Question

A particle performing uniform circular motion has angular momentum \(L\)., its angular frequency is doubled and its \(K . E\). halved, then the new angular momentum is \(\\{\mathrm{A}\\} 1 / 2\) \\{B \(\\} 1 / 4\) \(\\{\mathrm{C}\\} 2 \mathrm{~L}\) \(\\{\mathrm{D}\\} 4 \mathrm{~L}\)

Short answer

The new angular momentum is \(L' = \frac{1}{2} L\), which corresponds to option A.

Step-by-step solution

Define given variables

Initially, Angular momentum L Angular frequency ω Kinetic energy K.E After the changes, New angular frequency = 2ω New kinetic energy = (1/2) K.E

Introduce the formulas for angular momentum and kinetic energy

The angular momentum formula is: \(L = Iω\), where I is the moment of inertia and ω is the angular frequency. The kinetic energy formula for rotational motion is: \(K.E = \frac{1}{2} Iω^2\)

Express the moment of inertia in terms of angular momentum and angular frequency

Using the angular momentum formula, we can express moment of inertia as: \(I = \frac{L}{ω}\)

Express the initial kinetic energy in terms of the moment of inertia and angular frequency

Using the moment of inertia expression from step 3 and the kinetic energy formula: \(K.E = \frac{1}{2}(\frac{L}{ω})ω^2 = \frac{1}{2}Lω\)

Calculate the new angular momentum

The new kinetic energy is half of the initial, so: \(\frac{1}{2}K.E = \frac{1}{2}(\frac{1}{2}Lω) = \frac{1}{4}Lω\) Since the new angular frequency is 2ω: \(\frac{1}{2}K.E = \frac{1}{4}L(2ω) = \frac{1}{2}L(ω)\) New angular momentum \(L' = \frac{1}{2} L\) The new angular momentum is \(L' = \frac{1}{2} L\), which corresponds to option A.

Key concepts

Uniform Circular Motion

Uniform circular motion occurs when an object travels in a circular path at a constant speed. Although the speed is constant, the velocity is not, because the direction is continuously changing. As a result, there is always an acceleration towards the center of the circle, known as centripetal acceleration. This motion is governed by some key parameters that help us understand its dynamics:

• Angular Frequency (\(\omega\)): It represents how fast the object is rotating and is measured in radians per second.
• Angular Momentum (\(L\)): This is the quantity of rotation of the object, taking into account its moment of inertia and its angular frequency.
• Radius: The distance from the center of the circular path to the point where the object is moving.

Understanding these parameters is essential to solve problems involving uniform circular motion, especially those related to the changes in angular momentum when frequency and kinetic energy vary.

Angular Frequency

Angular frequency, often denoted by \(\omega\), is a measure of how quickly an object travels through its circular path. It is measured in radians per second (rad/s). Angular frequency differentiates from regular frequency in that it relates to the angle through which the object rotates, not just the number of revolutions.

• In uniform circular motion, angular frequency is constant.
• It is directly proportional to the speed of the rotating object and inversely proportional to the time period of the rotation.
• When the angular frequency doubles, the object covers twice the angular distance in the same time span, resulting in changes to other parameters like kinetic energy and angular momentum.

In our exercise, the particle's angular frequency was doubled. This affected how we calculated the new angular momentum, influencing both the rotational inertia and kinetic energy.

Kinetic Energy

Kinetic energy (K.E) in the context of rotational motion is the energy possessed by an object due to its rotation. In equation form, it is expressed as \(K.E = \frac{1}{2} I\omega^2\), where "I" represents the moment of inertia and "\(\omega\)" is angular frequency.

• Moment of inertia (I) is akin to mass in linear motion – it quantifies how the mass is distributed relative to the axis of rotation.
• In the exercise, when the angular frequency doubled, the kinetic energy was halved, showing the inverse relation in some cases of physics phenomena.
• This change helped in determining the new angular momentum, using given kinetic energy's altered state as a clue.

The interplay between kinetic energy and angular frequency is crucial in understanding the rotational dynamics and how they affect the object's kinetic energy when other factors change.

Moment of Inertia

The moment of inertia (I) is a fundamental concept in rotational dynamics. It measures an object's resistance to changes in its rotational motion. In mathematical terms, it depends on the object's mass distribution relative to the axis of rotation. The formula \(I = \frac{L}{\omega}\), derived from the angular momentum formula, connects it to angular frequency and momentum.

• Higher moment of inertia means the object is more resistant to change in its rotational state.
• It heavily influences how angular frequency and kinetic energy changes affect angular momentum.
• In the context of the exercise, recognizing this relationship was key to calculating the new angular momentum when the angular frequency was altered.

Understanding moment of inertia helps in analyzing rotational motion problems, providing insight into how energy, momentum, and rotational dynamics interplay.