How to find the shm equation(for small angles) for a right circular cone of radius r and height h immersed in water?

where do we use the fact that its area is a variable over height?

To derive the simple harmonic motion (SHM) equation for a right circular cone of radius \( r \) and height \( h \) immersed in water, we need to consider the forces acting on the cone and how its submerged area changes with height. This analysis will help us understand the oscillatory behavior of the cone when it is displaced from its equilibrium position.

Understanding the Forces at Play

When the cone is partially submerged in water, the buoyant force acting on it is determined by the volume of water displaced, which in turn depends on the submerged height of the cone. The buoyant force can be expressed using Archimedes' principle:

• The buoyant force \( F_b \) is equal to the weight of the displaced water.
• For a cone submerged to a height \( x \), the volume of the displaced water can be calculated using the formula for the volume of a cone: \( V = \frac{1}{3} \pi r^2 h \). However, since the radius of the submerged part changes with height, we need to find the radius at height \( x \).

Finding the Submerged Volume

For a cone, the radius at a given height \( x \) can be found using similar triangles. The relationship between the radius \( r_x \) at height \( x \) and the total height \( h \) is given by:

r_x = \frac{r}{h} x

Thus, the volume of the submerged part of the cone becomes:

V = \frac{1}{3} \pi r_x^2 x = \frac{1}{3} \pi \left(\frac{r}{h} x\right)^2 x = \frac{\pi r^2}{3h^2} x^3

Calculating the Buoyant Force

The buoyant force can now be expressed as:

F_b = \rho_w g V = \rho_w g \left(\frac{\pi r^2}{3h^2} x^3\right)

Here, \( \rho_w \) is the density of water and \( g \) is the acceleration due to gravity. The weight of the cone, \( W \), is constant and can be expressed as:

W = \rho_c V_c = \rho_c \left(\frac{1}{3} \pi r^2 h\right)

Setting Up the Equation of Motion

When the cone is displaced by a small angle \( \theta \), we can relate the vertical displacement \( x \) to the angle using the small angle approximation, where \( \sin(\theta) \approx \theta \). The net force acting on the cone when displaced is given by:

F_{net} = F_b - W = \rho_w g \left(\frac{\pi r^2}{3h^2} x^3\right) - W

For small displacements, we can linearize this equation. The restoring force can be approximated as:

F_{net} \approx -k x

where \( k \) is a constant that depends on the geometry of the cone and the properties of the fluid. This leads us to the standard form of the SHM equation:

m \frac{d^2x}{dt^2} + kx = 0

Final Thoughts on the Area Variation

The fact that the area of the submerged part of the cone varies with height is crucial in determining the volume of water displaced, which directly affects the buoyant force. This variation is what allows us to derive the relationship between the displacement and the restoring force, ultimately leading to the SHM equation. In practical applications, understanding this relationship helps in designing systems that involve floating or submerged bodies, such as buoys or underwater sensors.