Styrofoam's density is \(160 \mathrm{kg} / \mathrm{m}^{3} .\) What percent error is introduced by weighing a Styrofoam block in air (density \(1.2 \mathrm{kg} / \mathrm{m}^{3}\) ), which exerts an upward buoyancy force, rather than in vacuum?

If you've ever wondered why some objects float in water and others sink, the answer lies in the concepts of density and buoyancy. Density is defined as mass per unit volume and is a measure of how tightly matter is packed within an object.

For example, the density of Styrofoam is around \(160 \mathrm{kg} / \mathrm{m}^{3}\), which is less than that of water, explaining why it floats. On the other hand, buoyancy is a force that pushes objects up when they are submerged in a fluid. This force is equal to the weight of the fluid displaced by the object.

It's important to remember that buoyancy is not just about water. Air, although much less dense than water, also exerts a buoyant force on objects. In our exercise, the weight of the Styrofoam block in air is reduced by the air's buoyancy, which is less than it would be in a vacuum where there is no buoyant force.

The true weight of an object is the force due to gravity on the object's mass. It's different from the apparent weight because it does not take into consideration any buoyant force acting on the object.

To calculate the true weight of an object, like our Styrofoam block, you simply multiply the object's mass by the acceleration due to gravity (\(9.8 \mathrm{m/s}^{2}\)). For a block of Styrofoam with a known volume, we multiply this volume by the Styrofoam's density to find the mass and then multiply by the acceleration due to gravity to determine the true weight:
True Weight = Volume × Density × Gravity
It's important for students to strive for accuracy in this step, as this value forms the basis for further calculations.

Apparent weight is what we measure when we weigh an object in a fluid, like air or water. It's different from the true weight because it considers the buoyancy force acting on the object. To measure the apparent weight, you weigh the object normally while it's submerged in the fluid.

In our case with the Styrofoam block, the apparent weight is less than the true weight because of the upward buoyancy force exerted by the air. This can be calculated by subtracting the buoyancy force from the true weight:
Apparent Weight = True Weight - Buoyancy Force

Understanding this difference is crucial for fields such as material science and engineering, where precise weight measurements are necessary.

Percent error is a measurement of how accurate a value is, compared to a known or accepted value. It is used to understand the precision of a measurement and to identify any discrepancies that need to be addressed.

To calculate the percent error in the context of our Styrofoam block exercise, you take the difference between the true weight (accurate value) and the apparent weight (experimental value), divide by the true weight, and then multiply by 100:
Percent Error = \(\frac{|\text{True Weight} - \text{Apparent Weight}|}{\text{True Weight}} \times 100\%\)
It expresses the error as a percentage of the true value, giving a clear picture of the accuracy of the apparent weight measurement. This is especially useful in experiments where exact measurements are difficult to obtain, such as in fluid mechanics.