Work-Energy Principle (Edexcel A-Level Further Mathematics): Revision Notes

Example 1: Box on a Smooth Horizontal Surface

Problem

A box of mass 3 kg is pushed along a smooth horizontal surface with a constant force of 10 N over a distance of 5 m. The box starts from rest. Find:

1. The work done on the box.
2. The final speed of the box.

Step 1: Use the work formula:

Since the force acts along the displacement ($\theta = 0^\circ, \cos 0^\circ = 1$):

Step 2: Use the work-energy principle:

Since $KE_\text{initial} = 0$ (the box starts from rest):

Step 3: Write $KE_\text{final}$ in terms of the final velocity $v$:

Step 4: Solve for $v$:

Final Answer:

3. Work done: 50 J
4. Final speed: 5.77 ms⁻¹

Example 2: Object Sliding Down an Inclined Plane

Problem

A block of mass 4 kg slides 6 m down a smooth incline of 30°, starting from rest. Find:

5. The work done by gravity.
6. The block's speed at the bottom of the incline.

Step 1: Find the component of gravitational force along the incline:

Step 2: Calculate the work done:

Step 3: Use the work-energy principle:

Since $KE_\text{initial} = 0$

Step 4: Write $KE_\text{final}$ in terms of $v$:

Step 5: Solve for $v$:

Final Answer:

Work done: 117.6 J

Final speed: 7.67 ms⁻¹

Common Mistakes

1. Forgetting to resolve forces: For inclined planes, resolve gravity into components parallel and perpendicular to the incline.
2. Mixing units: Ensure consistent units, especially for force ($N$), displacement ($m$), and work ($j$).
3. Ignoring the angle $\theta$: Work depends on the cosine of the angle between force and displacement.
4. Misapplying work-energy principle: Use only the net work done by all forces, not just a single force.
5. Omitting starting conditions: Account for initial velocity in $KE_\text{initial}$

Key Formulas

1. Work Done by a Constant Force:

2. Kinetic Energy:

3. Work-Energy Principle:

4. Gravitational Work on an Incline: