/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 The radius of gyration of a body... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 1

The radius of gyration of a body depends upon (1) mass of the body (2) nature of distribution of mass (3) axis of rotation (4) none of these

Short Answer

Expert verified
The radius of gyration depends on mass, mass distribution, and axis of rotation.

Step by step solution

01

Understanding the concept

The radius of gyration represents how an object's mass is distributed with respect to an axis of rotation. It is related to the moment of inertia, which quantifies the rotational inertia of an object.
02

Identify the factors influencing the radius of gyration

The radius of gyration depends on both the mass of the body and how that mass is distributed about the axis of rotation. Factors (1) and (2) contribute to this dependency. Therefore, the nature of mass distribution and the mass itself are crucial in determining the radius of gyration.
03

Axis of rotation relevance

The axis of rotation influences the moment of inertia, which in turn affects the radius of gyration. Therefore, the axis of rotation (factor 3) also plays a role in determining the radius of gyration because it affects how the mass is rotationally distributed.
04

Conclusion

Considering the dependencies identified: (1) mass of the body, (2) nature of distribution of mass, and (3) axis of rotation are all correct. Hence, options (1), (2), and (3) are the proper factors related to the radius of gyration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Inertia
The moment of inertia is a fundamental concept when dealing with rotational motion. It helps us understand how difficult it is to change an object's rotational state. Think of it like mass in linear motion. Just like mass determines how hard it is to start or stop an object moving in a straight line, the moment of inertia measures the difficulty of starting or stopping rotation.

When calculating the moment of inertia, you need to consider:
  • The mass of the object
  • How that mass is spread out or distributed relative to a certain axis
These factors determine how much resistance an object has to being rotated. The farther the mass is from the axis of rotation, the larger the moment of inertia and thus greater resistance to rotation. In mathematical terms, the moment of inertia \( I \) can be expressed as \( I = \sum m_i r_i^2 \), where \( m_i \) is the mass element and \( r_i \) is the distance from the axis of rotation.
Axis of Rotation
The axis of rotation plays a crucial role in both the moment of inertia and the radius of gyration. It is the imaginary line around which the object spins. The choice of the axis affects how the object's mass is distributed relative to this line.

Different orientations of the axis will change:
  • How mass elements are positioned relative to the axis
  • The calculations of both the moment of inertia and the radius of gyration
For example, two objects with the same shape and size might have very different moments of inertia if their axes of rotation are located differently. This is because mass distribution is affected. Therefore, understanding the axis of rotation is key to studying rotational dynamics.
Mass Distribution
Mass distribution is all about how the weight of an object is spread out in space. This is a deciding factor in how the object spins or rotates around an axis. When a large portion of an object's mass is located far from the axis of rotation, it will have a greater moment of inertia, as discussed earlier. This can make it more difficult to rotate but also more stable once in motion.

Key points to consider with mass distribution include:
  • Uniform vs. non-uniform distribution
  • The shape and geometry of the object
In practical terms, engineers and designers often try to optimize mass distribution to achieve desired rotational characteristics. For example, a flywheel might have most of its mass distributed around the rim to enhance its moment of inertia, making it store more energy.

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Most popular questions from this chapter

A sphere is moving towards the positive \(x\)-axis witn a velocity \(v_{c}\) and rotates clockwise with angular speed \(\omega\) as shown in figure, such that \(v_{c}>\omega R\). The instantaneous axis of rotation will be 1) on point \(P\) (2) on point \(P^{\prime}\) 3) inside the sphere (4) outside the sphere

A uniform thin flat isolated disc is floating in space radius \(R\) and mass \(m\). \(A\) force \(F\) is applied to it at a distance \(d=(R / 2)\) from the centre in the \(y\)-direction. Treat this problem as twodimensional. Just after the force is spplied (1) acceleration of the centre of the disc is \(F / m\) (2) angular acceleration of the disk is \(F\rangle m R\) (3) acceleration of leftmost point on the disc is zero (4) point which is instantaneously unaccelerated is the riahtmost point

Two identical masses are connected to a horizontal thin massless rod as shown in the figure. When their distance from the pivot is \(x\), a torque produces an angular acceleration \(\alpha_{1} .\) If the masses are now repositioned so that they are at distance \(2 x\) each from the pivot, the same torque will produce an angular acceleration \(\alpha_{2}\) such that (1) \(\alpha_{2}=4 \alpha_{1}\) (2) \(\alpha_{2}=\alpha_{1}\) (3) \(\alpha_{2}=\frac{\alpha_{1}}{2}\) (4) \(\alpha_{2}=\frac{\alpha_{1}}{4}\)

From a complete ring of mass \(M\) and radius \(R\), a \(30^{\circ}\) sector is removed. The moment of inertia of the incomplete ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is (1) \(\frac{9}{12} M R^{2}\) (2) \(\frac{11}{12} M R^{2}\) (3) \(\frac{11.3}{12} M R^{2}\) (4) \(M R^{2}\)

Three point masses \(m_{1}, m_{2}\) and \(m_{3}\) are located at the vertice, of an equilateral triangle of side \(a\). What is the momert inertia of the system about an axis along the altitude of triangle passing through \(m_{1}\) ? 1) \(\left(m_{1}+m_{2}\right) \frac{a^{2}}{4}\) (2) \(\left(m_{2}+m_{3}\right) \frac{a^{2}}{4}\) (3) \(\left(m_{1}+m_{3}\right) \frac{a^{2}}{4}\) (4) \(\left(m_{1}+m_{2}+m_{3}\right) \frac{a^{2}}{4}\)
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