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A torsional pendulum involves the oscillation of a rigid body in which the restoring force is provided by the torsion in the string from which the rigid body is suspended. Ideally, the string should be massless; practically, its mass is much smaller than the rigid body's mass and is neglected.
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When a rigid body is hanging freely from a fixed pivot point and is displaced, it oscillates similar to a simple pendulum and is known as a physical pendulum. The period and angular frequency of a physical pendulum are obtained by using the small-angle approximation and drawing parallels with a spring-mass system. The small-angle approximation (sinθ=θ) is valid up to about 14°.
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Precession can be demonstrated effectively through a spinning top. If a spinning top is placed on a flat surface near the surface of the Earth at a vertical angle and is not spinning, it will fall over due to the force of gravity producing a torque acting on its center of mass. However, if the top is spinning on its axis, it precesses about the vertical direction, rather than topple over due to this torque. Precessional motion is a combination of a steady circular motion of the axis and the...
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Relative Motion Analysis using Rotating Axes-Problem Solving01:29

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Consider a crane whose telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2. Along with the rotation, the boom also extends linearly with a uniform speed of 5 m/s. The extension of the boom is measured at point D, which is measured with respect to the fixed point C on the other end of the boom. For the given instant, the distance between points C and D is 60 meters.
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Rotational Motion about a Fixed Axis01:26

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A rigid body's rotation around a fixed axis makes every point within it trace a circular path around a specific line or point. The term given to this type of spinning is defined by the angular position, symbolized by the angle θ. This angle is gauged from a static reference line to the revolving object. From this angular position, any variation is referred to as angular displacement, denoted by dθ. The extent of this displacement can be calculated in degrees, radians, or...
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Updated: Apr 6, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Balancing a retroreflector to minimize rotation errors using a pendulum and quadrature interferometer.

T M Niebauer, A Constantino, R Billson

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    Collocating the center of mass with the optical center is crucial for gravity meters. This study introduces a novel pendulum-based method using a quadrature interferometer for precise alignment, improving accuracy by an order of magnitude.

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    Area of Science:

    • Metrology
    • Optical Engineering
    • Gravimetry

    Background:

    • Corner-cube retroreflectors offer rotationally invariant optical path lengths.
    • Accurate gravity measurements require precise collocation of the proof mass center of mass (COM) and retroreflector optical center (OC).
    • Traditional mechanical balancing methods for COM-OC collocation lack sufficient accuracy.

    Purpose of the Study:

    • To develop a novel, high-accuracy method for collocating the COM and OC of a proof mass in ballistic absolute gravity meters.
    • To overcome the limitations of traditional mechanical balancing techniques.

    Main Methods:

    • Incorporating the proof mass into a pendulum system.
    • Utilizing a quadrature interferometer to measure apparent translation in the twist mode.
    • Detecting the signal generated by COM-OC mismatch in a resonance-free spectral region.

    Main Results:

    • Achieved COM-OC tuning accuracy of approximately 1 μm in all three axes.
    • Demonstrated that proof mass rotations of several degrees result in apparent laser beam translation of less than 1 nm.
    • Showcased an order of magnitude improvement in accuracy compared to traditional methods.

    Conclusions:

    • The novel pendulum and quadrature interferometer method significantly enhances the accuracy of COM-OC collocation.
    • This improved collocation is critical for reducing vertical position errors in free-falling proof masses during gravity measurements.
    • The technique offers a substantial advancement for high-precision gravimetry and related optical metrology applications.