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Chapter 9: Problem 59

What is moment of inertia in terms of angular momentum \((L)\) and kinetic energy \((K) ? \quad\) [UP SEE 2006] (a) \(\frac{L^{2}}{K}\) (b) \(\frac{L^{2}}{2 K}\) (c) \(\frac{L}{2 K^{2}}\) (d) \(\frac{L}{2 K}\)

Short Answer

Expert verified
The moment of inertia in terms of the given variables is \(I = \frac{L^2}{2K}\). Option (b).

Step by step solution

01

Understanding the Relationship

The moment of inertia (\(I\)) can be related to angular momentum (\(L\)) and kinetic energy (\(K\)) using the formulas for angular momentum and rotational kinetic energy. Angular momentum is given by \(L = I\omega\) and kinetic energy for rotational motion is \(K = \frac{1}{2} I \omega^2\). Our goal is to express \(I\) in terms of \(L\) and \(K\).
02

Express Angular Velocity

From the angular momentum equation \(L = I \omega\), express angular velocity \(\omega\) in terms of \(L\) and \(I\). We have \(\omega = \frac{L}{I}\).
03

Substitute Angular Velocity in Kinetic Energy

Substitute the expression for angular velocity \(\omega\) from step 2 into the kinetic energy equation \(K = \frac{1}{2} I \omega^2\). This gives us:\[K = \frac{1}{2} I \left(\frac{L}{I}\right)^2 = \frac{L^2}{2I}\].
04

Solve for Moment of Inertia

Rearrange the equation from step 3 to solve for \(I\). This gives:\[I = \frac{L^2}{2K}\].
05

Identify the Correct Option

Compare the expression \(I = \frac{L^2}{2K}\) with the options given in the exercise. Option (b) matches our derived expression.

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Key Concepts

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

Angular Momentum
Angular momentum is a fundamental concept in physics, especially when dealing with objects in rotational motion. It can be thought of as the rotational equivalent of linear momentum.
  • Angular momentum (\(L\)) is defined by the equation \(L = I\omega\), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.
  • The moment of inertia here acts similar to mass in linear motion, representing the object's resistance to changes in its rotational state.
  • Angular velocity represents how fast the object is spinning.
Angular momentum is a vector quantity, meaning it has both magnitude and direction. Typically, it points along the axis of rotation. This conserved quantity helps explain why, for instance, a spinning ice skater can control their spin by pulling in their arms.
Kinetic Energy
Kinetic energy comes into play in the realm of both linear and rotational motion, reflecting the energy due to an object's motion. For rotational motion:
  • Rotational kinetic energy (\(K\)) is given by the formula \(K = \frac{1}{2} I \omega^2\).
  • This formula is analogous to the linear kinetic energy expression \(K = \frac{1}{2} mv^2\), where \(v\) is velocity. In rotational terms, \(\omega\) (angular velocity) replaces linear velocity and \(I\) (moment of inertia) replaces mass.
  • It signifies the energy associated with the object's rotation.
Understanding kinetic energy in the context of rotational motion is vital for calculating how much work is done in spinning objects and helps in solving complex dynamics problems.
Rotational Motion
Rotational motion is the motion of objects around a central pivot or axis. It's an essential concept that extends our understanding of motion from straight lines to circles.
  • The study of rotational motion involves several key quantities: angular displacement, angular velocity, angular acceleration, and moment of inertia.
  • Angular displacement measures how much an object has rotated, relatively to a fixed point.
  • Angular velocity (\(\omega\)) tells us the rate of rotation, while angular acceleration explains how this rate changes over time.
  • Moment of inertia (\(I\)) is critical in rotational dynamics as it serves a role similar to mass, influencing how difficult it is to change an object鈥檚 rotational state.
Grasping rotational motion equips us to analyze a variety of phenomena in our everyday surroundings, from the spin of a bicycle wheel to the orbit of planets.

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Most popular questions from this chapter

Assertion When ice on polar caps of earth melts, duration of the day increases. Reason \(L=I \omega=I \cdot \frac{2 \pi}{T}=\) constant.

A particle of mass \(m\) is moving in \(y z\)-plane with a uniform velocity \(v\) with its trajectory running parallel to +ve \(y\)-axis and intersecting \(z\)-axis at \(z=a\). The change in its angular momentum about the origin as it bounces elastically from a wall at \(y=\) constant is [NCERT Exemplar] (a) \(\operatorname{mva} \mathbf{e}_{x}\) (b) \(2 \mathrm{mva} \mathrm{e}_{x}\) (c) \(y m v \mathbf{e}_{x}\) (d) \(2 y m v \mathbf{e}_{x}\)

An inclined plane makes an angle of \(30^{\circ}\) with horizontal. A solid sphere rolling down the inclined plane from rest without slipping has a linear acceleration equal to (a) \(5 \mathrm{~g} / 14\) (b) \(5 \mathrm{~g} / 4\) (c) \(2 g / 3\) (d) \(g / 3\)

A solid cylinder (SC), a hollow cylinder \((\mathrm{HC})\) and a solid sphere (S) of the same mass and radius are released simultaneously from the same height of incline. The order in which these bodies reach the bottom of the incline is (a) SC, HC, S (b) \(\mathrm{SC}, \mathrm{S}, \mathrm{HC}\) (c) \(\mathrm{S}, \mathrm{SC}, \mathrm{HC}\) (d) \(\mathrm{HC}, \mathrm{SC}, \mathrm{S}\)

A thin uniform \(\operatorname{rod} A B\) of mass \(m\) and length \(L\) is hinged at one end \(A\) to the level floor. Initially, it stands vertically and is allowed to fall freely to the floor in the vertical plane. The angular velocity of the rod, when its end \(B\) strikes the floor is \((g\) is acceleration due to gravity) (a) \(\left(\frac{m g}{L}\right)\) (b) \(\left(\frac{m g}{3 L}\right)^{1 / 2}\) (c) \(\left(\frac{g}{L}\right)\) (d) \(\left(\frac{3 g}{L}\right)^{1 / 2}\)
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