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A body consists of a solid hemisphere of radius r joined to a right circular cone of base radius r and perpendicular height h. The plane surfaces of the cone and hemisphere coincide and both solids are made of the same material. Show that the centre of gravity of the body lies on the axis of symmetry at a distance 3r^2-h^2/4(h+2r) from base of cone.
To show that the center of gravity of the body lies on the axis of symmetry at a specific distance from the base of the cone, we can use the principle of moments.
The center of gravity of a body is the point at which the entire weight of the body can be considered to act. In other words, if you were to balance the body on a small pivot at its center of gravity, it would remain in equilibrium.
To determine the position of the center of gravity, we need to consider the weight distribution of the body. The weight of the body is distributed throughout its volume and can be divided into small elemental parts.
Let's denote the weight distribution as w(x, y, z), where (x, y, z) represents the Cartesian coordinates of a specific point within the body. We assume that the weight distribution is uniform within each individual component (hemisphere and cone) of the body.
First, we can find the center of gravity for the hemisphere alone. The center of gravity of a solid hemisphere coincides with the center of the sphere. Therefore, the center of gravity of the hemisphere is located at a distance of r/2 from the base of the hemisphere along the axis of symmetry.
Next, we determine the center of gravity for the cone. The center of gravity of a right circular cone is located along the axis of symmetry at a distance of h/4 from the base of the cone.
Now, let's consider the combined system of hemisphere and cone. The total weight of the body is equal to the sum of the weights of the hemisphere and the cone. Since the two components are made of the same material, their densities are the same, and hence their weights are proportional to their respective volumes.
The volume of the hemisphere can be calculated using the formula (2/3)πr³, and the volume of the cone can be calculated using the formula (1/3)πr²h. Therefore, the weight distribution of the hemisphere and cone can be expressed as w_hemisphere(x, y, z) = (2/3)πr³ * (1/(4/3)πr²) = 3r/4 and w_cone(x, y, z) = (1/3)πr²h * (1/(4/3)πr²) = h/4.
Next, we need to determine the distance of the center of gravity of the combined system from the base of the cone. Let the distance be denoted as d.
Using the principle of moments, the sum of the moments of the weight distribution about any axis is equal to the moment of the total weight about the same axis.
Since we are interested in the position of the center of gravity along the axis of symmetry, we can consider moments about this axis. The moment of the hemisphere about the axis is zero since its center of gravity is already located on the axis. Therefore, the moment of the total weight about the axis of symmetry is equal to the moment of the cone about the axis.
The moment of the cone about the axis of symmetry is given by the formula: Moment_cone = w_cone * d = (h/4) * d.
Now, the total weight of the body is equal to the sum of the weights of the hemisphere and the cone: Total_weight = w_hemisphere + w_cone = 3r/4 + h/4 = (3r + h)/4.
According to the principle of moments, the moment of the total weight about the axis of symmetry is equal to the total weight times the distance of the center of gravity from the base of the cone. Therefore, we have:
Moment_total = Total_weight * d = (3r + h)/4 * d.
Since the moments of the total weight and the cone are equal, we can equate them:
(h/4) * d = (3r + h)/4 * d.
Simplifying this equation, we get:
(h/4) = (3r + h)/4.
Multiplying both sides by 4, we obtain:
h = 3r + h.
Subtracting h from both sides, we have:
0 = 3r.
Since this equation is true, it means that the distance of the center of gravity from the base of the cone, which is given by d, is equal to zero. In other words, the center of gravity of the body lies on the axis of symmetry, exactly at the base of the cone.
Therefore, the center of gravity of the body lies on the axis of symmetry at a distance of 0 from the base of the cone.