Problem 66 - Master Resource Book in JEE Main Physics

Question

Here, \(m=20 \mathrm{~kg}, l=1 \mathrm{~m}, r=0.2 \mathrm{~m}\) Moment of inertia about its geometrical axis is $$ \begin{aligned} I &=\frac{1}{2} m r^{2} \\ &=\frac{1}{2} \times 20(0.2)^{2}=0.4 \mathrm{~kg} \mathrm{~m}^{2} \end{aligned} $$

Step-by-Step Solution

Answer

The moment of inertia is \( 0.4 \text{ kg m}^{2} \).

Step 1: Identify Given Values

Identify the given parameters in the problem, which include mass \( m = 20 \text{ kg} \), radius \( r = 0.2 \text{ m} \) and length \( l = 1 \text{ m} \).

Step 2: Apply the Moment of Inertia Formula

The formula for the moment of inertia about an axis for a cylinder is given by \( I = \frac{1}{2} m r^2 \). We will use this to compute the moment of inertia.

Step 3: Substitute Values into the Formula

Substitute the values of \( m \) and \( r \) into the formula: \[ I = \frac{1}{2} \times 20 \times (0.2)^2 \]

Step 4: Calculate the Expression

First, compute \((0.2)^2 = 0.04 \). Next, calculate the product: \(20 \times 0.04 = 0.8 \). Finally, divide by 2 to obtain the final value: \( \frac{0.8}{2} = 0.4 \).

Step 5: Conclusion

Therefore, the moment of inertia about its geometrical axis is \( 0.4 \text{ kg m}^{2} \).

Key Concepts

Physics Problem Solving

Tackling problems related to physics often follows a structured path. One starts by clearly identifying the given quantities and the required result. This is essential to understanding what's being asked and how to reach the solution efficiently. It allows you to lay the groundwork for choosing the appropriate path and formulas to use.In our problem, we are given the mass \( m \), the radius \( r \), and the length \( l \) of a cylinder. Recognizing these quantities as part of the problem helps us choose the formula for the moment of inertia about its axis. For physics problems, visualizing the system can also aid in understanding.Another key step is substituting the known values into the right formula. It is crucial to simplify the problem by focusing on each step carefully, ensuring no arithmetic errors slip through. This approach helps reach the accurate solution, emphasizing understanding over memorization.

Cylinder Geometry

Cylinders are common geometric shapes in physics due to their symmetrical properties and relevance in rotational dynamics. Understanding the geometry of a cylinder is essential when analyzing problems involving rotational motion.For the calculations of the moment of inertia, we observe two important parameters:

• Radius \( r \): This is the distance from the axis of rotation, which greatly influences the distribution of mass.
• Mass \( m \): This constitutes the amount of matter within the cylinder, contributing to its inertia.

These elements contribute heavily to the calculation of the moment of inertia, which measures how resistant the cylinder is to rotational changes. The symmetry of a cylinder simplifies the process of these calculations, aligning with standard formulas used in physics.

Rotational Motion Concepts

Rotational motion is centered around how objects spin around an axis. A primary concept in studying such motion is the moment of inertia, symbolized by \( I \). It is analogous to mass in linear motion but in the context of rotation.Moment of inertia depends not only on the object's mass but how that mass is distributed relative to the axis of rotation. For a solid cylinder, like the one in our example, the formula for moment of inertia around its central axis is \( I = \frac{1}{2}mr^2 \). This equation highlights key principles of rotational dynamics:

• Mass is directly proportional to inertia, meaning more mass leads to greater resistance to change in rotational speed.
• The square of the radius plays a critical role, emphasizing how distance from the axis affects rotational inertia significantly more than in linear settings.

Understanding these concepts deepens your grasp on how and why objects behave as they rotate, forming the basis for more complex analyses in rotational systems.