Question

An iron horseshoe at room temperature is dunked into a cylindrical tank of water (radius of \(10.0 \mathrm{~cm}\) ) and causes the water level to rise by \(0.250 \mathrm{~cm} .\) When the horseshoe is heated in the blacksmith's stove from room temperature to a temperature of \(7.00 \cdot 10^{2} \mathrm{~K},\) worked into its final shape, and then dunked back into the water, how much does the water level rise above the "no horseshoe" level (ignore any water that evaporates as the horseshoe enters the water)? Note: The linear expansion coefficient for iron is roughly that of steel: \(11.0 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\)

Short answer

Answer: To find the new rise in the water level after the heated horseshoe is dunked back into the cylindrical tank, we need to follow the steps outlined in the solution provided. First, find the initial volume of the water in the tank, followed by calculating the volume of the horseshoe. Next, calculate the final volume of the horseshoe after heating using the linear expansion coefficient. Lastly, find the new rise in water level by calculating the difference in volume between the initial and final horseshoe volumes. By following these steps, we can find the new increase in the water level above the "no horseshoe" level.

Step-by-step solution

Find the initial volume of the water in the tank

To find the initial volume of the water in the tank, we'll use the formula for the volume of a cylinder, \(V = \pi r^2 h\). Here, \(r\) is the radius of the cylindrical tank, and \(h\) is the initial height of the water in the tank. We know the radius of the tank is \(10.0 \mathrm{~cm}\), and the water level rose by \(0.250 \mathrm{~cm}\) when the horseshoe was dunked in. Therefore, the initial height of the water in the tank is the difference in height, which is \(0.250 \mathrm{~cm}\).

Calculate the volume of the horseshoe

Now, let's find the volume of the horseshoe by using the initial volume formula for the cylinder tank \(V = \pi r^2 h\). The volume of the horseshoe (\(V_{hs}\)) can be found by multiplying the volume of the displaced water with the ratio of the density of iron to the density of water (\(\rho_{iron}/\rho_{water}\)). So, \(V_{hs} = V \cdot \frac{\rho_{iron}}{\rho_{water}}\).

Calculate the final volume of the horseshoe

To calculate the final volume of the horseshoe, we'll need to use the linear expansion coefficient (\(\alpha\)) for iron. The formula to find the final volume of the horseshoe (\(V'_{hs}\)) after heating is: \(V'_{hs} = V_{hs} (1+\alpha \Delta T)^3\), where \(\Delta T\) is the change in temperature, which is \((7.00 \cdot 10^{2} - T_{room})\).

Calculate the new rise in water level

To find the new rise in water level, we will find the difference in volume between the initial and final horseshoe volumes and then divide by the density ratio of iron to water to get the volume of the new displaced water. We can then use the formula for the volume of a cylinder to find the new height of water: \(\Delta h = \frac{\Delta V}{\pi r^2}\). We'll add this height to the initial height of \(0.250 \mathrm{~cm}\) to find the new rise in water level above the "no horseshoe" level. By following these steps, you'll be able to find the new water level in the cylindrical tank after the heated horseshoe is dunked back into it.