/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 Each of 100 identical blocks sit... [FREE SOLUTION] | 91Ó°ÊÓ

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Each of 100 identical blocks sitting on a frictionless surface is connected to the next block by a massless string. The first block is pulled with a force of \(100 \mathrm{N}\). a. What is the tension in the string connecting block 100 to block 99? b. What is the tension in the string connecting block 50 to block \(51 ?\)

Short Answer

Expert verified
The tension in the string connecting block 100 to block 99 is \(m \mathrm{N}\), where \(m\) is the mass of each block. The tension in the string connecting block 50 to block 51 is \(50m \mathrm{N}\).

Step by step solution

01

Determine the total system mass and acceleration

Since the question mentions that all blocks are identical, let's denote the mass of each block as ‘m’. Since there are 100 blocks, the total mass of the system is \(100m\). The total force applied to the system equals the mass times the acceleration (\(F = ma\)), so the total acceleration of the system is \(a=F/m =100 \mathrm{N}/(100m) = 1 \mathrm{m/s}^2\).
02

Calculate the tension in string between block 100 and block 99

The tension in the string between block 100 and block 99 (T100) is simply the force needed to pull block 100, which equals mass of block 100 times the acceleration. Since the acceleration is \(1 \mathrm{m/s^2}\), and block 100 has a mass of \(m\), then \(T100 = m * a = m * 1 \mathrm{m/s}^2 = m \mathrm{N}\).
03

Calculate the tension in string between block 50 and block 51

The tension in the string between block 50 and block 51 (T50) is the force required to pull the 50 blocks from block 50 to block 100. It equals the combined mass of these 50 blocks times the acceleration (\(T50 = 50m * a = 50m * 1 \mathrm{m/s}^2 = 50m \mathrm{N}\)).

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Key Concepts

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

Newton's laws of motion
Understanding Newton's laws of motion is crucial when examining the dynamics of an object, or in our case, a system of blocks connected by strings. The first law, often referred to as the law of inertia, states that an object at rest will stay at rest and an object in motion will stay in motion unless acted upon by an external force. This principle underlies why the blocks can remain either stationary or move with a consistent speed on a frictionless surface.

The second law provides the relationship between the force applied to an object, its mass, and the acceleration it undergoes. This is given by the formula \( F = ma \), where \( F \) is the force applied, \( m \) is the mass, and \( a \) is the acceleration. The third law states that for every action, there is an equal and opposite reaction, which explains the forces between the blocks and the strings in our system. In the textbook problem, we apply the second law to find the acceleration of the blocks and consequently determine the tension in the strings.
Massless string tension
In many physics problems, strings are considered to be massless to simplify calculations. When a string is massless, it means that the tension throughout the string is the same at all points. This concept is vital for the problem at hand because it allows us to assume that the force required to pull any one block is the same force that would be required to pull a system of blocks.

By treating the string as massless, we can easily deduce that the tension between any two blocks will be equivalent to the force necessary to accelerate the mass that the string is pulling forward, since the string itself does not contribute to the total mass of the system.
Frictionless surface physics
A frictionless surface is a theoretical concept used in physics to describe a surface over which objects can move without any resistance. In reality, all surfaces have some level of friction, but for simplicity, the concept of a frictionless surface allows us to focus on other forces acting upon the objects.

In the context of our scenario with the blocks, the frictionless surface signifies that the only horizontal force we need to consider is the pulling force applied; there won’t be any frictional force opposing the motion of the blocks. With this assumption, the calculation of acceleration becomes more straightforward, as it is solely determined by the applied force and the mass of the blocks.
Acceleration in physics
Acceleration is a fundamental concept in physics that describes how quickly an object changes its velocity. It is a vector quantity, meaning it has both magnitude and direction. The acceleration of an object is given by the change in velocity over time.

In a uniform linear motion scenario, where the force is constant and mass remains unchanged, calculating acceleration is rather simple. Referencing the formula \( F = ma \), we rearrange it to solve for acceleration as \( a = F/m \). This plays a central role in our exercise where the acceleration of the blocks was deduced by dividing the total force by the combined mass of the 100 blocks.
Force and motion
The relationship between force and motion is encapsulated in Newton's second law of motion. In essence, it's the force that causes an object to move or accelerates. In a system where multiple objects are involved, such as the string-connected blocks, we must consider the net force acting on the system to understand the resulting motion.

For our blocks on a frictionless surface, the motion we observe—the uniform acceleration—is due to the force applied on the first block. This force is transmitted through the tension in the massless strings to the other blocks, which produces a chain reaction, resulting in each block moving synchronously with the acceleration determined by the total force and system mass.

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