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.