Energy Graphs & Variable Forces
- Find work as the area under a force–displacement graph, including for non-constant forces
- Read a potential-energy curve to locate equilibrium points and classify their stability
- Use the fact that force is the negative slope of the potential-energy curve
Work is the area under an F–x graph
W = Fd cos θ only holds when the force is constant. When it is not, the general statement is that work is the area under the force–displacement graph. This is the same slope-and-area logic as the kinematics graphs, and it is how the exam asks about springs, rubber bands and any force given as a plot rather than a number. Area below the axis — a force opposing the displacement — counts as negative work.
The spring energy formula falls out of the graph
Hooke's law says the force needed to stretch a spring is F = kx, a straight line through the origin. Stretch it to displacement x and the area under that line is a triangle: ½ · base · height = ½ · x · kx = ½kx². That is the elastic potential energy. Seeing it as a triangle also explains why stretching a spring from 0 to 2 cm takes far less energy than stretching it from 2 cm to 4 cm — the second stretch sits under a taller part of the line.
Potential-energy curves: the shape tells you the motion
Plot U against position and you can read off the whole motion without solving anything. Draw a horizontal line at the total energy E. Wherever the curve is below that line, the difference E − U is the kinetic energy, so the object is moving. Wherever the curve meets the line, K = 0 and the object turns around — these are the turning points. Regions where U exceeds E are simply forbidden: the object can never get there.
Stable, unstable and neutral equilibrium
Where the U curve is flat, the slope is zero, so the force is zero — that is an equilibrium point. Its type depends on the curvature. A minimum (a valley) is stable: nudge the object either way and the force pushes it back, so it oscillates. A maximum (a hill) is unstable: any nudge and the force drives it further away. A flat stretch is neutral — displaced, the object simply stays put.
A force acting along the x-axis increases linearly from 0 N at x = 0 to 12 N at x = 4.0 m, then stays at 12 N until x = 7.0 m. How much work does it do over the full 7.0 m?
- 1.First segment is a triangle: area = ½ × 4.0 m × 12 N = 24 J.
- 2.Second segment is a rectangle: area = (7.0 − 4.0) m × 12 N = 36 J.
- 3.The force is along the motion throughout, so both areas are positive.
- 4.Total work = 24 J + 36 J.
On a potential-energy curve, the object is not physically rolling along the curve — the curve is a plot of energy against position, not a picture of a hill. It is a useful mental image because gravity makes real hills obey the same math, but do not read the vertical axis as height.
A spring with k = 200 N/m is stretched from its natural length to 0.10 m. How much elastic potential energy is stored?
On a potential-energy curve, an object sits at a local minimum of U. This equilibrium is:
A particle has total energy E = 10 J. At x = 3 m its potential energy is 4 J. What is its kinetic energy there?
When a graph question gives a force in newtons and a position axis in centimeters, convert before computing the area. Unit slips on graph-area questions are more common than conceptual errors, and the rubric does not distinguish between the two.
Answer the 3 checkpoints as you read.
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