A 10.0 -kg block of metal measuring 12.0 cm by 10.0 cm by 10.0 cm is suspended from a scale and

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

A 10.0 -kg block of metal measuring 12.0 $\mathrm{cm}$ by 10.0 $\mathrm{cm}$ by 10.0 $\mathrm{cm}$ is suspended from a scale and immersed in water as shown in Figure $\mathrm{P} 14.24 \mathrm{b}$ . The 12.0 -cm dimension is vertical, and the top of the block is 5.00 $\mathrm{cm}$ below the surface of the water. (a) What are the magnitudes of the forces acting on the top and on the bottom of the block due to the surrounding water? (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

Apparent Weight

The apparent weight of an object submerged in a fluid is the actual weight reduced by the buoyant force. This concept explains why objects seem lighter in water, and it is essential when interpreting readings from devices like spring scales in immersion problems.

Archimedes' Principle

Archimedes' principle states that a body wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body. This principle is fundamental for calculating the buoyant force acting on submerged objects.

Force Due to Fluid Pressure

The force exerted by a fluid on a submerged surface is the product of the local pressure and the area of that surface. Since pressure varies with depth, different surfaces of an object experience different forces, which is key to understanding the net effect on the object.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the weight of the fluid above the point of measurement. In problems involving submerged objects, the pressure increases with depth according to the relation P = P? + ?gh, where ? is the fluid density, g is the acceleration due to gravity, and h is the depth.

Transcript

00:02 So let us consider that this is the block and the height of this from the water surface is given.

00:23 Let's call that h and this is the length of the block.

00:31 Let's call that d.

00:32 We will not need the explicit surface area.

00:36 So let's just call that a.

00:40 And now we have got weight of the block.

00:47 We have got a force on this.

00:49 Surface which is f bottom and we have got a force on the top surface which is f top now this is being held up by the spring so there there is tension now force at the top surface will be given by the the area times pressure over there which is raw water g times h so using a is and with the with the pressure we should also add atmospheric pressure to that so it becomes that atmospheric pressure is 1 .013 times 10 10 to the 5 pascal plus row water g and height is 5 cm so 0 .05 meter multiplied by the surface area which is 10 cm or 0 .1 meter by 0 .1 meter.

02:43 So that gives the top surface.

02:46 Force at the top surface is 1 .079 times 10 to the 3 newton.

03:00 Now similarly, force at the bottom surface, we have to remove the we have to replace the formula stay as the same but the height becomes h plus d at that point plus atmospheric pressure d is 12 centimeter so that's 0 .1 -2 using the same values as in here we get the force at the bottom surface to be 1 .0297 times 10 to the 3 newton.

03:53 From there, we find that the difference in, before we find the difference in the ratings, what is the reading in the scale? the reading in the scale will be all the upward force minus all the downward force.

04:16 So that will be t plus f bottom, minus f top, or equals f top...

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