Problem 13 - Classical Mechanics - An Undergraduate Text

Question

Figure \(9.15\) shows a partially unrolled roll of paper on a horizontal floor. Initially the paper on the roll has radius \(a\) and the free paper is laid out in a straight line on the floor. The roll is then projected horizontally with speed \(V\) in such a way that the free paper is gathered up on to the roll. Find the speed of the roll when its radius has increased to \(b\). [Neglect the bending stiffness of the paper.] Deduce that the radius of the roll when it comes to rest is \(a\left(\frac{3 V^{2}}{4 g a}+1\right)^{1 / 3}\)

Step-by-Step Solution

Answer

The roll has a speed \(V' = \frac{M\cdot V\cdot a}{M+ \mu \cdot 2\pi (b-a)}\) when its radius has increased to \(b\), and comes to a stop when its radius is \(b = a\left(1+\frac{3V^{2}}{4\pi ga}\right)^{\frac{1}{3}}\).

Step 1: Understand the principle of conservation of angular momentum

Given that there are no external torques acting on the system, angular momentum is conserved. This means that the initial angular momentum when the roll is projected must be equal to the angular momentum after some paper is unrolled i.e., when the radius has increased to \(b\). The equation for conservation of angular momentum can be written as: \(I_{initial}\cdot \omega_{initial}=I_{final}\cdot \omega_{final}\). Where \(I\) and \(\omega\) represent the moment of inertia and angular velocity (proportional to speed), respectively.

Step 2: Set up the angular momentum equations

For a thin cylindrical shell (like our roll of paper), the moment of inertia \(I\) can be represented as \(I=M\cdot r^{2}\), where \(M\) is the mass and \(r\) is the radius. The mass of the paper being unrolled can be determined by considering the mass per unit length of the paper, which is consistent throughout the roll (\(M = \mu \cdot 2\pi r\), where \( \mu \) represents the mass per unit length). Similarly, the angular velocity \(\omega\) can be represented as \(V' / r\), where \(V'\) is the linear velocity. Therefore, the equation for conservation of angular momentum can be re-written in terms of these equations as: \((M\cdot a^{2})\cdot (V / a)=((M+ \mu \cdot 2\pi (b-a))\cdot b^{2})\cdot (V' / b)\). Simplifying this equation will give us the linear velocity \(V'\) when the radius has increased to \(b\).

Step 3: Solve the conservation of angular momentum equation for \(V'\)

Solving the conservation of angular momentum equation gives us \(V' = \frac{M\cdot V\cdot a}{M+ \mu \cdot 2\pi (b-a)}\). Here, \(V'\) represents the speed of the roll when its radius has increased to \(b\).

Step 4: Determine the radius when the roll comes to rest

To find the radius when the roll has come to rest, we need to solve the equation where \(V' = 0\). Doing so gives us: \((M+ \mu \cdot 2\pi (b-a)) = 0\). Solving this equation gives us \(b = a\left(1+\frac{3V^{2}}{4\pi ga}\right)^{\frac{1}{3}}\) which is the radius of the roll when it comes to rest.

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. Think of it as the rotational equivalent of mass. The larger the moment of inertia, the more difficult it is to change the object's rotational speed.
For a cylindrical shell, like the roll of paper in our exercise, the moment of inertia is calculated using the formula:

• \( I = M \cdot r^2 \)

where \( M \) is the mass and \( r \) is the radius.
This equation shows us that both the mass of the object and how far that mass is distributed from the axis of rotation contribute to its moment of inertia.
As the radius of the roll of paper increases from \( a \) to \( b \), its moment of inertia also increases. This is important for understanding how the speed of the roll changes as it gains more paper and hence more mass distributed further from its axis.

Angular Velocity

Angular velocity refers to how fast something is rotating. It's the rate of change of the angular position of a rotating body. For our roll of paper, angular velocity, \(\omega\), is an essential part of understanding how the roll's speed changes as its radius increases.
The relationship between linear velocity \( V \) and angular velocity \(\omega\) is given by:

• \( \omega = \frac{V}{r} \)

where \( r \) is the radius of the roll.
This equation shows that as the radius increases, the angular velocity decreases, assuming the linear velocity stays the same. However, in our exercise, the conservation of angular momentum ensures that changes in radius directly influence changes in angular velocity, keeping the product of the two [— especially combined with moment of inertia —] constant.

Cylindrical Shell

A cylindrical shell, like the roll of paper in this problem, is essentially a hollow cylinder. When considering its physics, it's crucial to remember that its mass is concentrated at a uniform distance from the central axis.
In the formula for the moment of inertia of a cylindrical shell:

• \( I = M \cdot r^2 \)

"\( M \)" represents not only as a fully solid object, but as one where the entire mass contributes evenly to the inertia. This characteristic allows us to calculate the inertia without considering the thickness or any forces that might affect a solid cylinder.
Understanding this concept helps in solving the exercise, because it tells us how the addition of more paper and a change in radius directly impacts the inertia without needing to alter the basic rotational physics principles involved.

Mass per Unit Length

Mass per unit length is a concept that simplifies the process of estimating mass distribution across a length of an object. For example, our roll of paper has a consistent mass per unit length [\( \mu \), representing how heavy each segment of a specific length is].
This consistency is crucial because as the paper rolls up, the overall mass increases, which directly affects the roll's moment of inertia:

• \( M = \mu \cdot 2\pi r \)

Here, \( \mu \) explains that the mass along the circumference of the roll increases with radius.
Using this principle, we can calculate how much more mass the roll gains as the radius changes from \( a \) to \( b \), helping us keep track of how these changes affect the inertia and the angular momentum. Understanding this ensures a correct application of the conservation of angular momentum formula in our problem.