Question

As mentioned on page \(13,\) the MCO was lost because of a mix-up in the units used to calculate the force needed to correct its trajectory. Ground-based computers generated the force correction file. On September \(29,1999,\) it was discovered that the forces reported by the ground-based computer for use in MCO navigation software were low by a factor of \(4.45 .\) The erroneous trajectory brought the MCO \(56 \mathrm{km}\) above the surface of Mars; the correct trajectory would have brought the MCO approximately \(250 \mathrm{km}\) above the surface. At \(250 \mathrm{km},\) the MCO would have successfully entered the desired elliptic orbit. The data contained in the force correction file were delivered in lb-sec instead of the required SI units of newton-sec for the MCO navigation software. The newton is the SI unit of force and is described in Appendix B. The British Engineering (gravitational) system uses a pound (lb) as a unit of force and \(\mathrm{ft} / \mathrm{s}^{2}\) as a unit of acceleration. In turn, the pound is defined as the pull of Earth on a unit of mass at a location where the acceleration due to gravity is \(32.174 \mathrm{ft} / \mathrm{s}^{2} .\) The unit of mass in this case is the slug, which is \(14.59 \mathrm{kg}\). Thus, BE unit of force \(=1\) pound \(=(\text { slug })\left(\mathrm{ft} / \mathrm{s}^{2}\right)\) Use this information to confirm that BE unit of force \(=4.45 \times\) SI unit of force 1 pound \(=4.45\) newton

Short answer

Yes, 1 pound (BE unit of force) is confirmed to be equivalent to 4.45 Newtons (SI unit of force).

Step-by-step solution

Identify the quantities given and the units that need conversion

The given problem mentions that 1 pound in the BE system equates to a slug multiplied by \( \frac{ft}{s^2} \), where a slug is \( 14.59 kg \). It also states that at a location where the acceleration due to gravity is \( 32.174 \frac{ft}{s^2} \), the Earth would pull a unit mass (slug) with a force of 1 pound. The aim is to confirm whether 1 pound equals to 4.45 newtons.

Write down the definition of a Newton (SI unit of force)

A newton is defined as the amount of force required to accelerate a one-kilogram mass by one meter per second squared. This can be written as: \(1 N = 1 \frac{kg \cdot m}{s^2}\).

Convert a pound (BE unit of force) to SI units

This step involves multiple subsets of conversion: Convert the slug to kilogram: \(1 slug = 14.59 kg\). Convert feet to meter: \(1 ft = 0.3048 m\). Substitute these conversion factors and plug in the given value of acceleration due to gravity into the equation for 1 pound (which equals to slug multiplied by \(\frac{ft}{s^2}\)): \(1 pound = 14.59 kg \cdot 32.174 (0.3048 \frac{m}{s^2}) = 4.45 \frac{kg \cdot m}{s^2}\). This is essentially 4.45 Newtons.

Confirm the conversion

As calculated in the previous step, a pound (BE unit of force) is equivalent to 4.45 Newtons (SI unit of force). Hence the conversion is confirmed.

Key concepts

SI Units

The International System of Units, or SI, is a globally accepted system used for scientific measurements. It ensures consistency and accuracy across different fields and locations.

• In the SI system, the unit of force is the newton (N).
• The unit of mass is the kilogram (kg).
• The unit of acceleration is meters per second squared (m/s²).

This system is crucial for scientists and engineers, as it allows them to communicate their findings and work collaboratively without misunderstandings resulting from different measurement systems.

British Engineering System

The British Engineering (BE) system, also known as the imperial system, is another measurement system used primarily in the United States.

• In the BE system, the unit of force is the pound (lb).
• The unit of mass is the slug.
• The unit of acceleration is feet per second squared (ft/s²).

This system is based on gravitational measures, making it different from the SI system. Understanding these differences is key when working on projects involving calculations from both systems.

Force Conversion

When working with forces, conversion between units is often essential, especially between SI and BE systems. This involves several conversion factors:

• The conversion between pounds and newtons is a common task.
• One slug, the BE unit of mass, is equivalent to 14.59 kg in SI units.
• 1 pound is defined in terms of the Earth's gravitational pull.

By using these conversions, problems like the Mars Climate Orbiter's trajectory can be solved correctly, preventing costly errors.

Newton

A newton is the SI unit for force and is crucial in physics and engineering.

• A force of 1 newton is enough to accelerate a mass of 1 kg at a rate of 1 m/s².
• This can be expressed by the formula: \(1 N = 1 \frac{kg \cdot m}{s^2}\).

Understanding this definition helps in converting other units to newtons, which is vital in ensuring correct calculations across various scientific tasks.

Pound

In the British Engineering system, the pound (lb) is used as a unit of force. It is important to comprehend how this unit works in converting to and from the SI system.

• The pound is defined by the force experienced by a unit mass (slug) under the Earth's gravity.
• Its value is dependent on a gravity of \(32.174 \frac{ft}{s^2}\).

Converting pounds to newtons involves using an understanding of both the pound and the slug.

Slug

The slug is the unit of mass in the British Engineering system. It's crucial in calculations involving forces:

• 1 slug is equivalent to 14.59 kg in the SI system.
• Used predominantly in contexts dealing with gravitational force.

Knowing how to convert slugs to kilograms and use them with feet and meters ensures accurate force calculations. This is especially significant in scenarios requiring precise engineering measurements, like space missions.