Introduction:
To find the period of small oscillations of a uniform rod rotating freely about a fixed axis through its end and perpendicular to its length, we can use the principles of rotational motion and harmonic motion.

Key Concepts:
1. Moment of Inertia of a Uniform Rod: The moment of inertia of a uniform rod rotating about an axis through its end and perpendicular to its length is given by the formula I = (1/3) * m * l^2, where m is the mass of the rod and l is the length of the rod.

2. Torque and Angular Acceleration: When an external torque is applied to the rod, it produces angular acceleration according to the equation τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

3. Simple Harmonic Motion: For small oscillations, the motion of the rod can be approximated as simple harmonic motion, where the restoring torque is proportional to the angular displacement. The equation for simple harmonic motion is τ = -k * θ, where τ is the restoring torque, k is the angular spring constant, and θ is the angular displacement.

Calculating the Period:
To find the period of small oscillations, we need to calculate the angular spring constant and relate it to the period.

1. Calculating the Angular Spring Constant:
From the equation τ = -k * θ, we can equate it with τ = I * α and solve for k.
-τ = I * α
-k * θ = I * (d^2θ/dt^2)
-k * θ = (1/3) * m * l^2 * (d^2θ/dt^2)
Comparing the coefficients, we get k = (1/3) * m * l^2.

2. Relating the Angular Spring Constant to Period:
For simple harmonic motion, the angular frequency (ω) is related to the angular spring constant (k) by the equation ω = √(k/I).

The period (T) of the small oscillations can be calculated using the equation T = 2π/ω.

Substituting the value of ω and rearranging the equation, we get T = 2π * √(I/k).
T = 2π * √(3l^2/3ml^2)
T = 2π * √(1/m)
T = 2π * √(1/g) [Using g = 9.8 m/s^2]

Conclusion:
The period of small oscillations of a uniform rod rotating freely about a fixed axis through its end and perpendicular to its length is given by T = 2π * √(1/g), where g is the acceleration due to gravity. This period is independent of the length and mass of the rod.