Problem 36 - Thomas Calculus

Question

A revealing experiment By experiment, you find that a force field \(\mathbf{F}\) performs only half as much work in moving an object along path \(C_{1}\) from \(A\) to \(B\) as it does in moving the object along path \(C_{2}\) from \(A\) to \(B .\) What can you conclude about \(\mathbf{F}\) ? Give reasons for your answer.

Step-by-Step Solution

Answer

The force field \( \mathbf{F} \) is not conservative because the work is path-dependent.

Step 1: Understanding the Problem

We need to analyze the behavior of the force field \( \mathbf{F} \) in terms of the work done on an object as it moves between two paths, \( C_{1} \) and \( C_{2} \), from point \( A \) to point \( B \). The key point is that the work done by \( \mathbf{F} \) over \( C_{1} \) is half that of \( C_{2} \).

Step 2: Concept of Conservative Force Field

Determine whether \( \mathbf{F} \) is conservative. In a conservative force field, the work done is path-independent and only depends on the initial and final positions. If \( \mathbf{F} \) were conservative, the work done over \( C_{1} \) and \( C_{2} \) would be the same.

Step 3: Implication of Different Work Values

Since work done over the two paths is different, \( \mathbf{F} \) cannot be conservative. Instead, it must be non-conservative. Non-conservative forces depend on the specific path taken, leading to different amounts of work over different paths between the same points.

Key Concepts

Force Field

A force field represents a region of space in which a force operates on an object. In physics, force fields are often represented by a vector that describes the magnitude and direction of the force acting at each point in the space.

Force fields can exist in various forms:

• Gravitational fields exert forces due to mass.
• Electric fields arise from electric charges.
• Magnetic fields result from moving charges or magnetic materials.

When analyzing a force field \( \mathbf{F} \), it's crucial to consider how it influences the movement of objects from one point to another.Understanding the work done by the force field helps determine its characteristics and behavior.

Work Done

The concept of work done in the context of vector calculus involves moving an object through a force field. Work done is calculated by integrating the force along the path of movement. If a vector field \( \mathbf{F} \) acts on a particle moving along a path \( C \), the work done \( W \) by the force is given by:
\[ W = \int_{C} \mathbf{F} \cdot d\mathbf{r} \]
where \( d\mathbf{r} \) represents an infinitesimal displacement along the path.

In simple terms, the work done is the product of the force along the direction of motion and the distance over which it acts.If more work is done over one path than another between the same points, the force field influencing these paths might differ, as shown in the initial problem.

Conservative Force Field

A conservative force field is a type of force field where the work done only depends on the initial and final positions, and not the path taken. In mathematical terms, the line integral of a conservative vector field around any closed loop is zero.

Some key properties of conservative force fields include:

• Work done is path-independent.
• They possess a potential energy function.
• The mechanical energy (kinetic plus potential) is conserved in the system.

If a force field \( \mathbf{F} \) were conservative, the work done moving an object from point \( A \) to point \( B \) would be equal on any path between these two points, as emphasized in the exercise.

Non-Conservative Forces

Non-conservative forces are forces where the work done depends on the path taken. They contrast with conservative forces, which are path-independent.

Characteristics of non-conservative force fields include:

• Work done varies with different paths between the same start and end points.
• Energy is not conserved because it is converted into other forms, such as heat.
• Examples include friction and air resistance.

In the given problem, since the work done on paths \( C_{1} \) and \( C_{2} \) is different, force field \( \mathbf{F} \) must be non-conservative, suggesting path-dependence in energy dynamics.