Work done by a force

Welcome to the World of Work!

In everyday life, we say we’re "working" when we study for Physics or sit at a desk. But in the world of Physics, work has a very specific, "hands-on" meaning. By the end of these notes, you’ll understand that work is all about the mechanical transfer of energy. If you apply a force and something moves, you’ve done work! It’s the bridge between forces and energy, and it's a foundational concept for everything in the "Energy and Fields" section.

1. What Exactly is "Work Done"?

In Physics, Work Done is defined as the product of the force and the displacement in the direction of that force.

Think of it as a way to measure how much energy you have given to (or taken from) an object by pushing or pulling it. If you push a shopping cart and it moves, you have transferred energy from your body to the cart. That transfer is "Work."

The Formula

The standard formula you need to know is:
\( W = Fs \cos \theta \)

Where:
• \( W \) is the Work Done (measured in Joules, J)
• \( F \) is the magnitude of the constant force (Newtons, N)
• \( s \) is the displacement (metres, m)
• \( \theta \) is the angle between the force and the direction of motion

The "In the Direction of" Rule

This is the most important part! Work is only done by the component of the force that acts along the line of motion.
• If you push a box horizontally and it moves horizontally, \( \theta = 0^\circ \). Since \( \cos 0^\circ = 1 \), the formula is simply \( W = Fs \).
• If you pull a suitcase at an angle, only the horizontal part of your pull is doing "work" to move it forward.

Quick Review: Work Done is the mechanical transfer of energy. No movement = No work!

2. Real-World Analogies

The "Wall Push" Challenge: Imagine you spend an hour pushing as hard as you can against a solid brick wall. You’ll be sweating and exhausted, but in Physics terms, you have done zero work on the wall. Why? Because the wall didn't move (\( s = 0 \)). Energy was spent by your muscles, but no energy was transferred to the wall.

The Waiter’s Tray: A waiter carries a heavy tray at a constant height while walking across a room. Surprisingly, the waiter does no work on the tray in the direction of motion. The force is upward (to balance gravity), but the motion is horizontal. Since the force and displacement are perpendicular (\( \theta = 90^\circ \)) and \( \cos 90^\circ = 0 \), no work is done by the lifting force!

Did you know? One Joule is roughly the energy required to lift a small apple (about 100g) one metre straight up!

3. Common Pitfalls to Avoid

Don't worry if this seems a bit confusing at first! Many students trip up on these three points:
1. Distance vs. Displacement: Always use displacement (the straight-line distance from start to finish) when calculating work done by a constant force.
2. The Units: Always convert your distance to metres and your force to Newtons before calculating. If a question gives you "kJ," remember that 1 kJ = 1,000 J.
3. Negative Work: Yes, work can be negative! If the force is acting against the motion (like friction slowing down a car), the angle \( \theta \) is \( 180^\circ \). Since \( \cos 180^\circ = -1 \), the work done is negative. This means energy is being removed from the object.

4. Finding Work from a Graph

Sometimes the force isn't constant. For example, when you stretch a spring, the harder you pull, the more it resists (Hooke's Law: \( F = kx \)). In these cases, you can't just multiply the final force by the distance.

Instead, look at a Force-Displacement Graph (F vs. s).
Key Rule: The area under the graph represents the Work Done.

• For a constant force, the graph is a rectangle (Area = Base \(\times\) Height).
• For a spring, the graph is a triangle (Area = \( \frac{1}{2} \times \text{base} \times \text{height} \)). This is why Elastic Potential Energy is \( \frac{1}{2} Fx \) or \( \frac{1}{2} kx^2 \).

Takeaway: If the force changes, use the area under the F-s graph to find the work done.

5. Work Done by a Gas

In the "Energy and Fields" and "Thermal Physics" sections, you'll often deal with gases expanding in a cylinder (like in a car engine). When a gas pushes a piston, it is doing work.

If the pressure \( p \) is constant, the work done by the gas is:
\( W = p \Delta V \)

Where:
• \( p \) is the pressure (Pascals, Pa)
• \( \Delta V \) is the change in volume (cubic metres, \( m^3 \))

Memory Trick: Think of "P-V" as "Pressure-Volume." If the volume increases (expansion), the gas does work on the surroundings. If the volume decreases (compression), work is being done on the gas.

6. Summary and Key Points

Work is the way forces move energy from one place to another. To master this chapter, keep these "Golden Rules" in mind:
• Work = Force \(\times\) Displacement (but only the parts that line up!).
• Units: Always use Joules (J).
• Perpendicular forces (like the normal force on a sliding box) do zero work.
• Graphs: Work Done = Area under the Force-Displacement graph.
• Energy Conservation: Work done on an object usually shows up as a change in its Kinetic Energy (\( E_k \)) or Potential Energy (\( E_p \)).

Keep practicing! Physics is like a sport—the more you exercise your "problem-solving muscles," the stronger they get. You've got this!