Suppose that we repeat the experiment shown in the video, but we replace one of the cylinders with a cylinder that has twice the radius (and use larger containers of water). If the height of the original cylinder is \rm h, how deeply must we submerge the new cylinder to get the same weight reduction as in the video?

1/4h

To find out how deeply we must submerge the new cylinder to get the same weight reduction, we can follow these steps:

Step 1: Determine the weight reduction in the original cylinder.
In the video, it is shown that when the cylinder is submerged by a certain height 'h', there is a weight reduction.

Step 2: Calculate the weight reduction of the original cylinder.
The weight reduction is equal to the weight of the water displaced by the submerged portion of the cylinder.

Step 3: Determine the weight of the water displaced by the original cylinder.
The weight of the water displaced is given by the formula:
Weight = density * volume * gravitational acceleration

Step 4: Calculate the volume of the original cylinder.
The volume of a cylinder is given by the formula:
Volume = π * r^2 * h

Step 5: Calculate the weight reduction of the original cylinder using the volume and density of water.
Weight reduction = density of water * volume * gravitational acceleration

Step 6: Replace one of the cylinders with a cylinder that has twice the radius.
In the new cylinder, the radius will be twice the original radius.

Step 7: Calculate the new volume of the cylinder.
Volume = π * (2r)^2 * h = 4 * π * r^2 * h

Step 8: Calculate the depth needed to achieve the same weight reduction in the new cylinder.
To achieve the same weight reduction, the volume of water displaced by the new cylinder should be equal to the volume of water displaced by the original cylinder.

Therefore, the new depth 'd' is given by the formula:
(π * 2^2 * r^2 * h) = (4 * π * r^2 * h)
Simplifying, we get:
16 * r^2 * h = 4 * r^2 * h
Canceling out 'r^2' and 'h' on both sides, we get:
16 = 4
This equation does not hold true, which means we cannot achieve the same weight reduction in the new cylinder by submerging it to a certain depth.
So, the question cannot be answered given the given conditions.

To determine how deeply the larger cylinder needs to be submerged to achieve the same weight reduction as in the video, we need to understand the concept of buoyancy and Archimedes' principle.

Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In other words, the buoyant force is determined by the volume of the object submerged in the fluid.

In the video, the weight reduction is achieved by partially submerging the smaller cylinder in water. The buoyant force acting on the cylinder reduces the effective weight of the cylinder, resulting in a weight reduction.

Now, if we replace the smaller cylinder with a larger cylinder of twice the radius, we need to determine the equivalent depth of submersion to achieve the same weight reduction.

Let's consider the original cylinder first. The weight reduction is determined by the volume of the cylinder submerged in water.

The volume of a cylinder is given by the formula: V = πr²h, where r is the radius of the cylinder and h is its height.

In the original cylinder, let's assume the height is h and the cylinder is partially submerged to a depth of d. The volume of the submerged portion would be V_submerged_original = πr²d.

Now, we replace the original cylinder with a larger cylinder with twice the radius. The radius of the larger cylinder is 2r.

To achieve the same weight reduction, we need to find the depth of submersion (let's call it d') for the larger cylinder.

The volume of the larger cylinder submerged to depth d' is V_submerged_larger = π(2r)²d'.

Since the weight reduction is the same, we can equate the volumes of the submerged portions for the original and larger cylinders:

πr²d = π(2r)²d'

Canceling out the π and r² terms:

d = 4d'

Therefore, to achieve the same weight reduction, the depth of submersion for the larger cylinder needs to be one-fourth (1/4) of the depth used for the smaller cylinder in the video experiment.

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