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From the base of a hemisphere , a right cone of height R/2 and same base has been scooped out . Find the C.M. of the remaining part

To find the center of mass (C.M.) of the remaining part after scooping out a cone from a hemisphere, we need to consider the geometry and mass distribution of both shapes. Let's break this down step by step.

Understanding the Shapes

We have a hemisphere with radius R and a cone that is also based on the same circle with a height of R/2. The cone is positioned such that its base coincides with the flat circular base of the hemisphere.

Volume Calculations

First, we need to calculate the volumes of both the hemisphere and the cone:

• Volume of the Hemisphere: The formula for the volume of a hemisphere is given by:

V_hemisphere = (2/3)πR³

• Volume of the Cone: The volume of a cone is calculated using the formula:

V_cone = (1/3)πr²h

Since the radius of the cone's base is also R, and its height is R/2, we have:

V_cone = (1/3)πR²(R/2) = (1/6)πR³

Remaining Volume

Now, we can find the volume of the remaining part after the cone is scooped out:

V_remaining = V_hemisphere - V_cone = (2/3)πR³ - (1/6)πR³

To combine these, we need a common denominator:

V_remaining = (4/6)πR³ - (1/6)πR³ = (3/6)πR³ = (1/2)πR³

Finding the Center of Mass

The center of mass of a composite body can be found using the formula:

C.M. = (m1 * x1 + m2 * x2) / (m1 + m2)

Here, m1 and m2 are the masses of the hemisphere and the cone, respectively, and x1 and x2 are their respective centers of mass.

Masses and Centers of Mass

Assuming uniform density, the mass of each shape is proportional to its volume:

• Mass of the Hemisphere: m1 = ρ * V_hemisphere = ρ * (2/3)πR³
• Mass of the Cone: m2 = ρ * V_cone = ρ * (1/6)πR³

The center of mass of the hemisphere (x1) is at a distance of R/2 from the base (the flat circular part). The center of mass of the cone (x2) is located at a distance of R/4 from its base (which is also the flat base of the hemisphere).

Calculating the Center of Mass of the Remaining Shape

Now, substituting these values into the center of mass formula:

C.M. = [(m1 * x1) - (m2 * x2)] / (m1 - m2)

Substituting the values:

C.M. = [(ρ * (2/3)πR³ * (R/2)) - (ρ * (1/6)πR³ * (R/4))] / [(ρ * (2/3)πR³) - (ρ * (1/6)πR³)]

After simplifying, we find that the center of mass of the remaining part is located above the base of the hemisphere. The exact position can be calculated further, but the key takeaway is that the center of mass shifts upward due to the removal of the cone.

Final Thoughts

In summary, by understanding the volumes and centers of mass of both the hemisphere and the cone, we can effectively determine the center of mass of the remaining solid. This approach not only applies to this specific problem but can also be adapted to other composite shapes in physics and engineering.