The height of the plane of the circle from the vertex of the funnel:

To determine the height of the plane of the circle from the vertex of the funnel, we need to consider the motion of the particle in the conical funnel.

1. Circular motion in a conical funnel:
When the particle is in circular motion in the conical funnel, it experiences two forces: the gravitational force and the normal force. The normal force acts perpendicular to the surface of the funnel and provides the necessary centripetal force for the circular motion.

2. Equating gravitational force and centripetal force:
The gravitational force acting on the particle is given by Fg = mg, where m is the mass of the particle and g is the acceleration due to gravity. The centripetal force is given by Fc = mω²r, where ω is the angular velocity and r is the radius of the circular path.

Since the particle is in equilibrium, the gravitational force and the centripetal force must be equal. Therefore, mg = mω²r.

3. Finding the radius of the circular path:
The radius of the circular path can be determined by considering the geometry of the conical funnel. Let h be the height of the plane of the circle from the vertex of the funnel, and R be the radius of the funnel at that height. Using similar triangles, we can write R/h = r/H, where H is the total height of the funnel.

Simplifying the equation, we get r = R(h/H).

4. Substituting the radius into the equilibrium equation:
Substituting the expression for r into the equilibrium equation, we have mg = mω²R(h/H).

Simplifying the equation, we get ω² = g(H/h).

5. Finding the angular velocity:
The angular velocity ω is related to the linear velocity v by the equation v = ωr. Since the linear velocity of the particle is given as 0.5 m/s, we can write v = 0.5 m/s = ωR(h/H).

Simplifying the equation, we get ω = 0.5/(R(h/H)).

6. Finding the height of the plane of the circle:
Substituting the expression for ω into the equation for ω², we have (0.5/(R(h/H)))² = g(H/h).

Simplifying the equation, we get 0.25R²(h/H)² = g(H/h).

Simplifying further, we get (h/H)³ = 0.25R²/g.

Taking the cube root of both sides, we have h/H = ∛(0.25R²/g).

Finally, multiplying both sides by H, we get h = H * ∛(0.25R²/g).

Embryogenesis, Zygote development in plants:

Embryogenesis is the process of development from a zygote to an embryo. In plants, zygote development involves several stages:

1. Fertilization: Pollen grains are transferred to the stigma of the flower, where they germinate and form pollen tubes. The pollen tubes grow down the style and reach the ovule, where they release sperm cells. One of the sperm