Question

Statement \(-1\) -If the cylinder rolling with angular speed- w. suddenly breaks up in to two equal halves of the same radius. The angular speed of each piece becomes \(2 \mathrm{w}\). Statement \(-2\) - If no external torque outs, the angular momentum of the system is conserved. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\) \(\\{\mathrm{B}\\}\) Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1 .\) \(\\{\mathrm{C}\\}\) Statement \(-1\) is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

Short answer

Answer: \(\\{A\\}\) Both Statement \(-1\) and Statement \(-2\) are true, and Statement \(-2\) is the correct explanation for Statement \(-1\).

Step-by-step solution

Analyzing Statement -1

Given a cylinder rolling with an angular speed w, suddenly breaking up into two equal halves with the same radius. The angular speed of each piece is said to become 2w. The moment of inertia of the whole cylinder can be expressed as \(I_{cylinder} = \frac{1}{2}MR^2\), where M is the total mass of the cylinder, and R is the radius. After breaking apart, the two equal halves of the cylinder have half the mass of the initial cylinder. Therefore, the moment of inertia of each half can be represented as \(I_{half} = \frac{1}{2}\left(\frac{M}{2}\right)R^2 = \frac{1}{4}MR^2\). According to the conservation of angular momentum, the initial angular momentum of the cylinder is equal to the sum of the angular momenta of the two halves: \(I_{cylinder}\omega = 2I_{half}\omega'\)

Analyzing Statement -2

Statement -2 says that if no external torque acts on the system, the angular momentum of the system is conserved. By definition, if no external torque is involved, the angular momentum of such a system is indeed conserved. Therefore, statement -2 is true.

Checking the relationship between Statement -1 and Statement -2

Now we need to check if statement -2 is the correct explanation for statement -1. For this, we can use the conservation of angular momentum equation derived in step 1: \(\frac{1}{2}MR^2\omega = 2\left(\frac{1}{4}MR^2\right)\omega'\) Simplifying the equation, we get: \(\omega = 2\omega'\) Hence, angular speed of each half becomes 2w. Since the conservation of angular momentum (statement -2) is the reason behind the resulting angular speed of the two halves (statement -1), it is the correct explanation for statement -1. So, our final answer is: \(\\{A\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\)

Key concepts

Moment of Inertia

The moment of inertia is a key concept in rotational motion and it acts like mass in linear motion.
It is the measure of an object's resistance to changes in its rotation speed around an axis.
Matter farther from the axis makes more contribution to the moment of inertia, making it harder to accelerate the rotation.
For a solid cylinder rolling around a central axis, the moment of inertia is given by the formula:\[I_{cylinder} = \frac{1}{2}MR^2\]Where:- \(I_{cylinder}\) is the moment of inertia of the cylinder,- \(M\) is the mass, and- \(R\) is the radius.When the cylinder splits into two equal halves, each has half the mass of the initial cylinder.
Thus, the moment of inertia for each half becomes:\[I_{half} = \frac{1}{2}\left(\frac{M}{2}\right)R^2 = \frac{1}{4}MR^2\]This concept is crucial in analyzing how rotational properties change when an object breaks into parts.

Angular Speed

Angular speed describes how fast something is rotating, typically measured in radians per second.
It indicates the change in the angular position over time and helps us understand how quickly an object spins.
In the context of the problem, both halves of the cylinder, after breaking, showed a change in their angular speed.
The original cylinder was spinning with angular speed \(\omega\)
After splitting, each half had an angular speed of \(2\omega\).
This change is due to the conservation of angular momentum, which we'll discuss later.
The relationship between the angular speeds of the original cylinder and its halves can be derived through:\[I_{cylinder}\omega = 2I_{half}\omega' \]Here, \(\omega'\) is the new angular speed of each half.
Simplifying this gives \(\omega = 2\omega'\), meaning \(\omega' = 2\omega\)
This illustrates how rotational dynamics adjust when objects change in structure.

External Torque

The concept of external torque is critical in understanding rotational motion.
Torque is the rotational equivalent of force\,, allowing us to quantify how much a force acting on an object causes it to rotate.
If the axe where force is acting is outside of the system we are considering, we say it is an "external torque".
When no external torque is acting on a system, its angular momentum remains constant, based on the principle of conservation of angular momentum.
This means any change in the internal configuration of the system, like breaking objects, doesn't affect the overall angular momentum.Thus, within the exercise, since no external torque acted when the cylinder split into halves, the total angular momentum conserved.The equation expressing this conservation is:\[I_{initial}\omega_{initial} = I_{final}\omega_{final}\]Where:- \(I_{initial}\) and \(I_{final}\) are the initial and final moments of inertia,- \(\omega_{initial}\) and \(\omega_{final}\) are the initial and final angular speeds.Recognizing the role external torque plays allows us to predict how systems behave under change, such as objects breaking apart while maintaining rotational velocity.