Problem 10 - An Introduction to Mechanics

Question

A chain of total mass \(M\) and length \(l\) is suspended vertically with its lowest end touching a scale. The chain is released and falls onto the scale. What is the reading of the scale when a length of chain, \(x\), has fallen? (Neglect the size of individual links.)

Step-by-Step Solution

Answer

The scale reads \( \frac{3Mgx}{l} \) when a length \( x \) has fallen.

Step 1: Understand the Problem

The chain of mass \( M \) and length \( l \) is released and falls onto a scale. We need to find the force exerted on the scale when a length \( x \) of the chain has fallen onto it.

Step 2: Determine the Mass of the Fallen Chain

The mass per unit length of the chain is \( \frac{M}{l} \). Thus, the mass of the portion of the chain that has fallen onto the scale is \( m = \frac{M}{l} \times x \).

Step 3: Calculate the Weight of the Fallen Chain

The weight of the fallen chain is the force due to gravity acting on the mass \( m \) that lies on the scale. This is given by \( W = m \cdot g = \frac{M}{l} \times x \times g \).

Step 4: Determine the Force of the Falling Chain

The chain falling onto the scale exerts an additional force due to the change of momentum. The velocity of the falling chain is \( v = \sqrt{2gx} \) (from energy conservation). The rate of change of momentum is \( \frac{d(mv)}{dt} = \left(\frac{M}{l}\right) v^2 = \left(\frac{M}{l}\right) \times (2gx) = \frac{2Mgx}{l} \).

Step 5: Combine Forces to Find the Scale Reading

The total force on the scale is the sum of the weight of the chain already on the scale and the force due to the falling chain. Thus, the reading of the scale is: \[ F_{total} = \frac{Mgx}{l} + \frac{2Mgx}{l} = \frac{3Mgx}{l}. \]

Key Concepts

physics problem solving

Solving physics problems involves breaking down a complex situation into more manageable parts, allowing us to better understand and calculate the various forces and motions at play. In the given exercise, we approach the problem using a structured method, focusing on finding the scale reading when a certain length of chain has fallen.
To begin, consider the physical situation: a chain is allowed to fall freely under the influence of gravity. The key is to dissect the scenario into smaller steps:

• First, identify the known quantities, such as the total mass of the chain ( \(M\), its length ( \(l\), and gravitational acceleration ( \(g\)).
• Next, determine the changing quantities, particularly the length of chain ( \(x\)), falling onto the scale.
• Finally, apply relevant physics principles to calculate the forces involved.

This methodical approach is beneficial when faced with complex problems, as it allows us to make logical connections and apply the right calculations, resulting in a clear and understandable solution. Utilizing a step-by-step strategy is crucial for successfully solving physics problems, aiding you in gaining not only the correct answer but also a deep understanding of the underlying physics principles.

momentum

Momentum is a fundamental concept in physics related to the quantity of motion an object possesses. It's given by the product of an object's mass and its velocity, expressed as \(p = mv\). When the chain falls, it gains momentum, contributing to the force exerted on the scale.
The chain falling onto the scale is a dynamic process, involving changes in momentum because the section of the chain that has fallen doesn't just rest on the scale—it impacts it. The rate of change of momentum is crucial for understanding the total force experienced by the scale.
Here's how it works:

• As each piece of the chain falls, its velocity increases due to gravitational acceleration.
• When the chain contacts the scale, there is an abrupt change in its momentum.
• This rate of change of momentum ( \(\frac{d(mv)}{dt}\)) is what adds to the total force over and above the static weight of the chain on the scale, calculated as \(\frac{2Mgx}{l}\) in this scenario.

Understanding momentum helps explain both static and dynamic forces in physical systems, illustrating why the scale reading goes beyond just the weight of the resting chain section.

energy conservation

Energy conservation is a principle stating that energy in a closed system remains constant, merely shifting from one form to another rather than being created or destroyed. In this problem, the potential energy of the chain (due to its height) is converted into kinetic energy as it falls onto the scale.
The falling chain demonstrates energy conservation visibly:

• Initially, the chain possesses gravitational potential energy, due to its elevation.
• As the chain falls, this potential energy decreases while its kinetic energy increases.
• The transition results in a velocity \(v = \sqrt{2gx}\), derived from equating potential energy lost to kinetic energy gained.

This continuous energy transformation, as the chain descends, shows how energy conservation predicts and connects the motion and impact of the falling chain on the scale. By understanding these changes, one can accurately predict and explain the physical behavior of the system, providing insight into both the calculation and the phenomenon itself.