Dynamics Lab Report - Radius of Gyration

Categories: Physics

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Introduction:

The study of dynamics and the principles governing the motion of objects is fundamental to understanding the physical world around us. In this lab report, we investigate the concept of the radius of gyration by conducting experiments with a rolling disc on an inclined plane. The radius of gyration is a crucial parameter in rotational motion, providing insights into how mass is distributed in a rotating body. By conducting this experiment, we aim to determine the radius of gyration of a disc rolling on an inclined plane and explore its practical applications.

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Apparatus:

Inclined Plane
Stopwatch
Vernier Caliper
Meter Rod

Theory:

Moment of Inertia:

Moment of inertia is the product of mass and the square of the moment arm. The moment arm is the perpendicular distance from the axis of rotation to the point of mass.

Moment of Inertia of Mass Distributions:

The moment of inertia of an n-point mass system at perpendicular distances from the axis of rotation is given by:

I = ∑ (m_i * r_i²)

Parallel Axis Theorem:

All moments of inertia are calculated along an axis of rotation that passes through the center of the object's mass.

Rigid Body:

If a mechanical device moves parallel to a fixed plane, its rotation occurs along an axis perpendicular to the plane. The moment of inertia of the mass in the system is a scalar called the polar moment of inertia.

Radius of Gyration:

The radius of gyration is the perpendicular distance from the axis of rotation to the point mass.

Applications:

The radius of gyration is useful in finding the stiffness of a column.

If the moments of 2-D gyration tensors are unequal, then the column will buckle along the axis with the smaller principal moment.

Procedure:

Fix the apparatus according to the provided instructions.
Check the balance on the higher and lower ends to remove errors. Unbalancing can cause deviations as the rolling ball touches the surface and may lead to reading errors.
Calculate the height and the distance of the rolling bar with a meter rod.
Take the ball to the given height and release it to roll to the end of the inclined bar.
Measure the time it takes for the ball to reach the end of the inclined plane.
Measure the time period and the radius of gyration for the given experiment.

Observation and Calculations:

Initial Height (h₁) (cm)	Final Height (h₂) (cm)	Time Taken by Disc (sec)	Angular Velocity (ω = 2S/rt) (rad/sec)	Radius of Gyration (exp) (cm)	Radius of Gyration (th) (cm)
19.7	10.8	19.9	19.53	12.42	0.707R
23.7	5.58	16.5	16.2	16.3	0.707R
17.2	5.58	22.9	22.8	22.8	0.707R
13	5.58	32.8	32.5	33.1	0.707R

Percentage Error: 6.12%

Discussion:

In this experiment, we aimed to determine the radius of gyration of a rolling disc on an inclined plane. We conducted multiple trials and calculated the radius of gyration both experimentally and theoretically. The percentage error of 6.12% indicates that there were sources of error in the experiment.

The experimental and theoretical values of the radius of gyration were found to be in agreement, which validates the accuracy of the measurements and calculations. The parallel axis theorem was applied to determine the theoretical value of the radius of gyration.

One possible source of error in this experiment is the unbalancing of the inclined plane, leading to deviations in the motion of the rolling disc. Additionally, human errors in timing measurements could have contributed to the overall error percentage.

Conclusion:

The objective of this experiment was to determine the radius of gyration of a disc rolling on an inclined plane. Through careful measurements and calculations, we obtained both experimental and theoretical values for the radius of gyration. The percentage error of 6.12% suggests that while there were sources of error, the experiment yielded reasonably accurate results.

This experiment emphasizes the importance of moment of inertia and its practical applications in understanding rotational motion. The concept of the radius of gyration is a valuable tool in engineering and physics, providing insights into the distribution of mass in rotating bodies.

In conclusion, this lab report has successfully demonstrated the principles of moment of inertia and the determination of the radius of gyration in a practical setting.