Unit 7 · Topic 7.1 · about 30 minutes · 7 parts · 13 scored checks, 2 of them multiple choice

AP Physics 1: Algebra-Based 7.1: Defining Simple Harmonic Motion (SHM)

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Part 1 Think first A bag bobbing on a spring

Commit to a guess first. Being wrong now makes the right idea stick better.

Predict first

An ideal spring hangs from a hook, and with nothing on it the spring is 20 cm long. You hang a bag of sand from it, and once the bag settles the spring is 30 cm long. You pull the bag down until the spring is 34 cm long and let go, and the bag bobs up and down. Air resistance is negligible. How long is the spring when the bag is at the middle of its bobbing?

In short. Simple harmonic motion is back-and-forth motion about an equilibrium position, driven by a restoring force whose size is proportional to the object's displacement from that position.

An equilibrium position is a place where the net force on the object is zero, and a restoring force points opposite to the object's displacement from it.
SHM is a special case of periodic motion. Its restoring force is proportional to the displacement, so $m a_{x} = - k Δ x$ . A ball that bounces back to the same height every time repeats its motion, but it is not in SHM.
A pendulum swinging through a small angle can be modeled as SHM, because gravity's torque about the pivot is then very nearly proportional to the angular displacement.

Part 2 Learn it Four short ideas, each checked as you go

2a. Motion that repeats around an equilibrium position

A block on a level, frictionless table is joined to a wall by a spring. Pull it away from the wall and let go. It slides back, shoots past where the spring is relaxed, slows, stops and comes back, over and over. Motion that repeats itself in equal intervals of time is periodic motion. A swinging pendulum and a ball that bounces back to the same height every time are periodic too.

Back-and-forth motion like the block's happens around an equilibrium position: a place where the net force on the object is zero. For the block it is where the spring is relaxed. There the spring exerts no force, and the table's push up balances gravity, as in 2.8 Spring Forces.

Move the block away from equilibrium and the spring pushes or pulls it back. A force exerted in the direction opposite to an object's displacement from an equilibrium position is a restoring force. Pull the block right and the restoring force points left; push it left and the restoring force points right.

Zero net force at equilibrium does not make the block stop there. It arrives moving, and with no net force its velocity is not changing at that instant, so it carries on through. On the far side the restoring force points back again and slows it down.

Quick check · Decide and justify

A block on a level, frictionless table is joined to a wall by a horizontal ideal spring. It was pulled right and released, and it is now passing through the spot where the spring is relaxed, moving left. Right is positive. What is the net force on the block at that instant?

Which reason supports your answer?

2b. When the restoring force is proportional: SHM

An ideal spring's restoring force has a special property: its size is proportional to the displacement from equilibrium. Twice as far from equilibrium, twice the force. With $Δ x$ the block's displacement from equilibrium, Hooke's law from 2.8 Spring Forces gives the net force on the block on its level table, and Newton's second law turns that into an acceleration:

m a_{x} = - k Δ x

The minus sign says the force, and so the acceleration, always points back toward equilibrium. The graph in the diagram at the top of this page shows it: net force against displacement is a straight line through zero that slopes down, with slope $- k$ .

Motion driven by a restoring force like this is simple harmonic motion, or SHM. It is a special case of periodic motion. Every SHM repeats, but plenty of repeating motions are not SHM. Take the ball that bounces back to the same height every time. While it is in the air the only force on it is gravity, $m g$ down, the same size at every height. Then the floor gives it one short, hard push up. The motion repeats, but the force is not proportional to how far the ball is from any point, so it is periodic and not simple harmonic.

Quick check · Sketch the graph

A block on a level, frictionless table is joined to a horizontal ideal spring with spring constant 40 N/m, whose other end is fixed to a wall on the left. A meter stick lies beside it, its numbers increasing to the right, and right is positive. With the spring relaxed, the block rests at the 30 cm mark. Sketch the net force on the block against its position on the meter stick, from 20 cm to 40 cm.

Drag a point up or down, or tap it and use the arrow keys. The shape is checked, and the graph must pass through the values the question gives.

2c. A block hanging from a spring

Hang a block from a spring and its equilibrium position moves. At the spring's relaxed length the spring exerts no force, so gravity is unbalanced and the block cannot stay there. It rests lower, with the spring stretched a distance $d$ from its relaxed length, where the spring's pull up balances gravity: $k d = m g$ .

The spring's own force always comes from its stretch measured from the relaxed length, never from where the block rests. What changes is the net force. Take up as positive and let $y$ be the block's displacement from equilibrium, so $y$ is negative below it. The spring is stretched $d - y$ , so it pulls up with $k (d - y)$ . Take away gravity, $m g = k d$ , and the net force is $- k y$ . Pull the block 0.10 m below equilibrium, $y = - 0.10$ m, and the net force is $+ 0.10 k$ , up, back toward equilibrium. Lift it 0.10 m above instead and the net force is $- 0.10 k$ , down.

So the net force on a hanging block is $- k$ times its displacement from equilibrium, with the spring's own $k$ , and it moves in SHM about that lower point. Watch the stretch just above equilibrium: the spring is still stretched there, so it still pulls up, but with less than the block's weight, and the net force points down.

Quick check · Calculate

A spring with spring constant 12 N/m hangs from a hook. You hang a 0.30 kg block from its lower end. How far below the spring's relaxed length is the block's equilibrium position? Give your answer in centimeters.

Your answer cm

2d. A pendulum at small angles

A pendulum's bob hanging straight down at rest has zero net force on it, so straight down is its equilibrium position. Pull the bob aside an angle $θ$ and let go. The string pulls along itself, straight through the pivot, so it exerts no torque about the pivot. Gravity does. From 5.3 Torque, with $r = L$ , the string's length, gravity's torque on the pendulum has size $m g L \sin θ$ , and it turns the pendulum back toward straight down. A torque opposite to the angular displacement is a restoring torque.

For small angles, $\sin θ$ is very nearly equal to $θ$ in radians. At 10°, $θ$ is 0.1745 rad and $\sin θ$ is 0.1736. So the restoring torque is very nearly proportional to the angular displacement, and a pendulum swinging through a small angle can be modeled as SHM.

At large angles the proportion breaks down. At 60°, $θ$ is 1.05 rad but $\sin θ$ is only 0.87, so the torque grows more slowly than the angle. A pendulum swung that far still swings back and forth periodically, but it is not in SHM.

Quick check · Multiple choice

A pendulum's bob is pulled aside and released. Its motion can be modeled as simple harmonic when it is pulled aside a small angle, but not when it is pulled aside a large one. Which statement explains why?

Part 3 Watch it done A hanging block lifted and let go

Watch each step, then finish the problem yourself.

A 0.80 kg block hangs at rest from an ideal spring with spring constant 40 N/m. Someone lifts the block 0.050 m above that resting position, holds it still, and lets go. Find the block's acceleration just after it is released.

Step 1 · Find the equilibrium position
Take up as positive. At the equilibrium position the net force on the block is zero, so the spring's pull up balances gravity: $k d = m g$ , which gives $d = \frac{(0.80) (10)}{40} = 0.20$ m. The block rests 0.20 m below the spring's relaxed length.
Why this step?
Why is the equilibrium position not where the spring is relaxed?
At the relaxed length the spring exerts no force, so gravity, 8.0 N down, would be the only force on the block. The net force there is not zero, and an equilibrium position is where it is.
Step 2 · Find the spring's force at the release point
Lifted 0.050 m above equilibrium, the block is $0.20 - 0.050 = 0.15$ m below the relaxed length. The spring is still stretched, so it still pulls up, with $(40) (0.15) = 6.0$ N. The stretch from the relaxed length sets the spring's force, not the 0.050 m.
Step 3 · List every force on the block
Gravity pulls down with $(0.80) (10) = 8.0$ N, and the spring pulls up with 6.0 N. Nothing else touches the block once it is let go.
Your step · Calculate

Finish it. A 0.80 kg block hangs at rest from an ideal spring with spring constant 40 N/m. It is lifted 0.050 m above that resting position and released from rest. Take up as positive. What is the block's acceleration just after it is released?

Your answer m/s^2

Part 4 Try it A block between two springs

A new situation. A 2.0 kg block on a level, frictionless table sits between two walls. It is joined to the left wall by one ideal spring and to the right wall by another, each with spring constant 50 N/m, and both springs are relaxed when the block is at the center. Pull the block aside and both springs act on it at once. Draw the forces first, then decide what kind of motion this is.

Try it · Draw the forces

A 2.0 kg block on a level, frictionless table is joined to the left wall by one ideal spring and to the right wall by another, each with spring constant 50 N/m. Both springs are relaxed when the block is at the center. The block is sliding back and forth, and at one instant it is 0.10 m right of the center, moving right. Build the block's free-body diagram at that instant, and set each arrow's length to compare with the others.

The dot is the block.

Pick each force exerted on the block, then choose the way it points and how long its arrow is compared with the others. The arrows are drawn for you.

Try it · Decide and justify

A block on a level, frictionless table is joined to the left wall by one ideal spring and to the right wall by another, each with spring constant 50 N/m, and both springs are relaxed when the block is at the center. The block is pulled aside and released, and it slides back and forth. Is its motion simple harmonic?

Which reason supports your answer?

Part 5 Explain it Diagrams, a graph and an equation for one block

Nothing you type leaves this page. Check your answer against the model and each scoring point.

Free response is 50% of the exam score. This one is shaped like Question 2, Translation Between Representations.

Free-response practice · Question 2 style

A 0.50 kg block hangs from the lower end of a vertical ideal spring with spring constant 25 N/m, whose upper end is fixed to a ceiling. The block is pulled down 0.10 m below its equilibrium position and released, and it bobs up and down between 0.10 m below and 0.10 m above that position. Take up as positive, let $y$ be the block's displacement from its equilibrium position, and use $g = 10$ m/s².

(a) Calculate how far below the spring's relaxed length the equilibrium position is. Then draw a free-body diagram of the block at each of three positions: $y = - 0.10$ m, $y = 0$ and $y = + 0.10$ m. Label each force with what exerts it, and draw the arrows' lengths to compare across all three diagrams.

(b) On paper, sketch a graph of the net force on the block, $F_{net, y}$ , against $y$ from $- 0.10$ m to $+ 0.10$ m, using the values from your diagrams.

(c) Starting from the spring's force, $k$ times its stretch from the relaxed length, and the gravitational force, derive an expression for $F_{net, y}$ at any position $y$ , in terms of $k$ and $y$ only.

(d) Justify that your diagrams, your graph and your expression agree, and use them to explain whether the block's motion is simple harmonic.

Your answer stays on this page. It is never sent or saved.

Model answer

(a) At equilibrium the net force is zero, so $k d = m g$ and $d = \frac{(0.50) (10)}{25} = 0.20$ m below the relaxed length. Each diagram has two arrows from the dot: the gravitational force exerted by Earth, 5.0 N down, the same length in all three, and the force exerted by the spring, up in all three, because the spring is stretched at every position. At $y = - 0.10$ m the stretch is 0.30 m and the spring pulls up with $(25) (0.30) = 7.5$ N, longer than gravity's arrow. At $y = 0$ the stretch is 0.20 m and the pull is 5.0 N, the same length as gravity's. At $y = + 0.10$ m the stretch is 0.10 m and the pull is 2.5 N, shorter than gravity's.

(b) From the diagrams, the net force is $7.5 - 5.0 = + 2.5$ N at $y = - 0.10$ m, 0 at $y = 0$ and $2.5 - 5.0 = - 2.5$ N at $y = + 0.10$ m. The graph is a straight line through those three points: it passes through the origin and slopes down.

(c) With up positive, a block at $y$ is $d - y$ below the relaxed length, so the spring pulls up with $k (d - y)$ and gravity pulls down with $m g$ . Then $F_{net, y} = k (d - y) - m g = k d - k y - m g$ . From (a), $k d = m g$ , so $F_{net, y} = - k y$ .

(d) The expression gives $- (25) (- 0.10) = + 2.5$ N, 0 and $- (25) (0.10) = - 2.5$ N at the three positions, the same net forces the diagrams give as the difference between the spring's arrow and gravity's. It is a straight line through the origin with slope $- k = - 25$ N/m, which is the graph, and it says the line stays straight between the three points. The net force is proportional to the displacement from equilibrium and points opposite to it, so it is a restoring force of the kind SHM needs, and the block's motion is simple harmonic. At $y = + 0.10$ m the spring still pulls up, but the net force points down, toward equilibrium, as the expression says.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Calculates $d = \frac{m g}{k} = 0.20$ m, and draws three diagrams with gravity 5.0 N down in each and the spring's force up in each, longer than gravity at $y = - 0.10$ m, equal at $y = 0$ and shorter at $y = + 0.10$ m.
Sketches a straight line of net force against $y$ through $+ 2.5$ N at $y = - 0.10$ m, 0 at $y = 0$ and $- 2.5$ N at $y = + 0.10$ m.
Writes the spring's force from its stretch from the relaxed length, $k (d - y)$ , subtracts $m g$ , and uses $k d = m g$ to derive $F_{net, y} = - k y$ .
Justifies the agreement by checking the expression against each diagram's net force at all three positions, and its slope, $- k = - 25$ N/m, against the graph.
Concludes the motion is SHM because the net force is proportional to the displacement from equilibrium and opposite to it, and notes that at $y = + 0.10$ m the spring still pulls up while the net force points down.

Part 6 Check yourself Two gliders on an air track

Both case questions use the glider table.

Two gliders ride on a level air track, so friction is negligible, and right is positive. Glider P, of mass 0.50 kg, is joined to a spring. Glider Q has no spring; it bounces back and forth between two stiff rubber bumpers, which touch it when its center is 5 cm either side of the middle of its track, and between the bumpers nothing pushes it along the track. A student uses a force sensor to hold each glider still at several displacements of its center from the middle of its track. The sensor's force is equal and opposite to the net force from everything else, so the table gives the net force on the glider at each displacement the moment the sensor lets go.

Net force along the track at each displacement from the middle
Displacement (cm)	Glider P (N)	Glider Q (N)
-6	+1.2	+2.4
-4	+0.8	0.0
-2	+0.4	0.0
0	0.0	0.0
+2	-0.4	0.0
+4	-0.8	0.0
+6	-1.2	-2.4

Case question 1 · Calculate

Using the glider table, in which a student measured the net force along a level air track on glider P, of mass 0.50 kg, at several displacements from the middle of its track, with right positive: what is glider P's acceleration when it is 4 cm to the right of the middle? Include its sign.

Your answer m/s^2

Case question 2 · Multiple choice

Using the glider table, in which a student measured the net force along a level air track on glider P, joined to a spring, and on glider Q, bouncing between two rubber bumpers, at several displacements from the middle of each track: which glider's motion is simple harmonic?

The rest stand on their own and do not use the case.

Check yourself · Spot the error

A pendulum's bob hangs from a string tied to a fixed pivot. A student explains when the pendulum's motion can be modeled as simple harmonic. Tap the first line that goes wrong.

Check yourself · Fill the statement

Fill in the facts about simple harmonic motion. Tap a blank, then a word or phrase.

Every simple harmonic motion is motion, though a motion can repeat without being simple harmonic. In SHM, moving an object twice as far from equilibrium makes the restoring force as large. A ball that bounces back to the same height every time is not in SHM, because while it is in the air the net force on it .

Part 7 Keep it Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 2.8 Spring Forces · Spot the error

A block on a level, frictionless table is joined to a horizontal ideal spring with $k = 100$ N/m, whose left end is fixed to a wall. Right is positive, and $Δ x$ is how far the spring's end is from where it sits when the spring is relaxed. A student pushes the block left, compressing the spring 0.05 m, then lets go, and explains the spring's force. Tap the first line that goes wrong.

From 5.3 Torque · Calculate

A child pushes on the rim of a playground merry-go-round, 1.5 m from its center, with a force of 120 N. She stands outside it and pushes in and across, so the angle between her push and the line from the center out to the point where she pushes is 140°. What is the size of the torque her push exerts about the center? Give your answer to the nearest 1 N·m.

Your answer N·m

In three lines

An equilibrium position is where the net force on an object is zero, and a restoring force points opposite to the object's displacement from it.
Simple harmonic motion is periodic motion whose restoring force is proportional to the displacement from equilibrium, $m a_{x} = - k Δ x$ . A hanging block on a spring is in SHM about the lower point where the spring's pull balances gravity.
A pendulum at a small angle is modeled as SHM because gravity's restoring torque is very nearly proportional to its angular displacement; at large angles it is periodic but not SHM.

Vocabulary

Periodic motion: Motion that repeats itself in equal intervals of time, such as a swinging pendulum or a block bouncing on a spring.
Simple harmonic motion: A special case of periodic motion in which the size of the restoring force on an object is proportional to its displacement from its equilibrium position.
Restoring force: A force exerted in a direction opposite to an object's displacement from an equilibrium position.
Equilibrium position: A location at which the net force exerted on an object or system is zero.
Restoring torque: A torque exerted in a direction opposite to an object's angular displacement from its equilibrium position, such as gravity's torque on a pendulum pulled aside.

Course alignment, for teachers

AP Physics 1: Algebra-Based topic 7.1, Unit 7: Oscillations.

Skill 1.C: Create qualitative sketches of graphs that represent features of a model or the behavior of a physical system.
Skill 2.B: Calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway.
Skill 3.B: Apply an appropriate law, definition, theoretical relationship, or model to make a claim.
Skill 3.C: Justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws.

This lesson maps all 5 essential knowledge statements of the topic to the part that teaches each one: 5 are checked by a scored question. The map is in the lesson file.