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Chapter 4: Problem 1

Two bodies with moments of inertia \(I_{1}\) and \(I_{2}\left(I_{1}>I_{2}\right)\) have equal angular momenta. If their kinetic energies of rotation \(\operatorname{are} E_{1}\) and \(E_{2}\), respectively, then (i) \(E_{1}=E_{2}\) (2) \(E_{1}E_{2}\) (4) \(E_{1} \geq E_{2}\)

Short Answer

Expert verified
The answer is (2) \(E_1 < E_2\).

Step by step solution

01

Understand Given Information

We are given that two bodies have moments of inertia denoted by \(I_1\) and \(I_2\), where \(I_1 > I_2\). Furthermore, they have equal angular momenta.
02

Equal Angular Momenta

Since the bodies have equal angular momenta, we write \( L_1 = L_2 \). By definition, angular momentum \( L \) is given by \( L = I \cdot \omega \), where \( \omega \) is the angular velocity. Thus, \( I_1 \omega_1 = I_2 \omega_2 \).
03

Express Angular Velocities

From \( I_1 \omega_1 = I_2 \omega_2 \), we can express \( \omega_1 \) as \( \omega_1 = \frac{I_2}{I_1} \omega_2 \). This shows that \( \omega_1 < \omega_2 \) because \( I_1 > I_2 \).
04

Write Rotational Kinetic Energy Formulas

The rotational kinetic energy \( E \) of a body is given by \( E = \frac{1}{2} I \omega^2 \). Therefore, \( E_1 = \frac{1}{2} I_1 \omega_1^2 \) and \( E_2 = \frac{1}{2} I_2 \omega_2^2 \).
05

Substitute Angular Velocities

Substituting \( \omega_1 = \frac{I_2}{I_1} \omega_2 \) into \( E_1 \), we get \( E_1 = \frac{1}{2} I_1 \left( \frac{I_2}{I_1} \right)^2 \omega_2^2 = \frac{1}{2} \frac{I_2^2}{I_1} \omega_2^2 \).
06

Compare Kinetic Energies

Note that \( E_2 = \frac{1}{2} I_2 \omega_2^2 \). Hence, \( E_1 = \frac{1}{2} \frac{I_2^2}{I_1} \omega_2^2 \) and \( E_2 = \frac{1}{2} I_2 \omega_2^2 \). Since \( I_1 > I_2 \), clearly \( \frac{I_2^2}{I_1} < I_2 \), thus \( E_1 < E_2 \).

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Key Concepts

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

Moments of Inertia
In the realm of rotational motion, "moments of inertia" are a crucial concept. They reflect how much a body resists angular acceleration. Think of it as a rotational equivalent of mass in linear motion. The more inertia a body has, the harder it is to change its state of rotation. Moments of inertia are affected by both the mass of the object and how far the mass is from the axis of rotation.
  • For a given body, with mass farther from the axis, the moment of inertia is greater.
  • The moment of inertia is represented by the variable \(I\).
Understanding moments of inertia is vital for problems involving rotational motion, as it allows predictions of how much force is needed to rotate an object. In the exercise, the two objects have moments of inertia \(I_1\) and \(I_2\) with \(I_1 > I_2\), meaning the first object is harder to rotate than the second.
Angular Momentum
"Angular momentum" is the rotational counterpart of linear momentum. It is conserved in a closed system without external torques, similar to how linear momentum is conserved. The angular momentum \(L\) of a body is calculated as the product of its moment of inertia and its angular velocity: \(L = I \cdot \omega\).
  • If the moment of inertia increases, the angular velocity must decrease, and vice versa if the angular momentum is to stay constant.
In the given exercise, both objects have equal angular momenta, \(L_1 = L_2\). This implies a relationship between their moments of inertia and angular velocities. We used the fact that \(I_1 \omega_1 = I_2 \omega_2\), establishing a connection that helps compare angular velocities and kinetic energies between the two bodies.
Angular Velocity
"Angular velocity" refers to how fast an object rotates or spins around its axis. It is a vector quantity, usually denoted by \(\omega\), and measured in radians per second. Angular velocity connects directly to how kinetic energy in rotation is evaluated through the equation \(E = \frac{1}{2} I \omega^2\).
  • If angular velocity is high, the rotational kinetic energy is higher, assuming the moment of inertia is constant.
Within the problem, we deduced that the first object has a lower angular velocity than the second, \(\omega_1 < \omega_2\), due to its higher moment of inertia, \(I_1 > I_2\). Therefore, despite having equal angular momentum, their velocities differ significantly, influencing their rotational kinetic energies and leading to \(E_1 < E_2\).

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Most popular questions from this chapter

A uniform rod is resting freely over a smooth horizontal plane. A particle moving horizontally strikes at one end of the rod normally and gets stuck. Then (1) the momentum of the particle is shared between the particle and the rod and remains conserved (2) the angular momentum about the mid-point of the rod before and after the collision is equal (3) the angular momentum about the centre of mass of the combination before and after the collision is equal (4) the centre of mass of the rod particle system starts to move translationally with the original momentum of the particle

Inner and outer radii of a spool are \(r\) and \(R\), respectively. A thread is wound over its inner surface and spool is placed over a rough horizontal surface. Thread is pulled by a force \(F\) as shown in figure. In case of pure rolling, which of the following statements are false? (1) Thread unwinds, spool rotates anticlockwise and friction acts leftwards. (2) Thread winds, spool rotates clockwise and friction acts leftwards. (3) Thread winds, spool moves to the right and friction acts rightwards. (4) Thread winds, spool moves to the right and friction does not come into existence.

A uniform disc is rotating at a constant speed in a vertical plane about a fixed horizontal axis, passing through the centre of the disc. A piece of disc from its rim detaches itself from the disc at the instant when it is on the same horizontal with the centre of the disc and moving upwards. Then about the fixed axis, the angular speed of the (1) remaining disc remains unchanged (2) remaining disc decreases (3) remaining disc increases (4) broken away piece decreases initially and later increases

A ring, cylinder and solid sphere are placed on the top of a rough incline on which the sphere can just roll without slipping. When all of them are released at the same instant from the same position, then (1) all of them reach the ground at the same instant (2) the sphere reaches first and the ring at the last (3) the sphere reaches first and the cylinder and ring reach together (4) none of the above

A uniform smooth rod (mass \(m\) and length \(l\) ) placed on a smooth horizontal floor is hit by a particle (mass \(m\) ) moving on the floor, at a distance \(l / 4\) from one end elastically \((e=1)\). The distance travelled by the centre of the rod after the collision when it has completed three revolutions will be (1) \(2 \pi l\) (2) cannot be determined (3) \(\pi l\) (4) none of these
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