Problem 559 - JEE Physics

Question

A wheel of mass \(10 \mathrm{~kg}\) has a moment of inertia of \(160 \mathrm{~kg} \mathrm{~m}\) radius of gyration will be \(\begin{array}{llll}\\{\mathrm{A}\\} 10 & \\{\mathrm{~B}\\} 8 & \\{\mathrm{C}\\} 6 & \\{\mathrm{D}\\} 4\end{array}\)

Step-by-Step Solution

Answer

The radius of gyration, \(k\), can be found using the formula \( k = \sqrt{\frac{I}{m}} \), where \(I\) is the moment of inertia and \(m\) is the mass of the object. Substituting the given values (\(m = 10 \, kg\) and \(I = 160 \, kg \cdot m^{2}\)), we get \( k = \sqrt{\frac{160 \, kg \cdot m^{2}}{10 \, kg}} \). Simplifying the equation, we get \( k = 4 \, m \). Hence, the radius of gyration of the wheel is 4 m. The correct answer is 'D'.

Step 1: Understand the formula

The formula to find the radius of gyration, \( k \), is given by \( k = \sqrt{\frac{I}{m}} \), where \( I \) is the moment of inertia and \( m \) is the mass of the object. This formula allows us to calculate the hypothetical distance at which the entire mass can be assumed to be concentrated without changing the moment of inertia.

Step 2: Substitute in the given values

In this case, we are given that mass \( m = 10 \, kg \) and moment of inertia \( I = 160 \, kg \cdot m^{2} \). Substitute these values into the formula to get \( k = \sqrt{\frac{160 \, kg \cdot m^{2}}{10 \, kg}} \).

Step 3: Simplify the equation

Our equation \( k = \sqrt{\frac{160 \, kg \cdot m^{2}}{10 \, kg}} \) can now be simplified. The \( kg \) units on the top and bottom cancel out and dividing 160 by 10 gives us 16, so the equation becomes \( k = \sqrt{16 \, m^{2}} \).

Step 4: Solve the equation

The square root of 16 is 4, so our final answer is \( k =4 \, m \). Thus, the radius of gyration of the wheel is 4 m. Therefore, the answer to the exercise is 'D'.

Key Concepts

Radius of Gyration

Understanding the radius of gyration is crucial in the context of rotational motion. It represents a distance at which the entire mass of an object can be imagined to be concentrated. This concept helps maintain the same moment of inertia. The radius of gyration, denoted by the symbol \( k \), is calculated using the physics formula \( k = \sqrt{\frac{I}{m}} \). Here, \( I \) represents the moment of inertia, and \( m \) denotes the mass of the object.
\( k \) is essential in simplifying the analysis of rotational bodies, because it allows us to consider a complex mass distribution as a simpler singular distribution while preserving rotational characteristics. It's particularly helpful in engineering and physics for analyzing structures that rotate, such as wheels or beams.

Mass

Mass is a fundamental property of matter that measures the amount of material in a body. It's typically measured in kilograms (kg). In physics, mass serves as a measure of resistance to acceleration when a force is applied. It's essential in calculations involving force, momentum, and energy.
In our exercise, the wheel has a mass \( m = 10 \text{ kg} \). This value is vital for calculating the radius of gyration, as it influences the outcome directly when using the related formula. Understanding how mass interacts with other properties like velocity and acceleration provides deeper insights into an object's behavior when subject to forces.

Physics Formula

Physics relies heavily on formulas to describe the principles and relationships between different physical quantities. The formula used to calculate the radius of gyration is \( k = \sqrt{\frac{I}{m}} \). This equation links the moment of inertia \( I \), a measure of an object's resistance to changes in its rotation, to mass \( m \) and the radius of gyration \( k \).
This formula simplifies the concept of distributing an object's mass along its axis of rotation. It is useful for mechanical and structural analysis, ensuring systems operate efficiently. Understanding the role of formulas, like the one for radius of gyration, helps create models that predict real-world behavior of materials and systems under various forces.

Calculation Steps

For effective problem-solving in physics, clear calculation steps are essential. Let's review the steps taken to determine the radius of gyration for a wheel.

• **Understand the formula:** Begin with the formula \( k = \sqrt{\frac{I}{m}} \). It connects mass and the moment of inertia to radius of gyration.
• **Substitute values:** Given mass \( m = 10 \text{ kg} \) and moment of inertia \( I = 160 \text{ kg} \cdot \text{m}^2 \), substitute to get: \( k = \sqrt{\frac{160 \text{ kg} \cdot \text{m}^2}{10 \text{ kg}}} \).
• **Simplify:** Simplifying, cancel the units and compute \( \frac{160}{10} = 16 \). This reduces the expression to \( k = \sqrt{16} \).
• **Solve:** The square root of 16 is 4, resulting in \( k = 4 \text{ m} \). This confirms that the radius of gyration is 4 meters, corresponding to option D.

By following these steps, solutions become methodical and assumptions clear, preventing errors and reinforcing comprehension of physical concepts.