Problem 70 - University Physics with Modern Physics

Question

A block with mass \(M\) is attached to the lower end of a vertical, uniform rope with mass \(m\) and length \(L\) A constant upward force \(\vec{\boldsymbol{F}}\) is applied to the top of the rope, causing the rope and block to accelerate upward. Find the tension in the rope at a distance \(x\) from the top end of the rope, where \(x\) can have any value from 0 to \(L .\)

Step-by-Step Solution

Answer

The tension at distance \(x\) from the top is \(T(x) = (M + \frac{m(L-x)}{L})g + (M + \frac{m(L-x)}{L})a\).

Step 1: Identify Forces and System Components

To find the tension at a distance \(x\) from the top of the rope, consider the forces acting on the portion of the rope below this distance. The forces include gravity acting on both the block and the rope's mass below \(x\), and the upward force \(\vec{\boldsymbol{F}}\).

Step 2: Define the System's Equations

Consider the portion of the rope starting from the end where the force \(\vec{\boldsymbol{F}}\) is applied down to the point \(x\). The total mass below this section is \(M + \frac{m(L-x)}{L}\). The weight of this mass is multiplied by gravity \(g\), acting downward.

Step 3: Apply Newton's Second Law

According to Newton's Second Law, the net force acting on the system is the difference between the upward force \(T(x)\) and the gravitational force \((M + \frac{m(L-x)}{L})g\). Thus, the equation is:\[ T(x) - (M + \frac{m(L-x)}{L})g = (M + \frac{m(L-x)}{L})a \]where \(a\) is the acceleration of the system.

Step 4: Solve for Tension \(T(x)\)

Rearrange the equation from Step 3 to solve for the tension \(T(x)\):\[ T(x) = (M + \frac{m(L-x)}{L})g + (M + \frac{m(L-x)}{L})a \]This result shows that tension depends on both the gravitational and inertial forces acting on the portion of the rope and block below point \(x\).

Step 5: Consider Limits as a Check

Check the limits to verify the solution: - As \(x \to 0\), the tension approaches the tension at the very top of the rope.- As \(x \to L\), the weight of the entire block plus rope acts against the upward force, thus ensuring correctness in the setup and calculations.

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics. It describes how an object’s motion changes when forces are applied. The law is often summarized by the formula: \[ F = ma \]where

• \( F \) is the net force applied to an object,
• \( m \) is the mass of the object,
• \( a \) is the acceleration produced.

According to this law, the acceleration of an object depends directly on the net force acting upon it and inversely on its mass.
In our exercise with the block and rope, Newton's Second Law helps us calculate the tension by considering the net force balancing upward and downward forces, ultimately determining how much the block and rope accelerate.

uniform rope

A uniform rope is one where its mass is distributed evenly throughout its length. This means that any portion of the rope has the same mass per unit length as any other portion. For our problem, this uniformity allows us to express the mass of the rope below a certain point as a fraction of the total mass.
Given the rope of mass \( m \) and length \( L \), the mass at any distance below \( x \) would be proportional to the length from that point on. Using this idea helps in calculating the total mass from the point where tension is being calculated to the rope's end at the block, facilitating the computation of forces and acceleration.

vertical force

Vertical forces in this problem include the applied force \( \vec{F} \), the weight of the rope, and the block’s weight. **Vertical force** often refers to any push or pull in the vertical direction.
When calculating tension, you must consider how these forces interact.

• The applied force \( \vec{F} \) works against gravity by pulling the block and rope upwards.
• The gravitational force pulls all components downward.

The net vertical force decides the system’s acceleration and plays a significant role in determining the tension at any point along the rope.

acceleration

Acceleration is the rate at which an object's velocity changes over time. In our exercise, it's crucial for determining how the mass of both the block and rope move when a force is applied. It is denoted by \( a \) in our equations.
The acceleration is uniform throughout the rope and block system due to Newton's Second Law, meaning every part of the system speeds up or slows down at the same rate.
This acceleration is dependent on the net force imparted by the applied vertical force minus gravity working against this motion. While calculating tension, we factor in this uniform acceleration to understand how the entire rope and block move cohesively.