Archimedes’ principle states that the upward buoyant force experienced by a solid object in a fluid is equal to the weight of the fluid displaced. In mathematical form, Archimedes’ principles states:

Here,  is the buoyant force,  is the mass of the fluid displaced, and  is the acceleration due to gravity. Recall that density is defined as mass divided by volume; therefore, mass of fluid can be written as:

 is the density of fluid and  is the volume of fluid displaced. Plugging this equation into Archimedes’ principle gives:

Notice that the density of the object is not part of Archimedes’ principle; therefore, the the density of the object doesn’t affect the buoyant force.

Both submerged and floating objects in a fluid will experience an upward buoyant force that is equal to the weight of the fluid displaced; Archimedes’ principle is valid for both submerged and floating objects.

To solve this question, you need to first compare the density of the object with the density of the fluid (water). The density of the object (according to the question) is  and the density of water is . Since the density of the object is less than the density of water, the object will float. To solve for the volume of fluid displaced, we need to use Archimedes’ principle:

Here,  is buoyant force,  is density of fluid,  is volume of fluid displaced, and  is acceleration due to gravity. We have two unknowns in this equation: the buoyant force and the volume of fluid displaced; therefore, we do not have enough information to solve for the volume of fluid displaced.

If the density of the object and the density of the fluid were equal, then we could have assumed that the weight of the fluid displaced (Buoyant force) equals the weight of the object. We could have then calculated the buoyant force and, subsequently, the volume of fluid displaced; however, we cannot make that assumption in this question because the density of the object is different from the density of the fluid.

The question states that the object sinks to the bottom of the bowl. Recall that objects with greater density than water will either submerge completely or sink, whereas objects with lesser density than water will float on the surface of water. The object in this question must have a greater density than water. 

The second part of the question is asking you to compare the buoyant force with the weight of the fluid displaced. Archimedes’ principle states that the upward buoyant force always equals the weight of fluid displaced. Archimedes’ principle is valid for floating, submerging, and sinking objects; therefore, the buoyant force on the object will equal the weight of the water displaced.

The density of the object is , whereas the density of water is ; therefore, the object will float at the surface of water. One of the implications of Archimedes’ principle for a floating object is that the percent of object submerged equals the ratio of the density of object to the density of fluid:

This means that 50% of the object is submerged and 50% of the object is above the water. 50% of the volume of the object, or , is submerged.

To solve this question, you need to know the definition of density:

The question gives us the mass of the block and the density of the block. We can use these values to solve for the volume of the block.

Density has to be converted from  to  (SI units). Recall that ; therefore, .

Solving for volume of the block gives us:

A liter is defined as ; therefore, .

Remember that the volume of a cube is calculated as follows:

Since the cube has the same length, width, and height, the equation above can be rewritten in terms of height as follows:

This means that the height of the cube can be calculated by taking the cube root of volume:

The length, the width, and the height of this cubical block is .

Density is defined as mass divided by volume. Recall that the solid phase is usually characterized by tightly packed molecules. As a result, solids usually take up less volume than liquids and, consequently, have greater densities than liquids.

The most common exception to this rule is water. Liquid water has a density of , whereas solid water (ice) has a density of .

Density of any substance (solid or liquid) is calculated as follows:

 is density of the substance,  is mass, and  is volume. The density of a substance does not depend on the weight of the object.

Changing a substance from its solid phase to its liquid phase only alters volume; it does not change the mass. Remember that the mass of a substance depends on the amount of molecules. Changing from solid to liquid will increase the spacing between the molecules (volume), but it will not change the amount of molecules (mass); therefore, the solid phase and the liquid phase of a sample of substance will have the same mass.

Increasing a substance’s temperature to its melting point will convert the substance from a solid to a liquid. This will increase the volume of the substance because the molecules within the substance will be farther apart from each other; however, the amount of molecules (mass of the substance) will remain unchanged.

Recall the definition of density:

Since the volume increases and the mass stays the same, the density of iron will decrease when you change the temperature to its melting point.

Density is defined as follows:

To alter density, we need to alter mass and/or volume. Remember that mass can only be altered by removing from or adding to the sample. Temperature and pressure do not change the amount of substance; therefore, they do not alter the mass of the substance. 

Changing temperature will alter the distance between the molecules within the substance. Increasing the temperature will increase the distance between the molecules, which will increase the volume taken up by the substance. Similarly, decreasing the temperature will decrease the distance between the molecules, which will decrease the volume taken up by the substance. This means that changing temperature will alter the volume of the substance and, subsequently, the density of the substance. 

Applying pressure also changes the distance between the molecules. Increasing the pressure will force the molecules to be closer together, whereas decreasing the pressure will enable the molecules to spread and separate from each other; therefore, pressure also alters density by changing the volume of the substance.

To answer this question, you need to solve for the stress experienced by the tube. Stress is defined as:

Since this is a tube, the area we are concerned with is the circular cross-sectional area of the tube. The question states that the diameter is . This means that the radius (which is half the length of the diameter) is . Recall that meters is the SI unit for distance; therefore, we need to convert the radius to meters:

This means that the radius of the tube, in meters, is . The area of a circle is:

Using this formula and our radius, we can find the cross-sectional area:

The stress experienced by the tube is:

The closest answer choice is .

The question states that the tube is being stretched by the force. This means that the tube is undergoing tension because you are pulling the tube and applying a force outwards. If the force was applied in the opposite direction (inwards), then the tube would be undergoing compression (the tube would shorten).

Strain of an object is defined as:

 is the final length and  is the initial length. The equation implies that increasing the initial length of an object () would decrease the strain. Similarly, the equation also implies that increasing the final length of an object () would increase the strain; therefore, increasing the initial length decreases strain, whereas increasing the final length increases strain.