Question

The centre of gravity for a solid hemisphere about its diametral base having diameter 120 cm and is symmetric about y-axis is (in cm):

• A. \(22.5\)
• B. \(90\)
• C. \(135\)
• D. \(45\)

Hint:

Be careful not to confuse a solid hemisphere with a hollow hemispherical shell: - Solid Hemisphere: Distance from base = \(\frac{3R}{8}\) - Hollow Hemispherical Shell: Distance from base = \(\frac{R}{2}\) Using the correct solid hemisphere formula yields 22.5 cm.

Answer 1

The Correct Option is A

Concept: For a uniform, homogeneous solid hemisphere of radius \(R\), the center of gravity lies along its central geometric axis of symmetry. Measured from the flat, circular diametral base plane along this axis of symmetry, the distance to the center of gravity is given by the standard formula: \[ \bar{y} = \frac{3R}{8} \]

Step 1: Finding the radius of the solid hemisphere.
The problem statement gives the total diameter of the circular diametral base: \[ D = 120 \text{ cm} \] The radius \(R\) is half of the total base diameter: \[ R = \frac{D}{2} = \frac{120}{2} = 60 \text{ cm} \]

Step 2: Finding the position using the symmetry condition.
The problem states that the solid hemisphere is symmetric about the \(y\)-axis.
• This symmetry means the center of gravity must lie directly on the vertical \(y\)-axis (\(\bar{x} = 0\)).
• The flat diametral base rests on the horizontal \(x\)-axis, meaning the distance up from the base corresponds to the \(\bar{y}\) coordinate.

Step 3: Calculating the \(\bar{y}\) coordinate value.
Substitute our calculated radius value \(R = 60 \text{ cm}\) into the center of gravity formula: \[ \bar{y} = \frac{3R}{8} \] \[ \bar{y} = \frac{3 \times 60}{8} \] \[ \bar{y} = \frac{180}{8} \] Dividing the numerator and the denominator by their common factor of 4: \[ \bar{y} = \frac{45}{2} = 22.5 \text{ cm} \] Thus, the distance from the diametral base to the center of gravity is exactly 22.5 cm, matching option (1).