Question

According to the theorem of parallel axis \(\mathrm{I}=\mathrm{I}_{\mathrm{cm}}+\mathrm{md}^{2}\) the graph between \(\mathrm{I} \rightarrow \mathrm{d}\) will be

Short answer

The graph between the moment of inertia (I) and distance (d) based on the parallel axis theorem will be a parabolic curve that opens upwards, showing a positive and quadratic relation between I and d. This is due to the equation \(I = I_{cm} + md^2\), where Icm and m are fixed values for a specific object.

Step-by-step solution

Understanding The Variables and Formula

The equation for parallel axis theorem is given by: \(I = I_{cm} + md^{2}\) where: - I: Moment of inertia about an axis parallel to the axis passing through the center of mass - Icm: Moment of inertia about the axis passing through the center of mass - m: Mass of the object - d: Distance between the two parallel axes The equation represents the relationship between the moment of inertia around an axis parallel to the one passing through the center of mass and distance between both axes, mass and moment of inertia around the center of mass.

Fixing the Known Values

In order to create a graph between I and d, we need to fix the values of mass (m) and moment of inertia about the center of mass (Icm). These values will remain constant so that we can understand how the moment of inertia (I) varies with distance (d). For example, let's choose m = 5 kg and Icm = 10 kg m^2. (Feel free to choose any other values for clarification)

Writing the Equation with Fixed Values

Now that we have our fixed values for mass and moment of inertia about the center of mass, let's substitute them into the formula: \(I = 10 + 5 * d^{2}\)

Creating a Table of Values

To create a graph between I and d, we'll need to create a table with a variety of values for distance (d) and use the equation to calculate the corresponding moment of inertia (I). The table may look like this: d | I --------|------- 0 | 10 1 | 15 2 | 30 3 | 55 4 | 90

Plotting the Graph

With the table of values for I and d, we can now plot the graph. On the horizontal x-axis, represent the distance (d), and on the vertical y-axis, represent the moment of inertia (I). Plot the points from the table and connect them with a smooth curve. Considering the equation has a quadratic form (\(I = 10 + 5d^2\)), the graph will be a parabolic curve that opens upwards. #Conclusion# The graph between the moment of inertia (I) and distance (d) will be a parabolic curve that opens upwards, based on the theorem of the parallel axis, with the fixed values for mass (m) and moment of inertia about the center of mass (Icm). The graph will show a positive and quadratic relation between I and d.

Key concepts

Moment of Inertia

The concept of "Moment of Inertia" is central to understanding why objects rotate the way they do. It quantifies how difficult it is to change the rotation of an object. Just like mass determines how difficult it is to change an object's state of motion linearly, moment of inertia is the rotational counterpart. In simpler terms, it tells us how much an object resists rotational acceleration around an axis. Two main factors influence it:

• The mass of the object.
• The distribution of mass relative to the axis of rotation.

In our exercise, we use the parallel axis theorem, which connects the moment of inertia about an axis through the center of mass (\( I_{cm} \)) and an axis parallel to it, located a distance (\( d \)) away. By adjusting the parallel axis theorem equation, we can see how changing the distance affects the moment of inertia.

Quadratic Relationship

In mathematics, a quadratic relationship describes a situation where one quantity varies with the square of another. If you've ever graphed an equation like \( y = x^2 \), you'll recognize the resulting shape is a parabola. This exercise demonstrates that relationship through the equation:\( I = 10 + 5d^2 \). Here:

• The moment of inertia (\( I \)) varies with the square of the distance (\( d \)).
• A constant term (\( 10 \)) represents the moment of inertia about the center of mass (\( I_{cm} \)).
• The coefficient of \( d^2 \) multiplies this distance, emphasizing the quadratic nature.

As \( d \) increases, \( I \) grows more rapidly because the dependency is not linear, but quadratic. This means that small increases in \( d \) lead to larger changes in \( I \).

Physics Graphing

Graphing in physics isn't just about plotting numbers on a chart; it's a visual way to make powerful connections between variables. For our exercise, creating a graph between moment of inertia \( I \) and distance \( d \) reveals more than raw data might suggest.Here's how graphing enhances understanding:

• The plotted points lay out how exactly \( I \) grows as \( d \) changes, forming a parabolic curve.
• It visualizes the quadratic relationship and makes it easier to predict how \( I \) might behave for values yet uncalculated.
• The graph gives an intuitive grasp of not only mathematical relationships but physical principles, like how shape or configuration affects rotational dynamics.

When you plot \( I \) against \( d \), you can see the effects of increasing \( d \) quickly because the graph's curve steepens.

Rotational Dynamics

"Rotational Dynamics" is the study of objects in rotational motion and the forces that affect them. It looks at the interplay between torques, moments of inertia, and angular motion. This part of physics describes how rotational motion changes under various influences, whether they arise from external or internal forces. In studying rotational dynamics, you encounter important concepts like:

• Torque, the rotational equivalent of force, which tends to change the rotational motion.
• Angular momentum, which relates to the conservation of rotational motion in isolated systems.
• And of course, the moment of inertia, representing the "rotational mass.”

Understanding these concepts helps in analyzing real-world systems like wheels, gears, and planets. The parallel axis theorem provides a practical way to measure how rotational dynamics change when the axis of rotation shifts, like in assemblies or shifting loads.