understand the concept of work, and recall and use work done = force × displacement in the direction of the force

Energy Conservation

Energy is a property of physical systems that can be transferred or transformed.

The law of conservation of energy states that in an isolated system the total energy remains constant, even though it can change form (e.g., from kinetic to potential energy).

Work – The Bridge Between Force and Energy

Work is the mechanism by which energy is transferred when a force acts on an object over a distance.

Think of it like pushing a friend on a swing: the push (force) over the swing’s travel (displacement) gives the friend energy to keep moving.

Key Formula

The work done by a constant force is given by:

\$W = \mathbf{F} \cdot \mathbf{d} = F\,d\,\cos\theta\$

• \$\mathbf{F}\$ – force vector (in newtons, N)
• \$\mathbf{d}\$ – displacement vector (in metres, m)
• \$\theta\$ – angle between force and displacement

When Work is Positive, Negative, or Zero

• Positive work (\$\theta < 90^\circ\$): energy is transferred to the object.
• Negative work (\$\theta > 90^\circ\$): energy is taken from the object.
• Zero work (\$\theta = 90^\circ\$): force is perpendicular to motion (e.g., gravity on a projectile in flight).

Example 1 – Pushing a Box

You push a box with a constant force of \$30\,\text{N}\$ in the direction the box moves.

The box slides \$5\,\text{m}\$.

Because the force is parallel to the displacement (\$\theta = 0^\circ\$), \$\cos\theta = 1\$.

\$W = 30\,\text{N} \times 5\,\text{m} \times 1 = 150\,\text{J}\$

Example 2 – Pulling a Rope at an Angle

You pull a rope with a force of \$20\,\text{N}\$ at \$30^\circ\$ above the horizontal.

The rope moves \$4\,\text{m}\$ horizontally.

Only the horizontal component of the force does work on the rope:

\$F_{\text{horizontal}} = 20\,\text{N} \cos30^\circ \approx 17.32\,\text{N}\$

\$W = 17.32\,\text{N} \times 4\,\text{m} \approx 69.3\,\text{J}\$

Connecting Work to Energy Conservation

When work is done on an object, its kinetic or potential energy changes.

The work-energy theorem states:

\$W_{\text{total}} = \Delta K\$

If non-conservative forces (like friction) do negative work, the mechanical energy decreases, but the total energy (including thermal, etc.) remains constant.

Quick Practice Problems

1. A 2 kg crate is pulled across a floor with a constant force of 10 N for 3 m.

   Calculate the work done. 🤔

2. A roller coaster car of mass 500 kg starts from rest at the top of a 30 m hill.

   Assuming no friction, find its speed at the bottom using energy conservation. 🎢

Summary Table – Work Scenarios

Remember: Work is the vehicle that moves energy around.

By mastering the work formula, you unlock the ability to predict how energy changes in any physical situation. 🚀