Problem 44 - University Calculus: Early Transcendentals

Question

Pacific's Big Boy locomotive could pull 6000 -ton trains with a tractive effort (pull) of \(602,148 \mathrm{N}\) \((135,375 \mathrm{lb})\). At this level of effort, about how much work did Big Boy do on the (approximately straight) \(605-\mathrm{km}\) journey from San Francisco to Los Angeles?

Step-by-Step Solution

Answer

The work done is approximately \(364,297,740,000 \text{ J}\).

Step 1: Identify Given Values

We are given the tractive effort (force) as \(F = 602,148 \text{ N}\) and the distance of the journey as \(d = 605 \text{ km}\). First, convert the distance to meters: \(d = 605 \times 1000 \text{ m} = 605,000 \text{ m}\).

Step 2: Use the Formula for Work

The formula to calculate the work (\(W\)) done by a force \(F\) over a distance \(d\) in the direction of the force is \(W = F \times d\). Substitute the given values into this formula: \(W = 602,148 \times 605,000\).

Step 3: Calculate the Work Done

Perform the multiplication from the formula: \(W = 602,148 \times 605,000 = 364,297,740,000 \text{ J}\). The work done by the locomotive is therefore \(364,297,740,000 \text{ Joules}\).

Key Concepts

Tractive Effort

Tractive effort is a key concept when discussing how vehicles like locomotives pull heavy loads. It refers to the pulling force that a vehicle can exert on a surface, ultimately moving a load over that surface. The Big Boy locomotive, for instance, boasted an impressive tractive effort of 602,148 N (Newtons). This mighty force enabled it to move heavy trains, due to the power of its engines and the friction between its wheels and the track. Tractive effort is crucial for understanding how high-speed and heavy-load transportation is achieved.

• The greater the tractive effort, the heavier the load a vehicle can pull.
• Sufficient tractive effort helps overcome initial resistance to motion.

Understanding this concept helps explain why some trains are able to haul larger loads than others. It's all about how effectively they can apply force to overcome gravity and friction.

Force

Force is a fundamental idea in physics and is essential for analyzing how objects move and interact. In the context of the Big Boy locomotive, the force exerted ( F) is the same as the tractive effort. It measures how hard the locomotive's engines work to pull trains along the tracks. In physics, force is measured in Newtons (N). Forces play various roles:

• They cause objects to start moving, stop moving, or change direction.
• Forces act in pairs —the concept of action and reaction.

With the locomotive, the force helps counteract resisting forces such as friction and gravity. This balance allows the locomotive to efficiently move over great distances, performing considerable work.

Distance

Distance plays a significant role when calculating how much work a force has done. It refers to the length of the path that an object travels, regardless of its direction. It's often measured in meters (m), though kilometers can be used for longer distances. In our exercise, the Big Boy's journey spanned 605,000 meters (or 605 kilometers) from San Francisco to Los Angeles. Converting distances to consistent units helps in calculating work more easily. Key factors related to distance include:

• Work is directly proportional to distance — more distance means more work done if the force remains constant.
• Understanding the path and conditions is crucial for accurate distance measurement.

Joules

Joules are the units of measurement for work and energy in physics. When a force is applied to move an object over a distance, the work done is measured in Joules (J). For the Big Boy locomotive, the calculation of work done resulted in 364,297,740,000 Joules. This shows how much energy was transferred during its long journey. A few important points about Joules:

• It quantifies the energy required or produced by forces acting over distances.
• One Joule is the work done when a force of one Newton displaces an object by one meter.

Understanding Joules helps in grasping how energy is used and conserved in different physical processes. With Joules, we can assess the efficiency and demand of movement and force.