Hooke's Law states that an ideal spring exerts a restoring force proportional to its displacement from equilibrium, written F = -kx, where k is the spring constant and the negative sign means the force always points back toward equilibrium.
Hooke's Law is the model for how ideal springs push and pull. Stretch or compress a spring a distance x from its natural (equilibrium) length, and it fights back with a force F = -kx. The spring constant k measures stiffness in newtons per meter, so a bigger k means a stiffer spring. The negative sign is doing real physics work, not decoration. It tells you the force is a restoring force that always points opposite the displacement, back toward equilibrium.
In AP Physics C, Hooke's Law is your first example of a force that changes with position. That makes it the perfect playground for calculus. Because the force isn't constant, finding the work it does requires an integral, and that integral gives you elastic potential energy, U = ½kx². Going the other direction, the force is the negative derivative of potential energy, F = -dU/dx. Hooke's Law is where the force-energy relationship in Topic 3.2 stops being abstract and starts being computable.
Hooke's Law lives in Topic 3.2, Forces and Potential Energy, inside Unit 3 (Work, Energy, and Power). The big idea of that topic is the two-way calculus link between conservative forces and potential energy functions. Hooke's Law is the cleanest case. Integrate F = -kx and you get U = ½kx²; differentiate U = ½kx² and you recover F = -kx. If you can run that loop with a spring, you can run it with any conservative force the exam throws at you, including weird ones like F = -bx³.
It also matters far beyond Unit 3. Any system where the net force looks like F = -kx undergoes simple harmonic motion, which means Hooke's Law is the launching pad for the entire oscillations unit. When the exam asks you to prove a system exhibits SHM, the move is always to show the restoring force (or torque) is proportional to displacement, which is Hooke's Law in disguise.
Keep studying AP Physics C: Mechanics Unit 3
Elastic Potential Energy (Unit 3)
U = ½kx² is literally the integral of Hooke's Law. The work done compressing a spring gets stored as elastic PE, and the parabolic shape of the U(x) graph comes straight from the linear force. If you ever forget the energy formula, integrate F = kx yourself in ten seconds.
Restoring Force and Simple Harmonic Motion (Unit 7)
SHM happens whenever the net force has the form F = -kx. Combine Hooke's Law with Newton's second law and you get ma = -kx, a differential equation whose solution is sinusoidal motion with ω = √(k/m). Every SHM derivation on the exam starts by spotting a Hooke's Law-style force.
Work-Energy Theorem (Unit 3)
Because the spring force varies with position, computing its work forces you to integrate, W = ∫F dx, instead of using W = Fd. Spring problems are the classic test of whether you actually understand work as an integral rather than a multiplication.
Spring Constant (Unit 3)
k is the slope of the force-versus-displacement line, which is exactly how lab-based questions expect you to find it. Graph applied force against stretch, fit a line, and the slope is your experimental spring constant.
Hooke's Law shows up in three main costumes. First, energy problems in Unit 3, where a block compresses a spring and you track ½kx² through conservation of mechanical energy. Second, calculus checks, where a question gives you U(x) and asks for the force (take -dU/dx) or gives you a position-dependent force and asks for work or potential energy (integrate). Third, oscillations FRQs in Unit 7, where you're asked to show a system undergoes SHM by deriving the differential equation from F = -kx, then extracting the angular frequency. Lab-design questions also love springs. A standard setup has you hang masses, measure stretch, plot force versus displacement, and identify k as the slope. Watch for non-ideal springs too. The exam sometimes hands you F = -bx³ or similar to test whether you memorized formulas or understand the force-energy calculus, since ½kx² only works when the force is actually linear.
Hooke's Law is the force equation, F = -kx, linear in x. Elastic potential energy is the energy equation, U = ½kx², quadratic in x. Doubling the stretch doubles the force but quadruples the stored energy. They're connected by calculus (U is the negative integral of F), but mixing them up, like writing U = kx, is one of the most common point-losers on spring FRQs.
Hooke's Law, F = -kx, says an ideal spring's force is proportional to its displacement from equilibrium, with the negative sign showing the force always points back toward equilibrium.
The spring constant k is the stiffness of the spring in N/m, and experimentally it's the slope of a force-versus-displacement graph.
Integrating Hooke's Law gives elastic potential energy U = ½kx², and differentiating U gives back the force via F = -dU/dx.
Because the spring force varies with position, you must use W = ∫F dx to find work; W = Fd does not apply.
Any system with a net restoring force of the form F = -kx undergoes simple harmonic motion with ω = √(k/m), which makes Hooke's Law the foundation of Unit 7.
If a problem gives a nonlinear spring force like F = -bx³, Hooke's Law and U = ½kx² no longer apply, and you have to integrate the actual force.
Hooke's Law states that an ideal spring exerts a force proportional to its displacement from equilibrium, F = -kx, where k is the spring constant. It's covered in Topic 3.2 (Forces and Potential Energy) and is the starting point for elastic potential energy and simple harmonic motion.
The negative sign means the spring force always points opposite the displacement, back toward equilibrium. That's what makes it a restoring force, and it's the reason a mass on a spring oscillates instead of flying off.
No. F = -kx is the force and U = ½kx² is the stored energy. The energy formula comes from integrating the force over the stretch, so force is linear in x while energy is quadratic. Doubling the stretch quadruples the energy.
No, it's an idealization. Real springs follow it only within their elastic limit, and AP exam questions sometimes give nonlinear forces like F = -bx³ specifically to test whether you'll integrate the actual force instead of blindly using ½kx².
Setting F = -kx equal to ma gives the differential equation a = -(k/m)x, whose solution is sinusoidal motion with angular frequency ω = √(k/m). Any system whose restoring force matches the Hooke's Law form undergoes SHM, which is the core argument behind most Unit 7 FRQ derivations.
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