Understanding Work Done and Angle between Force and Displacement

Work done in physics is a concept that describes the transfer of energy that occurs when a force is applied over a distance. It is a scalar quantity, meaning it only has magnitude and no direction. The amount of work done depends on three factors:

• The magnitude of the force applied.
• The magnitude of the displacement.
• The angle between the direction of the force and the direction of the displacement.

Formula for Work Done

The mathematical formula for work done (\(W\)) by a constant force (\(F\)) causing a displacement (\(d\)) is given by the dot product of the force and displacement vectors. This can be written as:

\(W = \vec{F} \cdot \vec{d}\)

In terms of magnitudes of force and displacement and the angle between them, this formula is:

\(W = Fd \cos\theta\)

Here:

• \(F\) is the magnitude of the force.
• \(d\) is the magnitude of the displacement.
• \(\theta\) is the angle between the direction of the force and the direction of the displacement.

How the Angle Affects Work Done

The term \(\cos\theta\) in the work done formula is crucial because it determines how much of the force is acting in the direction of the displacement. The value of \(\cos\theta\) varies depending on the angle \(\theta\).

The cosine function (\(\cos\theta\)) for angles between \(0^\circ\) and \(180^\circ\) ranges from 1 to -1.

• When \(\theta = 0^\circ\), \(\cos\theta = \cos(0^\circ) = 1\). This is the maximum possible value for \(\cos\theta\).
• When \(\theta = 90^\circ\), \(\cos\theta = \cos(90^\circ) = 0\).
• When \(\theta = 180^\circ\), \(\cos\theta = \cos(180^\circ) = -1\). This is the minimum possible value for \(\cos\theta\).

Determining Maximum Work Done

For a given force (\(F\)) and displacement (\(d\)), the work done (\(W = Fd \cos\theta\)) is directly proportional to the value of \(\cos\theta\). To maximize the work done, we need to maximize the value of \(\cos\theta\).

As seen from the properties of the cosine function, the maximum value of \(\cos\theta\) is 1, which occurs when the angle \(\theta\) is \(0^\circ\).

Therefore, work done is maximum when the angle between the direction of force and the direction of displacement is \(0^\circ\).

When the angle is \(0^\circ\), it means the force is applied exactly in the same direction as the displacement.

Analyzing the Options

Let's look at the given options and calculate the value of \(\cos\theta\) for each angle:

1. \(\theta = 0^\circ\): \(\cos(0^\circ) = 1\)
2. \(\theta = 45^\circ\): \(\cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.707\)
3. \(\theta = 60^\circ\): \(\cos(60^\circ) = 0.5\)
4. \(\theta = 90^\circ\): \(\cos(90^\circ) = 0\)

Comparing the cosine values:

• \(\cos(0^\circ) = 1\)
• \(\cos(45^\circ) \approx 0.707\)
• \(\cos(60^\circ) = 0.5\)
• \(\cos(90^\circ) = 0\)

The highest value of \(\cos\theta\) among the given options is 1, which corresponds to an angle of \(0^\circ\).

Thus, work done will be maximum when the angle between the direction of force and the direction of displacement is \(0^\circ\).

Revision Table: Work Done and Angle

Additional Information on Work Done

• Positive Work: When the force and displacement are in the same general direction (\(0^\circ \le \theta < 90^\circ\)), \(\cos\theta\) is positive, and the work done is positive. This means the force is doing work on the object, increasing its energy (like speeding it up).
• Negative Work: When the force and displacement are in opposite directions (\(90^\circ < \theta \le 180^\circ\)), \(\cos\theta\) is negative, and the work done is negative. This means the force is opposing the motion, reducing the object's energy (like friction slowing down a moving object).
• Zero Work: When the force is perpendicular to the displacement (\(\theta = 90^\circ\)), \(\cos\theta\) is zero, and the work done is zero. This happens when a force acts but causes no displacement (like pushing against a wall) or when a force acts perpendicular to the motion (like the force of gravity on an object moving horizontally).