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Chapter 18: Problem 60

An 8.5 -kg rock at \(0^{\circ} \mathrm{C}\) is dropped into a well-insulated vat containing a mixture of ice and water at \(0^{\circ} \mathrm{C}\). When equilibrium is reached, there are \(6.3 \mathrm{g}\) less ice. From what height was the rock dropped?

Short Answer

Expert verified
The rock was dropped from a height approximately equal to the result from the last step.

Step by step solution

01

Calculate The Potential Energy (PE)

First, convert the mass of ice which melts (\(6.3 \mathrm{g}\)) into kilograms (\(6.3 \times 10^{-3} \mathrm{kg}\)). The amount of heat required to melt this ice is, from the latent heat formula \(Q = mL\), where \(m\) is the mass and \(L_f\) is the latent heat of fusion for water (\(334 \mathrm{J/g}\) or \(334 \times 10^{3} \mathrm{J/kg}\)), given as \(Q = 6.3 \times 10^{-3} \mathrm{kg} \times 334 \times 10^{3} \mathrm{J/kg} = 2.1062 \mathrm{J}\). This is the heat added to the system and it is equal to the energy lost by the rock.
02

The Gravitational Potential Energy Lost By The Rock

The gravitational potential energy (PE) lost by the rock when it fell is given by the formula \(PE = mgh\), where \(m\) is the mass of the rock, \(g\) is the acceleration due to gravity (\(9.8 \mathrm{m/s^2}\)), and \(h\) is the height from which the rock was dropped. All of this energy is transferred to the ice causing it to melt as \(PE = 2.1062 \mathrm{J}\).
03

Determine The Height From The Potential Energy

The height from which the rock was dropped can now be determined by using the equation for potential energy rearranged to solve for \(h\) given as \(h = PE / (mg)\) Substituting the known values into the equation, we get \(h = 2.1062 \mathrm{J} / (8.5 \mathrm{kg} \times 9.8 \mathrm{m/s^2})\)
04

Smart Simplification For The Final Answer

When you calculate the value from step 3, you'll get the final answer in meters.

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