A steel cable is to support an elevator whose total (loaded) mass is not to exceed 3100 kg. If the maximum acceleration of the elevator is 1.8 m/s² , calculate the diameter of cable required. Assume a safety factor of 8.0.

Verified step by step guidance

1. Determine the maximum force the cable must support using Newton's second law: \( F_{max} = m(a + g) \), where \( m \) is the mass of the elevator (3100 kg), \( a \) is the maximum acceleration (1.8 m/s²), and \( g \) is the acceleration due to gravity (9.8 m/s²).
2. Apply the safety factor to the maximum force: \( F_{safe} = F_{max} \times \text{safety factor} \). The safety factor is given as 8.0.
3. Relate the safe force to the tensile strength of the steel cable: \( F_{safe} = \sigma \cdot A \), where \( \sigma \) is the tensile strength of steel (a typical value is around 500 \( \text{MPa} \), but confirm the exact value if provided), and \( A \) is the cross-sectional area of the cable.
4. Express the cross-sectional area \( A \) in terms of the cable's diameter \( d \): \( A = \frac{\pi d^2}{4} \). Substitute this into the equation for \( F_{safe} \) to solve for \( d \): \( d = \sqrt{\frac{4F_{safe}}{\pi \sigma}} \).
5. Substitute the known values for \( F_{safe} \), \( \pi \), and \( \sigma \) into the equation to calculate the required diameter \( d \). Ensure the units are consistent throughout the calculation (e.g., convert \( \text{MPa} \) to \( \text{N/m}^2 \) if necessary).

Key Concepts

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). In this context, the total force required to support the elevator includes both the gravitational force and the force due to acceleration. Understanding this law is crucial for calculating the tension in the cable needed to support the elevator's mass while accounting for its acceleration.

Tension in a Cable

Tension is the force transmitted through a cable when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the steel cable must be sufficient to support the weight of the elevator and provide the necessary upward acceleration. The tension can be calculated by considering both the weight of the elevator and the additional force required for acceleration, which is essential for determining the cable's specifications.

Safety Factor

The safety factor is a design criterion that provides a margin of safety in engineering applications. It is defined as the ratio of the maximum load a structure can handle to the intended load. In this case, a safety factor of 8.0 means the cable must be able to support eight times the calculated tension to ensure safety under maximum load conditions. This concept is vital for ensuring that the cable can withstand unexpected stresses without failure.