receives heat and heats up. Body 2 (hot) gives off heat and cools.

**3. The graph below shows the volume dependence of a temperature liquid. What is this liquid?**

**Solution:**

At 4 °C the liquid has the lowest volume and therefore the highest density.

When the liquid is cooled below 4 °C, its volume increases.

When the liquid is heated above 4 °C, its volume increases.

These strange phenomena are suffered only by water and are called water anomaly.

**4. A bimetallic blade is an assembly of two blades of different metals, fastened with rivets. They bend when heated, because they increase their length differently. Bimetallic blades are used as electrical switches in electrical appliances with thermostats. The pictures below show two bimetallic blades that bent when heated. Order the three metals A, B and C in ascending order of their length expansion. **

**Solution:**

In the first bimetallic blade, metal A increases its length more than metal B by heating.

In the second bimetallic blade, metal B increases its length more than metal C by heating.

The ascending order of linearly expanding metals is C, B, A.

Example: Fe, Cu, Al

**5. A body's density reduces its density by 20%. By what percentage does it increase in volume?**

**Solution:**

*We write the body data at room temperature with index 1, and those after heating with index 2.*

*We write the body density after heating (ρ _{2}) taking into account that it has decreased by 20% compared to the initial density (ρ_{1}):*

*We write the density formula taking into account that by heating a body, when expanding, only the volume of the body is increased, its mass remains the same:*

*All the terms of this equation are brought to the same denominator (10 ∙ V _{1} ∙ V_{2}) and the denominator is dropped. Also the mass m is reduced to all terms:*

**Heating body volume increased by 25%.**

**6. Why is 20 °C written on graduated scale measuring instruments (graduated vessels, roulette wheels, etc.)?**

Graduated scale measuring instruments dilation can change the distance between divisions leading to erroneous readings of measured quantities. They are calibrated for use at room temperature of 20 °C.

**7. Un corp de aluminiu de 270 g la temperatura de t _{0} = 20 °C are densitatea ρ_{0} = 2,7 g/cm^{3}. Încălzind corpul la temperatura t = 80 °C, volumul său crește cu ΔV = 420 mm^{3}. Ce densitate ρ are corpul la t = 80 °C?**

**7. An aluminum body of 270 g at a temperature of t _{0} = 20 °C has a density of ρ_{0} = 2,7 g/cm^{3}. By heating the body to a temperature t = 80 °C, its volume increases by ΔV = 420 mm^{3}. What density ρ has the body have at t = 80 °C?**

**Solution:**

*We transform the volume increase into cm ^{3}:*

*We write the density formula and subtract the initial body volume:*

*We write the body volume after heating, increased by ΔV:*

**V = V _{0} + ΔV**

*We write the density formula after heating or dilating the body:*

*We replace the numerical data in the obtained formula and calculate the body density at 80 °C:*