Determining Radius of Gyration of a Disc Rolling on an Inclined Plane

Objective:

The objective of this experiment is to determine the radius of gyration of a disc rolling on an inclined plane.

Apparatus:

1. Inclined Plane
2. Stop Watch
3. Vernier Caliper
4. Meter Rod

Theory

Moment of Inertia:

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.

Get quality help now

Writer Lyla

Verified writer

Proficient in: Physics

5 (876)

“ Have been using her for a while and please believe when I tell you, she never fail. Thanks Writer Lyla you are indeed awesome ”

+84 relevant experts are online

Hire writer

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.

This k is known as the radius of gyration.

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 the sum of the products of each mass and the square of its distance from the axis.

Parallel Axis Theorem:

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.

Radius of Gyration:

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.

Applications:

1. The gyration radius is useful in estimating the stiffness of a column. If the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment.
2. It can also be referred to as the radial distance from a given axis at which the mass of a body could be concentrated without altering the rotational inertia of the body about that axis.
3. In engineering, where continuous bodies of matter are generally the objects of study, the radius of gyration is usually calculated as an integral.
4. When it comes to translational motion, the idea of mass is straightforward: heavy objects are harder to push. In rotation, both the shape of an object and the point it's rotating about are critical in figuring out how easy or hard it is to spin.

Procedure:

1. Fix the apparatus according to the given instructions.
2. Check the balance on the higher and lower ends to remove any errors.
3. Start the experiment by calculating the height and the distance of the rolling bar using a meter rod.
4. Take the ball at the given height and release it to roll to the end of the inclined plane.
5. Measure the time it takes for the ball to reach the end of the inclined plane.
6. Measure the time period and calculate the radius of gyration for the given experiment.

Observation and Calculations:

Percentage Error = 6.12%

Comments:

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.