If Joule started by giving the weight in his machine 40  J40  J (4040 joules) of GPE, calculate how much thermal energy could be gained by the water.

1. Break Down the Problem

To solve this problem, we need to understand the relationship between gravitational potential energy (GPE) and thermal energy transfer. The key steps are:

1. Identify the initial GPE given.
2. Determine the conversion efficiency of GPE to thermal energy in water.

2. Relevant Concepts

The gravitational potential energy (GPE) can be expressed using the formula: $\text{GPE} = mgh$ where:

• $m$ = mass in kilograms,
• $g$ = acceleration due to gravity (approximately $9.81 \, \text{m/s}^2$),
• $h$ = height in meters.

In this scenario, we are given that the GPE is $40 \, \text{J}$.

3. Analysis and Detail

Assuming that all the GPE is converted into thermal energy (which would be an ideal scenario, potentially in a controlled experiment), the amount of thermal energy (Q) gained by the water can directly be equated to the GPE: $Q = \text{GPE}$ Thus, we have: $Q = 40 \, \text{J}$

4. Verify and Summarize

Since we are taking the ideal case where all the GPE converts to thermal energy, we find: $\text{Thermal Energy} = 40 \, \text{J}$

Final Answer

The thermal energy gained by the water is $40 \, \text{J}$.