Problem 3 - University Physics with Modern Physics

Question

\(\mathrm{A} 75.0\) -kg wrecking ball hangs from a uniform heavy-duty chain having a mass of 26.0 \(\mathrm{kg}\) . (a) Find the maximum and minimum tension in the chain. (b) What is the tension at a point three-fourths of the way up from the bottom of the chain?

Step-by-Step Solution

Answer

T_max = 989.8 N, T_min = 735 N, T_3/4 = 926.1 N.

Step 1: Understanding the Problem

We have a wrecking ball with a mass of 75.0 kg hanging from a chain with a mass of 26.0 kg. The goal is to find the maximum and minimum tension in the chain and the tension three-fourths of the way from the bottom.

Step 2: Calculate Total Weight

First, calculate the weight of the wrecking ball and the chain. The weight is given by the formula \( W = mg \), where \( m \) is the mass and \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity.For the wrecking ball:\[ W_{b} = 75.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 735 \, \text{N} \]For the chain:\[ W_{c} = 26.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 254.8 \, \text{N} \]

Step 3: Determine Maximum Tension

The maximum tension occurs at the top of the chain. At this point, the entire weight of the wrecking ball and the chain must be supported.\[ T_{\text{max}} = W_{b} + W_{c} = 735 \, \text{N} + 254.8 \, \text{N} = 989.8 \, \text{N} \]

Step 4: Determine Minimum Tension

The minimum tension occurs at the bottom of the chain, where only the wrecking ball exerts force.\[ T_{\text{min}} = W_{b} = 735 \, \text{N} \]

Step 5: Find Tension Three-Fourths Up the Chain

Calculate the mass of the three-fourths section of the chain, which is\[ m_{3/4} = \frac{3}{4} \times 26.0 \, \text{kg} = 19.5 \, \text{kg} \]The weight of the chain three-fourths up is:\[ W_{3/4} = m_{3/4} \times g = 19.5 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 191.1 \, \text{N} \]So, the tension three-fourths up must support the weight of the wrecking ball and three-fourths of the chain:\[ T_{3/4} = W_{b} + W_{3/4} = 735 \, \text{N} + 191.1 \, \text{N} = 926.1 \, \text{N} \]

Key Concepts

Wrecking Ball Physics

A wrecking ball is an intriguing application of the physics we encounter in our daily lives. When you think of a wrecking ball, imagine a massive metal sphere used to demolish structures, suspended from a chain.
This setup introduces several fundamental principles of physics, primarily gravity and tension. The wrecking ball hangs from a chain because of the gravitational force pulling it towards the Earth. In physics, this force is called weight, which is distinct from mass.
While mass refers to the amount of matter in the wrecking ball, weight is the force exerted by the mass due to gravity. In essence, when a wrecking ball is released, it converts gravitational potential energy into kinetic energy, illustrating the transformation between different forms of energy.

• The chain itself is a critical component because it must be strong enough to support both the weight of the ball and its dynamic swinging motion.
• As the wrecking ball swings and collides with a structure, it also introduces the concepts of momentum and force, showing how its mass and velocity can apply powerful impacts.

Understanding these key ideas is essential for anyone studying basic mechanics or wanting to understand how forces and energies interact in real-world scenarios.

Mass and Weight Calculations

When dealing with physics problems like those involving a wrecking ball, understanding the difference between mass and weight is crucial. Mass and weight, while related, are not the same thing.**Mass** refers to the amount of matter an object contains and is measured in kilograms (kg). For example, a wrecking ball may have a mass of 75.0 kg, while the chain supporting it might have a mass of 26.0 kg.
**Weight**, on the other hand, is the force exerted by gravity on that mass. This force can be calculated using the equation:\[W = mg\]where \(m\) is the mass in kilograms and \(g\) is the acceleration due to gravity, approximately \(9.8 \, \text{m/s}^2\).

• Calculating the weight of the wrecking ball as 735 N indicates the force it exerts downwards due to gravity.
• The chain's weight can be calculated similarly, showing how forces are distributed along its length.
• Evaluating these values is essential to understand how the tension in the chain varies from top to bottom.

This quantitative understanding helps us predict how structures will behave under real-world conditions. It also prepares us to analyze more complex systems involving multiple forces and masses.

Tension Variation in Chains

Tension in a chain, like the one holding a wrecking ball, varies along its length. The tension forces are responsible for keeping the wrecking ball suspended and safe. Tension is essentially a pulling force exerted along the chain, and it changes according to the section of the chain and the amount of weight it supports.

• The maximum tension occurs at the top of the chain because it must support both the weight of the wrecking ball and the weight of the entire chain. This is calculated as the sum of the weights, resulting in a force of 989.8 N in the given problem.
• The minimum tension is at the bottom, only needing to support the wrecking ball itself, quantified as 735 N.
• In between, tension gradually varies. For instance, three-fourths up the chain, it combines the wrecking ball weight and the weight supported up to that point, giving a tension of 926.1 N.

Understanding tension variation helps engineers design materials and structures that can withstand different forces, ensuring safety and efficiency.
By mastering this concept, students can apply knowledge to scenarios ranging from simple suspended objects to complex engineering systems where load distribution is critical.