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The objective of this experiment is to determine the radius of gyration of a disc rolling on an inclined plane.
Newton’s First Law tells us that "a body remains at rest or moves with uniform velocity in a straight line unless an external force acts on it." This concept gives rise to the idea of inertia, which is a property of mass. The more massive an object, the greater its inertia.
Rotational Inertia, also known as the moment of inertia (Im), is a measure of how different bits of mass contribute to rotational inertia, depending on their distances from the axis of rotation.
Im is defined as the ratio of the net angular momentum (L) of a system to its angular velocity (ω) around a principal axis:
I = L / ω
If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase.
This occurs when objects change their distribution of mass to spin faster, like figure skaters pulling in their arms.
The moment of inertia Im depends on both the mass (m) of a body and its geometry, defined by the distance (r) to the axis of rotation.
I = mr²
For an arbitrarily shaped body, Im can be calculated as the sum of all elemental point masses (dm) each multiplied by the square of their perpendicular distance (r) to an axis:
I = ∑(dm * r²)
In general, for an object of mass (m), an effective radius (k) can be defined for an axis through its center of mass such that Im = mk².
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This k is known as the radius of gyration.
The moment of inertia of an n-point mass system at perpendicular distances from the axis of rotation is given by the sum of the products of each mass and the square of its distance from the axis.
The parallel axis theorem is used to find the moment of inertia of an object when the axis of rotation is not through its center of mass. If the new axis of rotation is parallel to the axis through the center of mass, the moment of inertia (I) can be calculated using the following equation:
I = Icm + Md²
Where:
I is the new moment of inertia around the new axis of rotation,
Icm is the original moment of inertia around the axis through the center of mass,
M is the object's mass,
d is the distance between the old and new axes of rotation.
The radius of gyration (k) of a body about an axis of rotation is defined as the radial distance from the axis at which the mass of a body may be assumed to be concentrated, and at which the moment of inertia will be equal to the moment of inertia of the actual mass about the axis.
Mathematically, the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application.
| Initial Height (cm) | Final Height (cm) | Time taken by disc (sec) | Angular Velocity (ω = 2S/rt) (rad/sec) | Radius of Gyration (exp) (cm) | Radius of Gyration (th) (0.707R) (cm) |
|---|---|---|---|---|---|
| 19.7 | 10.8 | 19.9 | 19.8 | 19.53 | 19.8 |
| 12.42 | 4.9 | 5.33 | 23.7 | 5.58 | 16.5 |
| 16.2 | 16.3 | 16.3 | 16.3 | 14.42 | 5.15 |
| 22.8 | 22.8 | 22.8 | 22.8 | 10.29 | 5.7 |
| 32.5 | 34 | 33.1 | 7.13 | 5.0 | 5.33 |
Percentage Error = 6.12%
This lab report provides insights into the differences between inertia and moment of inertia (Im) and their significance in our lives. It also highlights the applications of Im in various real-life scenarios and how it is influenced by factors such as mass distribution and axis of rotation.
The radius of gyration (k) is introduced as a key concept, and its importance in estimating the stiffness of columns and understanding rotational motion is discussed.
The observed percentage error in the experiment is attributed to factors such as the unbalancing of the inclined surface due to apparatus issues and human errors. These sources of error can impact the accuracy of experimental results.
Determining Radius of Gyration of a Disc Rolling on an Inclined Plane. (2024, Jan 05). Retrieved from https://studymoose.com/document/determining-radius-of-gyration-of-a-disc-rolling-on-an-inclined-plane
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