51. Homogeneous rod of length L = 40 cm, fixed

The vibration of a homogeneous rod of length L = 40 cm, fixed perpendicular to the horizontal axis and under the influence of gravity, is studied. It is necessary to determine the distance l from the center of mass to the suspension axis at which the oscillation frequency of the rod is maximum. It is assumed that friction forces can be neglected.

To solve the problem, you need to use the formula for the oscillation period of a mathematical pendulum:

T = 2π√(l/g)

where l is the distance from the center of mass to the suspension axis, g is the acceleration of free fall.

The oscillation frequency is related to the period as follows:

f = 1/T

Thus, to determine the maximum oscillation frequency, it is necessary to find a distance l at which the period T is minimal. To do this, it is necessary to differentiate the formula for the period with respect to l:

dT/dl = -π/√(gl)

Equating the derivative to zero, we get:

l = (π^2*g)/T^2

Substituting the value of the period, we get:

l = (π^2*g)/(4π^2*(L/2)/f)^2 = (g*L^2)/(16*f^2)

Thus, the distance l from the center of mass to the suspension axis, at which the oscillation frequency is maximum, is equal to:

l = (g*L^2)/(16*f^2) = (9.81*0.4^2)/(16*f^2) ≈ 0.098/f^2

This formula allows you to determine the optimal distance for the maximum frequency of vibration of a homogeneous rod at small amplitudes and the absence of friction forces.

Homogeneous rod for small vibrations

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This product is a digital product in Microsoft Word 2003 format, containing a detailed solution to the problem of oscillation of a homogeneous rod of length L = 40 cm, fixed perpendicular to the horizontal axis and under the influence of gravity. The material provides a description of the theoretical calculations and practical calculations necessary to determine the distance l from the center of mass to the suspension axis, at which the oscillation frequency of the rod is maximum. The product is suitable for students studying in areas related to physics and mechanics, teachers who want to supplement their lectures with practical examples, as well as physics and science lovers who want to expand their knowledge.

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The product is a solution to a physics problem describing the vibrations of a homogeneous rod 40 cm long, fixed perpendicular to the horizontal axis under the influence of gravity. The problem requires determining the distance l from the center of mass to the suspension axis at which the oscillation frequency of the rod will be maximum. The problem statement assumes that friction forces can be neglected.

The solution to the problem is designed in Microsoft Word 2003 using the formula editor. The text contains a detailed description of all stages of the solution, including calculations and formulas. The solution is suitable for students and anyone interested in physics and mechanics.

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