Problem 57 - University Physics with Modern Physics

Question

A student tries to raise a chain consisting of three identical links. Each link has a mass of 300 \(\mathrm{g}\) . The three-piece chain is connected to a string and then suspended vertically, with the student holding the upper end of the string and pulling upward. Because of the student's pull, an upward force of 12 \(\mathrm{N}\) is applied to the chain by the string. (a) Draw a free-body diagram for each of the links in the chain and also for the entire chain considered as a single body. (b) Use the results of part (a) and Newton's laws to find ( ) the acceleration of the chain and (ii) the force exerted by the top link on the middle link.

Step-by-Step Solution

Answer

The chain's acceleration is 3.53 m/s². The force from the top link on the middle link is 4.00 N.

Step 1: Understand the Problem

We have a chain with three links, each having a mass of \(300 \text{ g}\), which is equal to \(0.3 \text{ kg}\) when converted into kilograms. Each link is connected vertically, and an upward force of \(12 \text{ N}\) is applied to the chain by the string. We need to draw free-body diagrams and use Newton's laws to calculate the acceleration of the chain and the force exerted by the top link on the middle link.

Step 2: Calculate the Total Mass of the Chain

The total mass of the chain is the sum of the masses of its three links.\[ m = 3 \times 0.3 \text{ kg} = 0.9 \text{ kg} \]

Step 3: Draw Free-Body Diagram for the Entire Chain

For the entire chain, consider the forces acting on it: the upward force \( F_{up} = 12 \text{ N} \) and the gravitational force (weight) \( F_{g} = m \cdot g \), where \( g = 9.8 \text{ m/s}^2 \).\[ F_{g} = 0.9 \text{ kg} \times 9.8 \text{ m/s}^2 = 8.82 \text{ N} \]The net force \( F_{net} = F_{up} - F_{g} = 12 \text{ N} - 8.82 \text{ N} = 3.18 \text{ N} \).The free-body diagram will show the chain with a downward force \( 8.82 \text{ N} \) and an upward force of \( 12 \text{ N} \).

Step 4: Calculate the Acceleration of the Chain

Using Newton's second law, \( F = ma \), where \( F \) is the net force and \( m \) is the mass,\[ a = \frac{F_{net}}{m} = \frac{3.18 \text{ N}}{0.9 \text{ kg}} = 3.53 \text{ m/s}^2 \]This is the acceleration of the entire chain.

Step 5: Draw Free-Body Diagram for an Individual Link

For each individual link, consider the forces acting on it. The link's weight is \( F_{g,link} = 0.3 \text{ kg} \times 9.8 \text{ m/s}^2 = 2.94 \text{ N} \). The top link has an upward force \( F_{up} = 4.94 \text{ N} \) (to account for its own weight and the next link's weight plus extra force for acceleration), and the middle link experiences a force from both the top and bottom links. Its net force is:\[ F_{net, middle} = m \cdot a = 0.3 \text{ kg} \times 3.53 \text{ m/s}^2 = 1.06 \text{ N} \]

Step 6: Calculate the Force Exerted by the Top Link on the Middle Link

For the middle link, the net force must account for its own weight and the force needed for its acceleration:\[ F_{top \to middle} - 2.94 \text{ N} = 1.06 \text{ N} \] (force exerted by the top link minus its weight equals its net force).\[ F_{top \to middle} = 4.00 \text{ N} \]

Key Concepts

Free-Body Diagram

A free-body diagram is a simple drawing that represents all the forces acting on an object. For this exercise, we need to focus on each link of the chain, as well as the entire chain itself. The main forces to depict are the gravitational force and the applied force.

• Gravitational Force: This is always directed downward. For the entire chain, the gravitational force is the total weight. For each individual link, it's just its own weight.
• Applied Force: This force acts upward and is due to the string being pulled. For the entire chain, this is 12 N, while for each link, it varies depending on how many links it's pulling directly.

The free-body diagram for the entire chain should show the net forces including the total gravitational pull of 8.82 N downwards, and the 12 N upwards from the string. For individual links, show how the top link must exert additional force to also pull the succeeding links upward.

Force and Acceleration

Newton's Second Law tells us that the acceleration of an object is due to the net force acting on it, divided by its mass. Essentially, it represents how the force affects movement. In our exercise, we calculate the acceleration of the chain by first finding the net force applied.

• Calculate the net force as the difference between the applied upward force (12 N) and the chain's total gravitational force (8.82 N).
• Use the formula: \[ a = \frac{F_{net}}{m} \] Here, \( F_{net} \) is 3.18 N and \( m \) is the chain's mass, 0.9 kg.
• So, \[ a = \frac{3.18 \text{ N}}{0.9 \text{ kg}} = 3.53 \text{ m/s}^2 \]This means each link of the chain is accelerating upwards at this rate.

The force needed to move something always has a relationship with its mass and the acceleration desired. This principle drives our calculations and understanding of force interactions.

Mass and Weight Calculation

Understanding mass and weight is crucial when dealing with problems involving forces. Mass reflects the amount of matter in an object and remains constant, while weight is a force considering the local gravitational field.

• Mass: In our scenario, each link has a mass of 0.3 kg, crucial for calculating individual weights and the total chain's weight.
• Weight: The weight can be calculated using:\[ F_g = m \cdot g \]where \( F_g \) is the gravitational force (or weight), \( m \) is mass, and \( g \) is the acceleration due to gravity (9.8 m/s²).
• For one link: \[ F_{g,link} = 0.3 \text{ kg} \times 9.8 \text{ m/s}^2 = 2.94 \text{ N}\]For the entire chain: \[ F_{g} = 3 \times 0.3 \text{ kg} \times 9.8 \text{ m/s}^2 = 8.82 \text{ N} \]

In any situation involving weights and forces, understanding these concepts provides a solid base to analyze motion and dynamics correctly.