The frequency of small oscillations for a system like this can be found using the formula for the frequency of a simple harmonic oscillator, which is f = (1/2π) * sqrt(k/m), where k is the spring constant and m is the mass of the object.

However, in this case, the pivot point is not at the center of the stick, so we need to take into account the moment of inertia I of the stick about the pivot point. The moment of inertia of a uniform stick about its center is (1/12) * m * l^2, but since the pivot point is a distance x from the center, we need to use the parallel axis theorem to find the moment of inertia about the pivot point. The parallel axis theorem states that I = I_cm + m * d^2, where I_cm is the moment of inertia about the center of mass, m is the mass, and d is the distance from the center of mass to the new axis. In this case, I = (1/12) * m * l^2 + m * x^2.

The frequency of small oscillations is then given by f = (1/2π) * sqrt(k/I). To maximize the frequency, we need to minimize the moment of inertia I. Taking the derivative of I with respect to x and setting it equal to zero, we find that x = 0 or x = ±l/√3. Since x must be positive, the value of x that yields the largest frequency of small oscillations is x = l/√3.

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The frequency of small oscillations for a system like this can be found using the formula for the frequency of a simple harmonic oscillator, which is f = (1/2π) * sqrt(k/m), where k is the spring constant and m is the mass of the object.

However, in this case, the pivot point is not at the center of the stick, so we need to take into account the moment of inertia I of the stick about the pivot point. The moment of inertia of a uniform stick about its center is (1/12) * m * l^2, but since the pivot point is a distance x from the center, we need to use the parallel axis theorem to find the moment of inertia about the pivot point. The parallel axis theorem states that I = I_cm + m * d^2, where I_cm is the moment of inertia about the center of mass, m is the mass, and d is the distance from the center of mass to the new axis. In this case, I = (1/12) * m * l^2 + m * x^2.

The frequency of small oscillations is then given by f = (1/2π) * sqrt(k/I). To maximize the frequency, we need to minimize the moment of inertia I. Taking the derivative of I with respect to x and setting it equal to zero, we find that x = 0 or x = ±l/√3. Since x must be positive, the value of x that yields the largest frequency of small oscillations is x = l/√3.

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The frequency of small oscillations for a system like this can be found using the formula for the frequency of a simple harmonic oscillator, which is f = (1/2π) * sqrt(k/m), where k is the spring constant and m is the mass of the object.

However, in this case, the pivot point is not at the center of the stick, so we need to take into account the moment of inertia I of the stick about the pivot point. The moment of inertia of a uniform stick about its center is (1/12) * m * l^2, but since the pivot point is a distance x from the center, we need to use the parallel axis theorem to find the moment of inertia about the pivot point. The parallel axis theorem states that I = I_cm + m * d^2, where I_cm is the moment of inertia about the center of mass, m is the mass, and d is the distance from the center of mass to the new axis. In this case, I = (1/12) * m * l^2 + m * x^2.

The frequency of small oscillations is then given by f = (1/2π) * sqrt(k/I). To maximize the frequency, we need to minimize the moment of inertia I. Taking the derivative of I with respect to x and setting it equal to zero, we find that x = 0 or x = ±l/√3. Since x must be positive, the value of x that yields the largest frequency of small oscillations is x = l/√3.

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