Explanation:

To find the angular frequency (ω) of small oscillations of a particle near the minimum of a vertical parabolic path with the equation x² = 4ay, you can use the formula for angular frequency in simple harmonic motion:

ω = √(k / m)

Where:

ω = Angular frequency

k = Spring constant

m = Mass of the particle

In this case, you need to determine the spring constant, k. To do that, you'll have to relate this parabolic path to simple harmonic motion. For small oscillations near the minimum of a parabolic path, the equation can be approximated as:

x ≈ A sin(ωt)

Where:

x = Displacement from the equilibrium position

A = Amplitude of the oscillation (equal to the distance from the minimum point to the equilibrium position)

ω = Angular frequency

t = Time

Now, let's relate this to the given parabolic equation, x² = 4ay. Near the minimum point (equilibrium position), x is approximately zero. So, we can write:

0 ≈ A sin(ωt)

This means that for small oscillations, sin(ωt) is approximately equal to zero. The smallest positive value of ωt that satisfies this condition is π (180 degrees). So, we can write:

sin(ωt) ≈ sin(π)

Now, sin(π) = 0, so:

0 ≈ 0

This equation holds true, confirming that ωt = π.

Now, you can find ω:

ω = π / t

To find t, you need to consider the period of oscillation. In simple harmonic motion, the period (T) is related to the angular frequency as:

T = 2π / ω

So, ω = 2π / T.

Now, you'll need information about the period of oscillation to calculate ω. If you have the period (T), you can substitute it into the equation to find ω ..