Question

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

Short answer

(a) Maximum tension: 989.8 N, Minimum tension: 735 N. (b) Tension 3/4 up: 798.7 N.

Step-by-step solution

Understanding the Problem

We have a wrecking ball with mass 75.0 kg and a chain with mass 26.0 kg. We need to find the maximum and minimum tensions in the chain and the tension three-fourths up from the bottom.

Finding the Weight of the Wrecking Ball

The weight of the wrecking ball can be calculated using the formula for weight: \[ W_{ball} = m_{ball} \times g \]where \( m_{ball} = 75.0 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \).The weight is \( W_{ball} = 75 \times 9.8 = 735 \text{ N} \).

Calculating Total Weight of Wrecking Ball and Chain

Calculate the total weight of the chain using the same formula for weight: \[ W_{chain} = m_{chain} \times g \]where \( m_{chain} = 26.0 \text{ kg} \).The weight is \( W_{chain} = 26 \times 9.8 = 254.8 \text{ N} \).The total weight \( W_{total} = W_{ball} + W_{chain} = 735 + 254.8 = 989.8 \text{ N} \).

Determining Maximum Tension

The maximum tension in the chain occurs at the top, where it must support the total weight of the wrecking ball and the chain. Therefore, the maximum tension is equal to the total weight of both objects:\[ T_{max} = 989.8 \text{ N} \].

Determining Minimum Tension

The minimum tension occurs at the bottom of the chain, where the tension only needs to support the wrecking ball's weight:\[ T_{min} = 735 \text{ N} \].

Finding Tension Three-Fourths Up the Chain

To find the tension at a point three-fourths from the bottom:1. Calculate the weight of the chain portion above this point. The bottom quarter of the chain's mass is \( m = \frac{26}{4} = 6.5 \text{ kg} \).2. The weight of this portion: \( W_{portion} = 6.5 \times 9.8 = 63.7 \text{ N} \).3. Add the weight of the wrecking ball:\[ T_{three ext{-}fourths} = 735 + 63.7 = 798.7 \text{ N} \].

Key concepts

Understanding Tension in the Chain

Tension is an essential concept in physics, especially when dealing with objects that hang or are supported by cables or chains. In the context of our exercise, tension refers to the force exerted along the chain that is holding up the wrecking ball. The maximum tension in the chain occurs at the top, while the minimum tension is at the bottom. This is because the chain must support the weight of the wrecking ball and partially, or fully, the weight of the chain itself.
To calculate these tensions, we need to consider:

• The maximum tension occurs where the chain supports the entire weight of both the wrecking ball and the chain itself. This is the stress at the top of the chain.
• The minimum tension is found at the bottom, where it only supports the weight of the wrecking ball.
• At any given point along the chain, particularly when asked for specific points such as three-fourths up, the tension must support the wrecking ball and the chain's weight above that point.

Understanding these calculations helps in applying the concept of tension to similar problems involving hanging objects and provides clarity on how forces distribute through such systems.

The Role of Gravity

Gravity is the force that pulls two masses towards each other, and in this case, it acts on both the wrecking ball and the chain. It is an essential factor to consider when calculating weights and forces in physics problems. The acceleration due to gravity is typically denoted as \( g \), which is approximately \( 9.8 \text{ m/s}^2 \) on Earth.
The key points to remember about gravity in this exercise are:

• Gravity acts uniformly on both the wrecking ball and the chain, affecting their combined weight.
• The weight of any object can be calculated using the formula \( W = m \times g \), where \( m \) is the object's mass. Here, we use this formula to determine the weight of both the wrecking ball and the chain separately.
• The total force, or weight, due to gravity is necessary for determining the tension in the chain at various points as it must balance this gravitational force to keep the system static.

Grasping the role of gravity allows for a deeper understanding of how it influences tension and stability in structures like chains holding heavy objects.

Insights into Wrecking Ball Mechanics

A wrecking ball is a large, heavy object used in demolition that relies on both weight and motion to be effective. When analyzing it hanging from a chain, understanding its physical properties and the forces acting upon it is crucial.
Here's what you need to know:

• A standard wrecking ball is shaped spherical to maximize its impact area. The round shape allows it to exert force over a wide area, making it efficient in demolition.
• When calculating physical forces like tension and gravity, the mass of the wrecking ball is considered a point mass for simplicity unless otherwise specified.
• Static forces like the tension in the chain and dynamic forces, if the ball is swinging, need to be understood to safely and effectively operate equipment involving a wrecking ball.

Understanding the mechanics involved helps not just in solving physics problems but also in comprehending real-world applications of physics principles.