Question

Explain to your perplexed friend how you can bring together twelve ob jects of mass 1.01 and end up with one object of mass \(12.00 .\)

Short answer

To solve the puzzle, either there might be a physical loss of mass in the system due to some mechanisms, such as evaporation or chemical reactions, that makes the system loses 0.12. Or, it could be a mathematical rounding precision issue, where the individual weights of the objects were less than 1.01, but were rounded up. Then when all 12 objects were put together and rounded, their total weight became 12.00.

Step-by-step solution

Understanding Conservation of Mass

The first step is to understand the principle of conservation of mass which states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time. Hence, if the system is not closed and energy or matter is allowed to escape, then the mass of the system can decrease. In this case, the system is considered to be open.

The Possible Mass Loss

Given 12 objects each of mass 1.01, for the system to end up with mass 12.00, there must be a loss of mass. The total initial mass is 12.12 (1.01*12). Hence, the loss of mass would be 12.12 - 12.00 which is equal to 0.12. It means that, due to some mechanisms (evaporation, chemical reaction, etc.), some mass got expelled from the system.

Rounding Precision

Apart from an actual loss, we could also explore rounding precision. It could be that the original objects did not weigh exactly 1.01, but something slightly less. When rounding up, each of these 'less than 1.01' weights round up to 1.01, but when all 12 objects are put together and their total mass is rounded, the total weight is 12.00. Note that this scenario is more in the realm of mathematics and rounding errors, rather than physics.

Key concepts

Mass Loss

Mass loss can occur in a system that is not closed, which allows matter or energy to escape. When you bring twelve objects, each with a mass of 1.01, and sum them up, you initially get a total mass of 12.12. However, if the final mass reads 12.00, it suggests that some mass has not been accounted for. This could take place due to natural processes such as evaporation, where some material converts into vapor and escapes, or through chemical reactions where some matter transforms into gas and is lost to the surroundings. In such scenarios, the initial mass of 12.12 reduces by a mass loss of 0.12 to become the 12.00 observed. Thus, when dealing with open systems, it's crucial to consider the effects of both visible and invisible mass changes that can lead to loss.

Rounding Errors

Rounding errors are discrepancies that emerge from approximations during measurements or calculations. Suppose the objects don't exactly weigh 1.01, but slightly less, like 1.006. When you round each individual mass to the nearest hundredth, they all appear as 1.01. If you calculate based on these rounded numbers, you might expect the total to equal 12.12 for twelve objects. However, if you consider their actual values, the total could, upon final rounding, equal exactly 12.00. In practical terms:

• Rounding done for simplicity requires careful consideration of precision.
• The more times rounding occurs (e.g., each object's mass), the larger the potential error in the final sum.
• Understanding when and how rounding is applied can help minimize error impacts in results.

Thus, the way rounding is applied might lead to perceived discrepancies in mass, showcasing why exactitude in scientific systems is essential.

Open System

An open system is one in which energy or matter can be exchanged with the surroundings. This characteristic distinguishes it from a closed system, which is isolated from such exchanges. In the context of mass conservation, an open system means:

• The system can gain additional mass from energy or matter that comes in.
• Conversely, the system can lose mass when substances are released into the environment, such as water evaporating or gases being produced during a reaction.
• This type of system requires constant monitoring to track changes accurately.

For instance, a setup involving heated objects might result in a loss of some mass due to evaporation, which wouldn't happen in a closed system. Knowing the nature of your system helps explain why, even with original expectations of mass retention, the observed values might differ due to these exchanges.