A 10.0-kg block of metal measuring 12.0 cm ×10.0 cm ×10.0...

A $10.0-\mathrm{kg}$ block of metal measuring $12.0 \mathrm{~cm} \times$ $10.0 \mathrm{~cm} \times 10.0 \mathrm{~cm}$ is suspended from a scale and immersed in water, as shown in Figure $\mathrm{P} 15.23 \mathrm{~b}$. The 12.0-cm dimension is vertical, and the top of the block is $5.00 \mathrm{~cm}$ from the surface of the water. (a) What are the forces acting on the top and on the bottom of the block? (Take $P_{0}=1.0130 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}$.) (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above. It is defined by the relationship P = P? + ?gh, where P? is the surface pressure, ? is the density of the fluid, g is the acceleration due to gravity, and h is the depth. This concept is essential for understanding how pressure varies with depth in any fluid, influencing the forces acting on submerged objects.

Buoyant Force

The buoyant force is the net upward force exerted by a fluid on an immersed object. According to Archimedes’ principle, this force is equal to the weight of the fluid displaced by the object. The concept is crucial when analyzing fluid statics because it explains how and why objects experience an apparent reduction in weight when submerged, as the pressure difference between the bottom and top of the object produces this net upward force.

Pressure Difference in Fluids

The pressure difference in a fluid arises because pressure increases with depth. For a submerged object, the pressure at the bottom is higher than at the top by an amount that depends on the vertical distance between these surfaces and the fluid's density. This difference directly results in an upward net force, which is manifest as the buoyant force acting on the object.

Gravitational Force

Gravitational force is the force due to gravity acting on a mass, calculated by the product of the mass and the gravitational acceleration (mg). In the context of immersed objects, gravitational force provides the downward pull that must be counteracted by the buoyant force, and it plays a vital role in determining the net force acting on the object when combined with the fluid’s pressure forces.

Transcript

00:03 This question covers the concept of the pressure and another concept is the concept of the buoyancy and so to solve this problem first we need to find out the height of the top of the surface of the cube and hd is given and that is 5 centimeters from the top of the water level and the height of the bottom of the surface is 5 cm plus the height of the box and that is 12 cm or the height of the bottom is 17 centimeters from the top of the water surface so for part a the net pressure at the top of the surface is equivalent the atmospheric pressure plus row times g into the height of the top and the net force on the top is equivalent to the net pressure into the cross -extended area 8 so the net force on the top is equivalent to the atmospheric pressure that is 1 .013 into 10 -5 newton per meter square plus the density of the water that is 100 sorry 1 ,000 kilogram per meter cube into g and that is 9 .8 meters per meter square plus the density of water that is 100, sorry 1 ,000 kilograms second square into ht that is the height of the top and that is 5 cm or we can write 0 .05 meter into the cross sectional area and the cross sectional area is 0 .1 meter into 0 .1 meter into 0 .1 meter so the force on the top of the surface is 1 .17 .9 meter now for part p the force on the bottom of the surface is equivalent to the net pressure that is 1 .013 into 10 .5 newton per meter square plus the atmospheric pressure plus the density times the gravitational acceleration times the depth and the depth is 17 centimeter or we can write 0 .17 meters into the cross -extional area and that is 0 .1 meters into 0 .1 meter the force exerted by the water on the bottom is 1029 .66 newton.

03:00 Now part c the bian's force is equivalent to the density of water that is 1 ,000 kilograms per meter cube into the volume of the replaced water and that is 0 .12 meters into 0 .12 meters into 0 .1 meter into 0 .1 meter into the gravitational acceleration that is 9 .8 meters per second square.

03:29 The bion's force is equivalent to 11 .76 newton...

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