Question

The general equation for work is \(W=F d \cos \theta\). For what angle is the work \(W=F d\) ? For what angle is the work \(W=-F d\) ?

Short answer

The angle for \(W=Fd\) is \(0^\circ\) and for \(W=-Fd\) is \(180^\circ\).

Step-by-step solution

Understand the Work Formula

The formula for work is given by \(W = F d \cos \theta\), where \(W\) is the work done, \(F\) is the force applied, \(d\) is the distance moved by the point of application of the force in the direction of the force, and \(\theta\) is the angle between the force and the direction of the motion.

Work Without Angle Influence

We need to find the angle where the work done simplifies to \(W = Fd\). This is essentially \(W = Fd \cos \theta = Fd\), which implies that \(\cos \theta = 1\).

Solve for Angle with \(\cos \theta = 1\)

The condition \(\cos \theta = 1\) holds when \(\theta = 0^\circ\). Thus, the angle must be zero degrees, meaning the force is in the same direction as the displacement.

Negative Work Condition

Now, find the angle where the work done simplifies to \(W = -Fd\). This means \(W = Fd \cos \theta = -Fd\), which implies \(\cos \theta = -1\).

Solve for Angle with \(\cos \theta = -1\)

The condition \(\cos \theta = -1\) holds when \(\theta = 180^\circ\). Therefore, the angle must be 180 degrees, indicating that the force is in the opposite direction to the displacement.

Key concepts

Understanding the Work Formula

In physics, work is an important concept that describes how energy is transferred by a force acting over a distance. The basic formula for work is given by \[ W = F d \cos \theta \]

• \(W\) is the work done, measured in joules.
• \(F\) represents the force applied, often measured in newtons.
• \(d\) is the distance over which the force is applied, measured in meters.
• \(\theta\) is the angle between the direction of the applied force and the direction of the displacement.

This formula shows that work depends not only on the magnitude of the force and the distance moved but also on the angle between the force and the displacement.
A common task, as in the original exercise, is determining the angle \(\theta\) for specific work conditions, such as when \(W = Fd\) or \(W = -Fd\). Here, \(\cos \theta\) plays a critical role in defining how the force contributes to the work done.

The Role of Angle in Physics

The angle \(\theta\) in the work formula is crucial because it impacts how much of the force actually contributes to the work done. When the angle is zero (\(\theta = 0^\circ\)), the force is entirely in the direction of the displacement. This means that:

• \(\cos 0^\circ = 1\)
• The work done simplifies to the maximum value, \(W = Fd\)

When \(\theta = 180^\circ\), the force is directly opposite to the displacement. In this scenario:

• \(\cos 180^\circ = -1\)
• The work is negative, \(W = -Fd\)

Such negative work typically means that the force is working against the movement, like friction slowing down an object.
Understanding how different angles influence work helps you evaluate various real-world situations where force and movement are involved.

Force and Displacement Relationship

Force and displacement have a direct relationship with work. The primary factors are:

• Magnitude of the force, \(F\)
• The distance travelled in the direction of the force, \(d\)

When you apply a force perfectly in the direction of displacement, you maximize the work done, as computed by \(W = Fd\). If any components of the force act perpendicularly to the displacement, they do not contribute to the work done directly.
Consider, for example, pushing a box across the floor. If you push directly parallel to the floor, you're maximizing your effort conversion into work. Any deviation from this path means that a portion of your applied force doesn't contribute to the desired motion.

The Direction of Force

The direction of the force relative to the displacement path is a defining factor in calculating work. The angle \(\theta\) between the force's direction and the direction of the displacement determines how much of the force is effective.

• A zero-degree angle means full efficiency of the force, as all the force contributes to the work.
• A 180-degree angle implies complete opposition, resulting in the force working against the motion.
• Any angle in between adjusts the effectiveness of the force according to \(\cos \theta\).

Thus, to assess how forces operate in real life, always consider their direction and angle relative to the object's motion. Geometry and trigonometry become vital tools for these evaluations, enabling precise calculations of work and energy in various systems.