Volume Calculation: Converting a Sphere into a Cylindrical Wire

Understanding the principles of volume conservation is fundamental in many scientific and engineering applications. In this article, we will explore a practical scenario where a solid sphere is melted and formed into a cylindrical wire, and we will calculate the length of the wire. This problem is essential for students and professionals alike, as it highlights the importance of volume conservation and the interplay between geometry and physics.

Problem Statement

A solid sphere with a radius of 18 cm is melted down and drawn into a long cylindrical wire of uniform thickness of 6 mm. What is the length of the wire?

Step-by-Step Solution

To find the length of the wire, we need to equate the volume of the sphere to the volume of the cylindrical wire. This involves a series of calculations that demonstrate the principles of volume conservation.

Step 1: Calculate the Volume of the Sphere

The formula for the volume of a sphere is given by:

V43πr3

where r is the radius of the sphere.

Given that the radius r is 18 cm:

V43π18?cm?3

Calculating 18?cm?3:

18?cm?35832cm?3

Substituting back into the volume formula:

V43π58327776πcm?3

Step 2: Calculate the Volume of the Cylinder

The volume of a cylinder is given by:

Vπr2h

where r is the radius of the base of the cylinder and h is the height or length of the cylinder.

The thickness of the wire is 6 mm, which is equivalent to 0.6 cm. The radius of the wire is half of the thickness:

r0.620.3cm

Now we can express the volume of the cylinder in terms of its height:

Vπ0.3?0.3?h0.09πhcm?3

Step 3: Set the Volumes Equal

Since the volume of the sphere is equal to the volume of the cylinder:

7776π0.09πh

We can cancel π from both sides:

77760.09h

Step 4: Solve for h

Now divide both sides by 0.09:

h77760.0986400cm

Step 5: Convert to Meters

To convert the length to meters:

h86400cm864m

Conclusion

The length of the wire formed from the melted sphere is 864 meters. This problem demonstrates the principles of volume conservation and the interplay between geometry and physics. Understanding such concepts is crucial for students and professionals in the fields of engineering and science.

Related Keywords:

volume calculation sphere to cylinder conversion volume conservation

Original Contribution:

The volume of the sphere is 4/3 π183 cubic cm. The radius of the wire is 0.3 cm. Its volume is π0.32h. Since volumes are equal, set these 2 expressions equal to each other and solve for h. The final length of the wire is 86400 cm, which is 0.537 miles.

Note: This is more of a solid geometry question than a physics one. You should presume that the density of the wire and the density of the sphere are the same to make this a problem about volumes rather than a physics one.