AP Physics C: Mechanics · Topic 2.8

Topic 2.8: Spring Forces

Q: Does AP Physics 1 cover springs in series and parallel?

No. AP Physics 1's Topic 2.8 has a single learning objective, 2.8.A, with three essential knowledge statements covering the ideal spring and Hooke's law. AP Physics C: Mechanics keeps that objective and adds 2.8.B, describe the equivalent spring constant of a combination of springs exerting forces on an object, together with three sub-statements giving the series rule, the parallel rule, and the fact that a series combination is softer than its softest member. That extra objective and its boundary statement are the entire difference between the two versions of the topic.

Q: How do you find the equivalent spring constant of springs in series?

Add the reciprocals. Essential knowledge 2.8.B.1.i of the AP Physics C: Mechanics framework states that the inverse of the equivalent spring constant of a set of springs in series is equal to the sum of the inverses of the individual spring constants. The reason is that springs joined end to end all carry the same force, so each stretches by that force divided by its own constant and the extensions add. Statement 2.8.B.1.ii gives the check: the result must be smaller than the smallest constituent spring constant. Two springs of 120 and 300 newtons per metre in series give 85.7 newtons per metre.

Q: Can AP Physics C ask about springs in both series and parallel?

No. The boundary statement under Topic 2.8 reads in full that AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel. So a pure series chain is examinable, a pure parallel bank is examinable, and a network mixing the two is not. AP Physics 1 prints no boundary statement on this topic, because it has no combination objective for one to bound.

Q: Are the spring combination formulas on the AP Physics C equation sheet?

No. Both are labelled derived equations in the framework, and the CED's Required Equations page explains that derived equations are provided for reference and guidance or to demonstrate the final results of derivations expected of students on the exam, and that not all equations in the framework appear on the sheet. The AP Physics C: Mechanics Table of Information prints only Hooke's law for this topic, the spring force equals negative k times the change in position. You are expected to be able to produce both combination rules from Hooke's law and an argument about whether the springs share a force or share a displacement.

Q: What does the minus sign in Hooke's law mean?

It records the direction, not a negative magnitude. Essential knowledge 2.8.A.3 of the AP Physics C: Mechanics framework says the force exerted on an object by a spring is always directed toward the equilibrium position of the object and spring system. Since the displacement in Hooke's law is measured from the relaxed length, the minus sign says the force points opposite to that displacement: stretch the spring and it pulls back, compress it and it pushes back. That is what makes it a restoring force, and it is the property that leads to simple harmonic motion in Unit 7.

Q: What makes a spring ideal in AP Physics C?

Two conditions, both in essential knowledge 2.8.A.1. An ideal spring has negligible mass and exerts a force that is proportional to the change in its length as measured from its relaxed length. The same statement defines a nonideal spring as one that either has nonnegligible mass or exerts a force that is not proportional to the change in its length. The AP Physics C: Mechanics exam conventions state that springs and strings are assumed to be ideal unless otherwise stated, so unless a question says otherwise you may use Hooke's law and ignore the spring's own mass.

Q: What happens to the spring constant if you cut a spring in half?

Each half has twice the original constant. Think of the whole spring as two identical halves in series: applying the series rule to two springs of constant k each gives an equivalent of k over two, which must equal the constant of the original whole spring. So each half has constant 2k. Physically, the half has half as much length to stretch under the same force, so it stretches half as far and is twice as stiff. The same reasoning generalises: cutting a spring into n equal pieces gives each piece a constant of n times the original.

Unit 2: Force and Translational Dynamics20-25% of the multiple-choice section

An ideal spring exerts a force proportional to its stretch or compression from its relaxed length, directed back toward equilibrium. AP Physics C adds an objective AP Physics 1 lacks: combining springs. Series makes them softer, parallel stiffer, and mixed arrangements are out of scope.

AP Physics: Unit 2 (topics 2.8 Spring Forces). AP Physics C: Mechanics Unit 2, Topic 2.8. TWO learning objectives, where AP Physics 1's version of this topic has only one. 2.8.A, describe the force exerted on an object by an ideal spring, supported by 2.8.A.1 (an ideal spring has negligible mass and exerts a force proportional to the change in its length as measured from its relaxed length; a nonideal spring either has nonnegligible mass or exerts a force that is not proportional to that change), 2.8.A.2 (Hooke's law, F_s = -k delta x) and 2.8.A.3 (the force is always directed toward the equilibrium position of the object and spring system). 2.8.B, describe the equivalent spring constant of a combination of springs exerting forces on an object, has NO counterpart in AP Physics 1. It carries 2.8.B.1 (a collection of springs may behave as a single spring with equivalent constant k_eq), 2.8.B.1.i (DERIVED: the inverse of the series equivalent constant is the sum of the inverses of the individual constants), 2.8.B.1.ii (the series equivalent is SMALLER than the smallest constituent spring constant) and 2.8.B.1.iii (DERIVED: the parallel equivalent is the sum of the individual constants). Boundary statement, in full, including the exception clause: AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel. AP Physics 1 prints no boundary statement on its version. Neither derived equation is printed on the AP Physics C: Mechanics equation sheet; only Hooke's law is. No calculus is involved in either objective. Suggested skills are 1.A, 2.B, 2.D and 3.C, which share nothing with AP Physics 1's 1.B, 2.A, 2.C and 3.A for the same topic; notably skill 2.A is NOT among the C course's suggested skills here. The exam-conventions box states that springs and strings are assumed to be ideal unless otherwise stated. Topic 2.8 does not appear in the CED's sample multiple-choice alignment table, is not among the seven objectives of sample free-response question 2, and has none of the unit's six optional sample instructional activities.

AP Physics C has two objectives here, AP Physics 1 has one

AP Physics 1's Topic 2.8 is a single learning objective, 2.8.A, with three essential-knowledge statements. AP Physics C: Mechanics keeps that objective unchanged and adds 2.8.B: describe the equivalent spring constant of a combination of springs exerting forces on an object.

That second objective is the whole of the difference, and it comes with the topic's only boundary statement.

Statement 2.8.B.1 sets it up: a collection of springs that exert forces on an object may behave as though they were a single spring with an equivalent spring constant $k_{\text{eq}}$ . Then three sub-statements.

2.8.B.1.i: the inverse of the equivalent spring constant of a set of springs in series is equal to the sum of the inverses of the individual spring constants. Derived equation

\frac{1}{k_{\text{eq, series}}} = \sum_i \frac{1}{k_i} = \frac{1}{k_1} + \frac{1}{k_2} + \dots

2.8.B.1.ii: the equivalent spring constant of a set of springs arranged in series is smaller than the smallest constituent spring constant.
2.8.B.1.iii: the equivalent spring constant of a set of springs arranged in parallel is the sum of the individual spring constants. Derived equation

k_{\text{eq, parallel}} = \sum_i k_i = k_1 + k_2 + \dots

Statement 2.8.B.1.ii is worth keeping because it is a check rather than a formula. Once you have computed a series combination, look at the answer: it must be smaller than every individual constant in the set. If it is not, you have used the parallel rule by mistake. Two springs of 120 N/m and 300 N/m in series come out at 85.7 N/m, below the 120 N/m of the softer one, which is exactly what 2.8.B.1.ii promises.

No calculus appears in either objective. What appears is a structure AP Physics 1 never asks about, and a boundary statement that says precisely how far it goes.

The boundary statement, quoted whole

Topic 2.8 prints one boundary statement, under objective 2.8.B:

"AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel."

So: pure series, in scope. Pure parallel, in scope. A network with some springs in series and others in parallel, out of scope. AP Physics 1 prints no boundary statement on Topic 2.8, because it has no 2.8.B for one to bound.

That exception clause is the part that matters and the part most likely to be dropped in a summary. It rules out the mixed-network problems that circuit analysis makes routine in the electricity units, and it means you never need a reduction algorithm here. Every Topic 2.8 combination question is one application of one of the two derived equations.

Neither derived equation is printed on the AP Physics C: Mechanics equation sheet. What the sheet prints for this topic is Hooke's law and nothing else:

\vec{F}_s = -k\Delta\vec{x}

So you are expected to be able to produce both combination rules from Hooke's law and a physical argument about what is shared between the springs. That argument is short enough to be worth carrying rather than memorising, and the first worked example runs it.

What the CED requires of Topic 2.8

Two learning objectives, seven statements, one boundary statement. Suggested skills 1.A, 2.B, 2.D and 3.C.

Objective 2.8.A: describe the force exerted on an object by an ideal spring.

2.8.A.1: an ideal spring has negligible mass and exerts a force that is proportional to the change in its length as measured from its relaxed length. A nonideal spring either has nonnegligible mass or exerts a force that is not proportional to the change in its length as measured from its relaxed length.
2.8.A.2: the magnitude of the force exerted by an ideal spring on an object is given by Hooke's law, $\vec{F}_s = -k\Delta\vec{x}$ .
2.8.A.3: the force exerted on an object by a spring is always directed toward the equilibrium position of the object and spring system.

Objective 2.8.B: describe the equivalent spring constant of a combination of springs exerting forces on an object, with 2.8.B.1 and its three sub-statements, set out above.

Two details in 2.8.A repay attention.

The word ideal buys two things, and 2.8.A.1 names both. Negligible mass, and a force strictly proportional to the extension. A nonideal spring fails on either count. The exam-conventions box on the AP Physics C: Mechanics Table of Information makes the default explicit: "Springs and strings are assumed to be ideal unless otherwise stated."

The minus sign is about direction, and 2.8.A.3 says what direction. The force is always directed toward the equilibrium position of the object and spring system. So $\Delta\vec{x}$ is measured from the relaxed length, and the force points back at it. Stretch the spring and it pulls back; compress it and it pushes back. The sign is not a hint that the force is negative, it is a statement that the force and the displacement from equilibrium point opposite ways. That property is what makes it a restoring force and what makes Unit 7 possible.

Why the two combination rules come out the way they do

Both rules follow from one question: what do the springs share?

Series. Springs joined end to end, so the same force is transmitted through all of them. Each stretches by its own amount under that shared force, and the total extension is the sum:

\Delta x_{\text{total}} = \Delta x_1 + \Delta x_2 = \frac{F}{k_1} + \frac{F}{k_2} = F\left(\frac{1}{k_1} + \frac{1}{k_2}\right)

Since the combination behaves as a single spring, $\Delta x_{\text{total}} = F/k_{\text{eq}}$ , and dividing out $F$ gives 2.8.B.1.i. The physical reading is that adding a spring in series adds more length that can stretch, so the combination is softer. That is why 2.8.B.1.ii holds: adding any positive $1/k$ to the sum makes $k_{\text{eq}}$ smaller than any single term alone could.

Parallel. Springs side by side between the same two points, so they all stretch by the same amount and their forces add:

F_{\text{total}} = F_1 + F_2 = k_1\Delta x + k_2\Delta x = (k_1 + k_2)\Delta x

Comparing with $F_{\text{total}} = k_{\text{eq}}\Delta x$ gives 2.8.B.1.iii. Adding a spring in parallel adds another thing resisting the same stretch, so the combination is stiffer.

The pair of questions to ask on any diagram: is the force shared or is the displacement shared? Shared force means series and reciprocals; shared displacement means parallel and a plain sum.

One warning about the word parallel. It is about the mechanical arrangement, not about whether the springs look parallel on the page. Two springs attached to opposite sides of a block, one pulling and one pushing, both stretch or compress by the same displacement of the block, so they act in parallel and their constants add, even though they point in opposite directions. This case turns up constantly in Unit 7 oscillator problems.

A useful special case: $n$ identical springs of constant $k$ give $k/n$ in series and $nk$ in parallel. Cutting a spring in half gives two springs each of constant $2k$ , because the half has half the length to stretch under the same force. That last one is a favourite of question writers and follows directly from the series rule read backwards.

Traps this topic sets

Measuring $\Delta x$ from the wrong place. Statement 2.8.A.2 measures the change in length from the relaxed length, and 2.8.A.1 says the same. For a spring hanging vertically with a mass on it, the mass hangs at a new equilibrium where the spring is already stretched by $mg/k$ . Displacements for the oscillation are measured from that new equilibrium; extensions for Hooke's law are measured from the relaxed length. Keep the two straight by naming which zero you are using before you write an equation.

Thinking a hanging mass changes the spring constant. It does not. It moves the equilibrium position and leaves $k$ alone, which is exactly why the period of a vertical spring oscillator is the same as the horizontal one.

Using the kinematic equations on a spring. The force varies with position, so the acceleration is not constant, and $v^2 = v_0^2 + 2a(x - x_0)$ and its partners do not apply. Energy methods work, and the sheet prints $U_s = \frac{1}{2}k(\Delta x)^2$ for that purpose.

Reversing the two combination rules. Series is the reciprocal one, parallel is the plain sum. If you cannot remember which, derive it in ten seconds from what is shared, or use 2.8.B.1.ii as a check: a series answer must be smaller than the smallest individual constant.

Solving a mixed network anyway. The boundary statement says the course does not expect it. If a diagram looks like a mixed network, look again; you have probably misread which springs share a displacement.

Assuming Hooke's law always holds. Statement 2.8.A.1 defines a nonideal spring as one that either has nonnegligible mass or exerts a force that is not proportional to the change in its length. A question that describes a spring as nonideal, or gives you a force-extension graph that curves, has told you to stop using $F = k\Delta x$ . On a curved graph the local slope is still meaningful, but the single number $k$ is not.

The Hooke's law glossary entry, the spring constant glossary entry and the restoring force glossary entry cover the vocabulary.

If you want the algebra-based version of this topic

Objective 2.8.A is shared word for word, so if you want Hooke's law and the meaning of the minus sign at algebra-based pace, AP Physics 1 Topic 2.8: Spring Forces and Hooke's Law is the page for you. If you are taking AP Physics C: Mechanics, objective 2.8.B is only here, and so is the boundary statement that scopes it.

	AP Physics 1 Topic 2.8	AP Physics C Topic 2.8
Objectives	2.8.A only	2.8.A and 2.8.B
Statements	3	7
Hooke's law	$\vec{F}_s = -k\Delta\vec{x}$	identical
Springs in series	not in the course	2.8.B.1.i, derived equation
Springs in parallel	not in the course	2.8.B.1.iii, derived equation
Mixed networks	not in the course	explicitly excluded
Boundary statement	none	one, on mixed arrangements
Suggested skills	1.B, 2.A, 2.C, 3.A	1.A, 2.B, 2.D, 3.C
Calculus involved	none	none

The suggested-skill lists share nothing at all here, which is unusual. AP Physics 1 lists 1.B and 3.A, the graphing and experimental-design pair that fits measuring a spring constant from a hanging-mass data set. AP Physics C lists 1.A, 2.B, 2.D and 3.C, which fits drawing the arrangement, computing an equivalent constant, and predicting how it changes when a spring is added.

Also relevant: Topic 2.6, where statement 2.6.E.3 produces a gravitational force of exactly this linear restoring form inside a uniform sphere, and the simple harmonic motion guide for where springs go next.

How Topic 2.8 is tested

Unit 2 carries 20 to 25 percent of the multiple-choice section across about 15 to 25 class periods, the highest minimum weighting of any unit in AP Physics C: Mechanics (only Unit 3, at 15 to 25 percent, reaches the same 25 percent ceiling). Topic 2.8 holds two of the unit's nineteen learning objectives, and two of its eleven derived equations.

The suggested skills are 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.B, calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws.

Skill 2.D is the one that fits objective 2.8.B most naturally, and it carries 10 to 15 percent of the multiple-choice section. A question that asks what happens to the period of a mass-spring oscillator when a second identical spring is added in parallel is a pure 2.D question: the constant doubles, and since $T_s = 2\pi\sqrt{m/k}$ is printed on the sheet, the period falls by a factor of $\sqrt{2}$ . No numbers required.

Skill 2.A, derive a symbolic expression, is not among Topic 2.8's suggested skills, even though the two combination rules are labelled derived equations. AP Physics 1 does list 2.A for its version of the topic. Read that as the C exam preferring to hand you the arrangement and ask for a value or a comparison, rather than asking you to produce the general rule from scratch. Being able to produce it anyway is still the safest preparation, because the rule is not printed.

Topic 2.8 does not appear in the CED's sample multiple-choice alignment table or among the seven objectives of sample free-response question 2, and none of the unit's six optional sample instructional activities is on it. That is not a reason to skip it: sample sets are small, and every learning objective is examinable.

Where this material goes next is Unit 7. The equation sheet prints $U_s = \frac{1}{2}k(\Delta x)^2$ for the energy and $T_s = 2\pi\sqrt{m/k}$ for the period of a mass-spring oscillator, and an equivalent constant from 2.8.B slots into either of them without further work. That is a common way a combination question is dressed up: the springs are the setup, and the period or the energy is the question.

The Unit 2 hub sets this topic against the other nine, and the simple harmonic motion practice set covers the surrounding material at algebra-based level.

Two springs, series and parallel, and the period each gives

Two ideal springs have constants $k_1 = 120$ N/m and $k_2 = 300$ N/m. A $1.5$ kg mass is hung from them, first with the springs in series and then with them in parallel. Derive each equivalent constant, find the extension in each case, and find the period of small oscillations. Use $g = 9.8\ \mathrm{m/s^2}$ .

Series first. Ask what is shared: the springs are end to end, so the same force $F$ passes through both, while the extensions add.
Each spring obeys Hooke's law separately, so $\Delta x_1 = F/k_1$ and $\Delta x_2 = F/k_2$ , and the total is $\Delta x = F(1/k_1 + 1/k_2)$ . Setting that equal to $F/k_{\text{eq}}$ gives 2.8.B.1.i, $\dfrac{1}{k_{\text{eq}}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$ .
Compute: $\dfrac{1}{k_{\text{eq}}} = \dfrac{1}{120} + \dfrac{1}{300} = 0.008333 + 0.003333 = 0.011667$ , so $k_{\text{eq, series}} = 85.7$ N/m.
Check against 2.8.B.1.ii: the series result must be smaller than the smallest constituent constant. $85.7 < 120$ , so it passes. If you had got a number above 300 you would have used the parallel rule by mistake.
Parallel next. Now the springs are side by side between the same two points, so they share the displacement and their forces add: $F = k_1\Delta x + k_2\Delta x = (k_1 + k_2)\Delta x$ , giving 2.8.B.1.iii, $k_{\text{eq, parallel}} = 120 + 300 = 420$ N/m.
Extensions. The weight is $W = (1.5)(9.8) = 14.7$ N. In series, $\Delta x = 14.7/85.7 = 0.1715$ m. In parallel, $\Delta x = 14.7/420 = 0.0350$ m.
Check the series extension by adding the individual ones, which is where the rule came from: $14.7/120 = 0.1225$ m and $14.7/300 = 0.0490$ m, and $0.1225 + 0.0490 = 0.1715$ m. It matches.
Periods, using the printed line $T_s = 2\pi\sqrt{m/k}$ . In series: $T = 2\pi\sqrt{1.5/85.7} = 2\pi\sqrt{0.01750} = 2\pi(0.13229) = 0.831$ s.
In parallel: $T = 2\pi\sqrt{1.5/420} = 2\pi\sqrt{0.003571} = 2\pi(0.059761) = 0.375$ s.
Check the ratio without recomputing. Since $T \propto 1/\sqrt{k}$ , the ratio of periods should be $\sqrt{420/85.7} = \sqrt{4.90} = 2.214$ , and $0.831/0.375 = 2.214$ . That is the skill 2.D route to the same answer, and it needs no calculator.
One structural note. Every quantity here scales with the equivalent constant and nothing else, so once you have $k_{\text{eq}}$ the springs stop mattering. That is what 2.8.B.1 means when it says the collection may behave as though it were a single spring.

Series: $k_{\text{eq}} = 85.7$ N/m, extension $0.172$ m, period $0.831$ s. Parallel: $k_{\text{eq}} = 420$ N/m, extension $0.0350$ m, period $0.375$ s. The series combination is softer than either spring alone, as 2.8.B.1.ii requires.

Three springs in series, and the arrangement the CED excludes

Three ideal springs with constants $k_1 = 200$ N/m, $k_2 = 200$ N/m and $k_3 = 400$ N/m are joined end to end. A force of $24$ N stretches the chain. Find the equivalent constant, the total extension and the extension of each spring. Then say what happens if the third spring is instead placed in parallel with the first two.

Identify the arrangement. End to end means series, so the same 24 N passes through all three and the extensions add.
Apply 2.8.B.1.i to three springs: $\dfrac{1}{k_{\text{eq}}} = \dfrac{1}{200} + \dfrac{1}{200} + \dfrac{1}{400} = 0.005 + 0.005 + 0.0025 = 0.0125$ , so $k_{\text{eq}} = 80$ N/m.
Check against 2.8.B.1.ii: $80 < 200$ , the smallest individual constant, so it passes.
Total extension: $\Delta x = F/k_{\text{eq}} = 24/80 = 0.30$ m.
Individual extensions, each from Hooke's law with the shared 24 N: $24/200 = 0.12$ m, $24/200 = 0.12$ m, and $24/400 = 0.06$ m.
Add them: $0.12 + 0.12 + 0.06 = 0.30$ m, matching the total. That agreement is the check worth doing, because it is the assumption the series rule is built on.
Read the distribution. The stiffest spring stretches least, and the extension of each is inversely proportional to its constant. The 400 N/m spring takes exactly half the extension of each 200 N/m spring, since it is twice as stiff and carries the same force.
Now the second part. Suppose the two 200 N/m springs stay in series with each other, and the 400 N/m spring is placed in parallel across that pair. The boundary statement for this topic reads that AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel.
So that configuration is out of scope for the exam. There is nothing wrong with the physics, and the answer would come from combining the 200 N/m pair in series to 100 N/m and then adding 400 N/m in parallel to get 500 N/m, but the CED says you will not be asked for it. Recognising the arrangement and knowing it is excluded is what the boundary statement is testing.
What is in scope is the pure case: all three in parallel gives $200 + 200 + 400 = 800$ N/m, ten times the series value of 80 N/m. Same three springs, two orders of arrangement, a factor of ten in stiffness.

In series, $k_{\text{eq}} = 80$ N/m, total extension $0.30$ m, with individual extensions $0.12$ m, $0.12$ m and $0.06$ m. All three in parallel would give $800$ N/m. A mixed series and parallel arrangement is explicitly excluded by the Topic 2.8 boundary statement.

Frequently asked questions

Does AP Physics 1 cover springs in series and parallel?

No. AP Physics 1's Topic 2.8 has a single learning objective, 2.8.A, with three essential knowledge statements covering the ideal spring and Hooke's law. AP Physics C: Mechanics keeps that objective and adds 2.8.B, describe the equivalent spring constant of a combination of springs exerting forces on an object, together with three sub-statements giving the series rule, the parallel rule, and the fact that a series combination is softer than its softest member. That extra objective and its boundary statement are the entire difference between the two versions of the topic.

How do you find the equivalent spring constant of springs in series?

Add the reciprocals. Essential knowledge 2.8.B.1.i of the AP Physics C: Mechanics framework states that the inverse of the equivalent spring constant of a set of springs in series is equal to the sum of the inverses of the individual spring constants. The reason is that springs joined end to end all carry the same force, so each stretches by that force divided by its own constant and the extensions add. Statement 2.8.B.1.ii gives the check: the result must be smaller than the smallest constituent spring constant. Two springs of 120 and 300 newtons per metre in series give 85.7 newtons per metre.

Can AP Physics C ask about springs in both series and parallel?

No. The boundary statement under Topic 2.8 reads in full that AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel. So a pure series chain is examinable, a pure parallel bank is examinable, and a network mixing the two is not. AP Physics 1 prints no boundary statement on this topic, because it has no combination objective for one to bound.

Are the spring combination formulas on the AP Physics C equation sheet?

No. Both are labelled derived equations in the framework, and the CED's Required Equations page explains that derived equations are provided for reference and guidance or to demonstrate the final results of derivations expected of students on the exam, and that not all equations in the framework appear on the sheet. The AP Physics C: Mechanics Table of Information prints only Hooke's law for this topic, the spring force equals negative k times the change in position. You are expected to be able to produce both combination rules from Hooke's law and an argument about whether the springs share a force or share a displacement.

What does the minus sign in Hooke's law mean?

It records the direction, not a negative magnitude. Essential knowledge 2.8.A.3 of the AP Physics C: Mechanics framework says the force exerted on an object by a spring is always directed toward the equilibrium position of the object and spring system. Since the displacement in Hooke's law is measured from the relaxed length, the minus sign says the force points opposite to that displacement: stretch the spring and it pulls back, compress it and it pushes back. That is what makes it a restoring force, and it is the property that leads to simple harmonic motion in Unit 7.

What makes a spring ideal in AP Physics C?

Two conditions, both in essential knowledge 2.8.A.1. An ideal spring has negligible mass and exerts a force that is proportional to the change in its length as measured from its relaxed length. The same statement defines a nonideal spring as one that either has nonnegligible mass or exerts a force that is not proportional to the change in its length. The AP Physics C: Mechanics exam conventions state that springs and strings are assumed to be ideal unless otherwise stated, so unless a question says otherwise you may use Hooke's law and ignore the spring's own mass.

What happens to the spring constant if you cut a spring in half?

Each half has twice the original constant. Think of the whole spring as two identical halves in series: applying the series rule to two springs of constant k each gives an equivalent of k over two, which must equal the constant of the original whole spring. So each half has constant 2k. Physically, the half has half as much length to stretch under the same force, so it stretches half as far and is twice as stiff. The same reasoning generalises: cutting a spring into n equal pieces gives each piece a constant of n times the original.