Question

Cables and ropes are made of several strands that contain individual wires or threads. The term "7 \(\times 19\) cable" refers to a cable with 7 strands, each containing 19 wires. Suppose a manufacturer uses the function \(d=\sqrt{\frac{b}{30}}\) to relate the diameter, \(d,\) in millimetres, of its \(7 \times 19\) stainless steel aircraft cable to the safe working load, \(b\) in kilograms. a) Is a cable with a diameter of \(6.4 \mathrm{mm}\) large enough to support a mass of \(1000 \mathrm{kg} ?\) b) What is the safe working load for a cable that is \(10 \mathrm{mm}\) in diameter?

Short answer

a) Yes, 1228.8 kg > 1000 kg. b) 3000 kg.

Step-by-step solution

Understand the Problem

Identify the given values and the required unknowns. For part (a), we need to determine if a cable with a diameter of 6.4 mm can support a mass of 1000 kg. For part (b), we need to find the safe working load for a cable with a diameter of 10 mm.

Apply the Given Formula

Use the formula provided: \[ d = \sqrt{\frac{b}{30}} \] where \(d\) is the diameter in millimetres and \(b\) is the safe working load in kilograms. We need to rearrange and use this formula appropriately for both parts (a) and (b).

Solve for Part (a)

We need to determine if a cable with a diameter of 6.4 mm can support a mass of 1000 kg.1. Start by expressing \(b\) in terms of \(d\) using the formula: \[ b = 30d^2 \]2. Substitute \(d = 6.4\) mm into the formula: \[ b = 30 \times (6.4)^2 = 30 \times 40.96 = 1228.8 \, \text{kg} \]3. Compare the calculated safe working load with 1000 kg. Since 1228.8 kg > 1000 kg, the cable with a diameter of 6.4 mm can support the mass.

Solve for Part (b)

We need to determine the safe working load for a cable with a diameter of 10 mm.1. Use the formula again: \[ b = 30d^2 \]2. Substitute \(d = 10\) mm into the formula: \[ b = 30 \times (10)^2 = 30 \times 100 = 3000 \, \text{kg} \] Therefore, the safe working load for a cable with a diameter of 10 mm is 3000 kg.

Key concepts

safe working load

The term 'safe working load' (SWL) refers to the maximum load that a cable or rope can handle safely during regular operations. In the context of cables, this value is crucial as it ensures safety and prevents equipment failure.
To determine the SWL of a cable, we often use empirical formulas that consider the material properties and the cable's construction. In our exercise, we used the formula:
\[ d = \sqrt{\frac{b}{30}} \]
where:

• d is the cable diameter in millimetres
• b is the SWL in kilograms

It's important to note that exceeding the SWL can lead to potential hazards, including cable snapping or structural failure. Engineers and safety specialists always factor in SWL to ensure reliability and safety in critical applications like aircraft or construction industries. Make sure to always check for SWL in any load-bearing equipment you use.

diameter calculation

Calculating the diameter of a cable is a fundamental step in ensuring it can support a specified load. Using the same formula:
\[ d = \sqrt{\frac{b}{30}} \]
Here, we can rearrange it to solve for the diameter when the SWL is known:
\[d = \sqrt{\frac{b}{30}}\]
To determine if a cable is strong enough, follow these steps:

• Calculate the square of the given diameter.
• Multiply the result by the empirical factor (in this case, 30).

For instance, if a 6.4mm diameter cable is assessed to support 1000kg, the calculation would be:
\[b = 30 \times d^2 = 30 \times (6.4 \text{ mm})^2 = 1228.8 \text{ kg}\]
Since 1228.8 kg is greater than 1000 kg, the cable can indeed support the load.
Understanding diameter calculations helps in selecting the right cable for specific applications, thereby ensuring safety and functionality.

cable strength

Cable strength refers to the ability of a cable to endure forces without breaking or deforming. Its strength is dependent on multiple factors, including the material used, the construction of the strands, and the diameter of the cable.
Cables are often made with multiple strands, as indicated by the '7 x 19' specification, meaning 7 strands each containing 19 wires. This intricate construction boosts the overall strength and flexibility of the cable.
In our exercise, to find the safe working load for a 10mm diameter cable, we used:
\[ b = 30 \times d^2 \]
Plugging in our value:
\[b = 30 \times (10 \text{ mm})^2 = 3000 \text{ kg}\]
Thus, a 10 mm cable can support a load of up to 3000 kg.
Choosing the correct cable strength is critical to preventing accidents and ensuring that applications function correctly. Always factor in the load requirements and choose cables with appropriate strength ratings.