A cylinder of 50 mm diameter is 250 mm high and has a semi-ball at the top as shown in the sketch (FIGURE 2) below. Calculate the centre of gravity of the body from the table on which it is placed.

QUESTION 5.2: CENTRE OF GRAVITY

Given: Cylinder diameter $D = 50 \text{ mm}$, so radius $r = 25 \text{ mm}$ Cylinder height $h_c = 250 \text{ mm}$ Hemisphere diameter $D = 50 \text{ mm}$, so radius $r = 25 \text{ mm}$

Step 1: Calculate the volume and centroid of the cylinder from its base (the table). The volume of the cylinder is $V_c = \pi r^2 h_c$. $V_c = \pi (25 mm)^2 (250 mm) = 156250\pi mm^3$ The centroid of the cylinder from its base is $y_c = \frac{h_c}{2}$. $y_c = \frac{250 mm}{2} = 125 mm$

Step 2: Calculate the volume and centroid of the hemisphere from the base of the cylinder (the table). The volume of the hemisphere is $V_h = \frac{2}{3} \pi r^3$. $V_h = \frac{2}{3} \pi (25 mm)^3 = \frac{2}{3} \pi (15625) mm^3 = \frac{31250}{3}\pi mm^3$ The centroid of a hemisphere from its flat base is $\frac{3}{8}r$. Since the hemisphere is placed on top of the cylinder, its flat base is at $h_c = 250 \text{ mm}$ from the table. So, the centroid of the hemisphere from the table is $y_h = h_c + \frac{3}{8}r$. $y_h = 250 mm + \frac{3}{8}(25 mm) = 250 mm + 9.375 mm = 259.375 mm$

Step 3: Calculate the overall centre of gravity $\bar{Y}$ of the composite body from the table using the formula $\bar{Y} = \frac{V_c y_c + V_h y_h}{V_c + V_h}$. $\bar{Y} = \frac{(156250\pi mm^3)(125 mm) + (\frac{31250}{3}\pi mm^3)(259.375 mm)}{156250\pi mm^3 + \frac{31250}{3}\pi mm^3}$ We can cancel $\pi$ from all terms: $\bar{Y} = \frac{(156250)(125) + (\frac{31250}{3})(259.375)}{156250 + \frac{31250}{3}}$ $\bar{Y} = \frac{19531250 + 2701822.9166...}{156250 + 10416.6666...}$ $\bar{Y} = \frac{22233072.9166...}{166666.6666...}$ $\bar{Y} \approx 133.40 mm$ The centre of gravity of the body from the table is $\boxed{133.40 mm}$.

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