Question

Moment of inertia of a sphere of mass $\mathrm{M}$ and radius $\mathrm{R}$ is $\mathrm{I}$. keeping mass constant if graph is plotted between $\mathrm{I}$ and $\mathrm{R}$ then its form would be.

Short answer

The graph between the moment of inertia (I) and radius (R) of a sphere while keeping the mass constant (M) would be a parabolic curve with the equation: $I(R)={\displaystyle \frac{2}{5}}M{R}^2$

Step-by-step solution

Moment of Inertia of a Solid Sphere

The moment of inertia of a given object depends on its mass distribution. For a solid sphere of mass M and radius R, the moment of inertia is given by the formula: $I={\displaystyle \frac{2}{5}}M{R}^2$

Study the relationship between I and R

To analyze the relationship between I (moment of Inertia) and R (radius) while keeping the mass M constant, we can rewrite the formula as: $I(R)={\displaystyle \frac{2}{5}}M{R}^2$ Note that M is a constant value, so the graph of I with respect to R will be determined by the R^2 term.

Graphing I(R)

To plot I as a function of R, we see that I increases with R^2. This means that the graph will be a parabolic curve, with the vertex at the origin (0,0) and opening upwards (since R^2 is always positive). The graph will have the form of y = kx^2, where k is a constant, in this case, k = (2/5) * M. Therefore, the graph between the moment of inertia (I) and radius (R) of a sphere while keeping the mass constant M would be a parabolic curve with the equation: $I(R)={\displaystyle \frac{2}{5}}M{R}^2$

Key concepts

Physics Education

Understanding physics requires breaking down complex concepts into simpler ideas that students can easily relate to. One such concept is the moment of inertia, especially when dealing with various shapes and mass distributions. In physics education, teaching about solid spheres, their mass distribution, and how these relate to real-world applications helps students create better mental models of the physical world.

Visual aids, such as graphs, play a crucial role in helping students grasp complicated concepts by providing a visual representation of relationships in formulas.

• Clear step-by-step solutions simplify intricate problems.
• Teaching methods that incorporate both theoretical and practical experiences build a stronger foundation in understanding physics.
• Encouraging students to ask questions and explore different scenarios enhances critical thinking.

Understanding the moment of inertia in a solid sphere is one of the key building blocks in mastering physics.

Solid Sphere

A solid sphere is a three-dimensional object where every point on its surface is equidistant from its center. It is an essential object of study in physics because of its symmetrical properties, making it easier to derive mathematical patterns. The moment of inertia of a solid sphere can be calculated using the formula:

$$I={\displaystyle \frac{2}{5}}M{R}^2$$

where $M$ is the mass of the sphere and $R$ is its radius. This formula shows how the mass and radius of a sphere affect its rotational inertia.

• This property is critical in various engineering applications.
• The symmetrical nature simplifies calculations compared to irregular objects.
• Inertia plays a significant role in dynamics, involving spinning and rotation.

Understanding solid spheres helps in exploring how mass distribution can affect motion and handle practical rotational dynamics scenarios.

Mass Distribution

Mass distribution refers to how the mass is spread across an object, and it dramatically influences an object's moment of inertia. For a solid sphere, the mass is evenly distributed, leading to unique rotational characteristics. The formula for a sphere's moment of inertia—$I={\displaystyle \frac{2}{5}}M{R}^2$—directly incorporates mass distribution by considering mass (M) and radius (R) squared.

The even distribution of mass in a spear means its moment of inertia depends less heavily on its shape changes and more on size and mass consistency.

• Larger spheres have higher inertia due to increased mass distribution over a larger radius.
• The uniform distribution simplifies calculations and assists engineers and physicists in problem-solving.
• Studying how mass affects motion empowers students to understand real-world physics applications, from simple machines to complex systems.

Insight into mass distribution not only leads to a better grasp of rotational inertia but extends to other fields like material science and mechanics.

Parabolic Graph

A parabolic graph is formed when plotting the relationship between the moment of inertia $I$ and the radius $R$ of a solid sphere, keeping mass constant. This is described by the equation:

$$I(R)={\displaystyle \frac{2}{5}}M{R}^2$$

Such graphs, with a parabolic curve, demonstrate how inertia increases with the square of the radius. The general equation for this parabola is in the form of $y=k{x}^2$, where $k$ is the constant, $k={\displaystyle \frac{2}{5}}M$ for the current scenario.

• The graph will always open upwards as $R^2$ is positive.
• At the point (0,0), the graph touches the origin where both I and R are zero.
• The steepness of the parabola depends on the mass of the sphere (M).

Understanding parabolic graphs help students visualize how variables like $I$ and $R$ interact and beholds several real-world applications, such as in projectile motion and optics.