Mechanics & Motion

From the swing of a pendulum to the flight of a rubber band, mechanics is all around us. It is the physics of motion and force — the foundation of how things move, why they fall, and what keeps them in place. In this chapter, we’ll explore motion through smartphone sensors like the accelerometer and gyroscope, investigate Newton’s laws with simple setups, and experiment with concepts like friction, momentum, and oscillation. Many of the experiments here require little more than your smartphone and curiosity, but they open the door to understanding the principles that govern everything from the smallest falling pebble to the largest planetary orbit.

Mechanics is the branch of physics that studies how things move and the forces that cause those movements. From falling apples to orbiting moons, it’s the foundation of nearly every physical phenomenon. What makes it especially suitable for exploration with smartphones is that motion and force leave measurable traces: vibrations, accelerations, positions, velocities — all of which modern devices are equipped to record.

Free Fall & Acceleration

Measuring Acceleration with Free Fall (MECH-01)

Sensors Used: Accelerometer
What’s Measured: Vertical acceleration during free fall

Description
Drop your smartphone gently onto a soft surface — or better yet, let it fall inside asafe enclosuresuch as a padded box or sling. Use a sensor app to recordaccelerometer dataduring the fall. If done correctly, you’ll see a short period where the phone registers near-zero acceleration — a brief moment oftrue free fall.

This simple experiment vividly demonstrates the concept ofapparent weightlessness, just like what astronauts experience in orbit.

Measuring Gravitational Acceleration with Video and Multiflash Photography (MECH-02)

Sensors Used: Camera (video), flashlight (strobe mode)
What’s Measured: Acceleration over time, time of fall

Description
This experiment demonstrates how to measure the acceleration due to gravity (g) using two different visual techniques. In both methods, a falling object is recorded, and its motion is analyzed to extract position, time, and acceleration data.

Method 1 - High-Speed Video Analysis: Drop a small object (like a ball) in front of a smartphone camera recording at high frame rate. Use frame-by-frame analysis to measure the object’s vertical displacement over time. Plotting position vs. time allows for determination ofgusing the equation of motion.

Method 2 - Multiflash (Strobe) Photography: Use a smartphone app with a flashing light or an external strobe while a second camera captures a long-exposure photo of the object’s fall. Each flash illuminates the object’s position at fixed intervals, creating a sequence of bright spots. The increasing spacing between flashes visually shows constant acceleration.Students can measure distances between dots and apply kinematic equations to calculateg.

References: [1] “Multiflash photographs of free fall,”http://practicalphysics.org/multiflash-photographs-free-fall.html

Inclined Plane (MECH-03)

Sensors Used: Accelerometer
What’s Measured: Acceleration as a function of incline angle; acceleration profiles during motion

Description
Roll your phone (or a toy car with the phone on it) down a smooth inclined plane. Record the acceleration and calculate how it changes with the steepness of the slope. This experiment makes it easy to explore trigonometry in physics — usinga=gsin⁡(θ)to relate angle and acceleration.

Alternative: Have an inclined plane. Have a toy car. Put phone on toy car and measure acceleration. Or place your phone on a skateboard, rolling toy, or a simple cart, and push it gently on flat ground and on a slight incline. Use the accelerometer to measure acceleration profiles and compare distances covered over time. This lets students analyze friction, net forces, and kinetic energy without fancy lab setups.

References: [1] “Investigating of Motion on a Sloping Surface,” http://practicalphysics.org/investigating-motion-sloping-surface.html

Jumping, Walking, Running, and Biking Acceleration (MECH-04)

Sensors Used: Accelerometer
What’s Measured: Acceleration patterns during jumping, walking, running, and biking; airtime; step frequency; stride dynamics

Description
Hold the phone while jumping vertically (on a soft surface). The accelerometer will show a spike during takeoff, a flat section near zero during midair (free fall), and a strong jolt on landing. You can measure the airtime and use it to estimate the jump height — a cool combination of biomechanics, motion, and energy. Or hold your phone or wear it while walking, running, or biking. The accelerometer will show regular pulses — a direct measurement of the biomechanics of motion. You can estimate stride length, step frequency, or even design your own DIY pedometer.

Acceleration in Elevators and Airplanes (MECH-05)

Sensors Used: Accelerometer
What’s Measured: Vertical acceleration during elevator motion; longitudinal acceleration during airplane takeoff, cruising, and landing; optional: jerk (rate of change of acceleration); vibration patterns and motion phases

Description
This experiment uses your phone’s accelerometer to explore how we experience and measure acceleration in common transportation systems. Whether riding an elevator or flying in an airplane, these motions offer hands-on insight into Newton’s laws, inertial frames, and perceived “weight” changes.

Part 1 - Elevator Acceleration:Take your smartphone into a modern elevator. Using a sensor app, log the vertical acceleration as the elevator starts, moves, and stops. You’ll see characteristic spikes that correspond to upward and downward acceleration. From this, you can estimate net forces and compare trip profiles between floors. Bonus: calculate jerk — the rate of change of acceleration — to measure “smoothness.”

Part 2 - Airplane Ride Acceleration:Record data during takeoff, cruising, and landing of a commercial flight. Acceleration patterns show a clear buildup during takeoff, minimal change during cruising, and deceleration on landing. Background vibrations and turbulence may also be visible in the signal — a real-world example of non-uniform motion.

Pendulums, Oscillations & Springs

Pendulum Period and Simple Harmonic Motion (MECH-06)

Sensors Used: Accelerometer, Gyroscope
What’s Measured: Oscillation period, angular motion, acceleration at key points in the swing; optional: estimate of gravitational acceleration (g)

Description
Make a pendulum using a string and your phone as the mass. Tie your phone securely to a string—or better yet, place the phone in a bag and tie the bag to a string. Gently swing it like a pendulum. Use the phone’s accelerometer or gyroscope to measure the oscillations and analyze the period. Investigate how the period depends on the length of the pendulum — and whether it depends on the mass.

From the measured period and the length of the pendulum, you can estimate the gravitational acceleration,gg.

Alternatively, measure the phone’s acceleration at the lowest and highest points of the swing. Observe the conversion between potential and kinetic energy during the motion.

References: [1] “Investigation of a Simple Pendulum,” http://practicalphysics.org/investigation-simple-pendulum.html

Mass-Spring Oscillator (MECH-07)

Sensors Used: Accelerometer, Camera (slow-motion video as an alternative)
What’s Measured: Oscillation period and frequency; vertical acceleration; moments of apparent weightlessness; estimation of gravitational acceleration (g)

Description
Attach your smartphone securely to a spring or elastic band and hang it vertically. Let it oscillate up and down like a mass-spring system, and record the motion using either the phone’saccelerometerorslow-motion video.

Use the collected data to study simple harmonic motion. Analyze theoscillation period and frequency, and explore how they depend on themassand thespring constant(or effective stiffness of the rubber band). At the top of each bounce, you may observe moments of apparent weightlessness, when the phone’s acceleration briefly approaches zero.

From the measured period and effective length of the system, you can also estimate thegravitational acceleration,gg, using equations of motion for oscillating systems.

References: [1] “Investigating a Mass-Spring Oscillator,” http://practicalphysics.org/investigating-mass-spring-oscillator.html

Rubber Bands in Series and Parallel: Measuring Effective Spring Constants with a Smartphone (MECH-08)

Sensors Used: Accelerometer, Camera (high-speed video), or Motion Sensor App
What’s Measured: Oscillation period; changes in effective spring constant across single, series, and parallel rubber band configurations

Description
This experiment explores how combining elastic elements — like rubber bands or springs — inseriesorparallelaffects the overall spring constant.

A smartphone is suspended from one or two rubber bands and set into vertical oscillation. Using a motion sensor app or high-speed video, students measure theperiod of oscillationfor different configurations: a single rubber band, two in series, and two in parallel.

By analyzing the period and applying the relationship

T=2πmkeff,T = 2\pi \sqrt{\frac{m}{k_{\text{eff}}}},

students can determine how the effective spring constantkeffk_{\text{eff}}relates to the individualkk-values. The results confirm that:

In series:1keff=1k1+1k2\frac{1}{k_{\text{eff}}} = \frac{1}{k_1} + \frac{1}{k_2}
In parallel:keff=k1+k2k_{\text{eff}} = k_1 + k_2

This provides a hands-on connection between harmonic motion and the mechanical properties of materials.

Oscillations & Mechanical Waves

Demonstrating Standing Waves on a Rubber Cord (MECH-09)

Sensors Used: Camera for video analysis
What’s Measured: Standing wave patterns (node/antinode positions); resonant frequencies; relationship between tension, frequency, and wavelength

Description
In this experiment, students generatestanding wavesby attaching one end of a rubber cord to amechanical vibrator, while keeping the other end fixed or under tension. At specific driving frequencies, the cord resonates and forms distinctnodesandantinodes, creating a vivid visual of wave interference.

This demonstration highlights the relationship betweenwave speed,frequency, andwavelength, and offers a direct way to observe howresonanceoccurs in mechanical systems. By adjusting the tension or frequency, students can explore different standing wave modes and better understand the underlying physics of oscillatory motion.

References: [1] “Standing waves on a rubber cord,”http://practicalphysics.org/Standing-waves-rubber-cord.html

Visualizing Mechanical Waves with a Spring (MECH-10)

Sensors Used: Camera (slow-motion or high-speed video)
What’s Measured: Wave speed, reflection behavior, type of wave (transverse vs. longitudinal), and effects of tension, amplitude, and length on wave properties

Description
This experiment introduces bothtransverseandlongitudinal wavesusing a stretched spring or slinky. It allows students to directly compare how these two types of mechanical waves propagate and interact with boundaries.

To observetransverse waves, stretch the spring along a smooth floor or tabletop. Displace one end sideways and release to create pulses or continuous waves that travel perpendicular to the spring’s length. Watch how the waves reflect at the ends, and look for interference patterns if multiple waves overlap.

To explorelongitudinal waves, compress and release one end of the spring along its axis. The coils will compress and expand in the direction of wave travel, demonstrating how particle motion isparallelto the wave direction — in contrast to the transverse case.

Record the motion using your smartphone inslow-motion or high-speed videomode. Playback frame-by-frame to measure wave speed, observe reflections, and identify energy transfer in each mode. Students can also vary thetension,amplitude, orlengthof the spring to investigate how these factors affect wave speed and wavelength.

This hands-on experiment clearly demonstrates the difference betweenwave types, while reinforcing key principles likewave velocity,medium dependence, andboundary behavior— all within a single apparatus.

References: [1] “Transverse waves on a spring,”http://practicalphysics.org/Transverse-waves-spring.html

Hooke’s Law & Material Properties

Stretching and Compressing Materials: Investigating Hooke’s Law (MECH-11)

Sensors Used: Camera (photo or video), Ruler app
What’s Measured: Extension (change in length) under applied force; force-extension relationship to determine spring constant and identify elastic vs. plastic behavior

Description
This classic mechanics experiment lets students explore how different materials respond to applied forces by stretching or compressing them. By suspending materials such as springs, rubber bands, or even strands of spaghetti, and applying measured weights, students can observe how length changes with force.

They then evaluate whether the deformation followsHooke’s Law, and investigate the concepts ofstress,strain,elastic limit, andspring constant. The experiment also highlights the distinction betweenelasticandplasticbehavior in materials.

A smartphone camera orruler appcan be used to capture extension data precisely. By plottingforce versus extension, students can extract slopes, identify linear regions, and determine material properties quantitatively.

References: [1] “Stretching and compressing materials,”http://practicalphysics.org/stretching-and-compressing-materials.html

Stretching Copper Wire: Observing Elastic and Plastic Deformation (MECH-12)

Sensors Used: Optional: Camera, Ruler app
What’s Measured: Change in wire length under increasing load; transition from elastic to plastic deformation; yield point and breaking threshold

Description
In this hands-on experiment, students investigate how a copper or silver wire responds to increasing tension, revealing bothelasticandplasticdeformation. As small weights are gradually added, the wire first stretches in a reversible, elastic way. Once it reaches itsyield point, the wire begins to deform permanently — showing plastic behavior — and eventually breaks.

This simple yet powerful setup provides a clear visual introduction to key concepts such asHooke’s Law,stress and strain,yield strength, and theelastic limit. It also opens the door to deeper topics likeductility,work hardening, andmaterial failure.

References: [1] “Stretching copper wire (qualitative),”http://practicalphysics.org/stretching-copper-wire-qualitative.html

Collisions & Momentum

Investigating 1D Collisions: Mass, Elasticity, and Conservation Laws (MECH-13)

Sensors Used: Camera (overhead video), Motion-tracking app
What’s Measured: Pre- and post-collision velocities; momentum transfer; energy conservation in elastic vs inelastic collisions; velocity distribution with varying masses

Description
This experiment explores the physics ofone-dimensional collisions, with two key variables:** themass of the colliding bodiesand thetype of collision(elastic vs inelastic). Students use low-friction carts on tracks or similar objects to model interactions and measure motion before and after impact.

Part 1 - Equal vs Unequal Masses (Elastic Collisions):Using carts of equal mass, launch one toward another at rest and observe the outcome. In an idealelastic collision, the moving cart should come to rest, while the second cart moves away with the same speed — a demonstration of bothmomentumandkinetic energy conservation. Repeat the experiment using unequal masses to observe how velocity distribution changes.

Part 2 - Elastic vs Inelastic Collisions (Magnet vs Putty):To reduce energy loss, mountrepelling magnetson the carts for a nearlyperfectly elasticinteraction without physical contact. Then replace the magnets withmodeling clay or putty, creating aninelastic collisionwhere the carts stick together. Compare how kinetic energy is conserved (or not), and analyze how momentum behaves in both cases.

Record each collision from above using your phone’s camera. Use video analysis or a motion-tracking app to extract velocity data and verify theconservation of momentumand the presence or absence ofenergy loss.

Investigating Two-Dimensional Collisions Using Coins (MECH-14)

Sensors Used: Camera (overhead video), Video analysis app
What’s Measured: Pre- and post-collision velocities and angles; momentum components in two dimensions; deflection angles; comparison of elastic vs inelastic outcomes

Description
In this experiment, students exploremomentum conservation and energy transferin two-dimensional collisions by sliding coins across a smooth, flat surface. One coin is propelled toward a stationary one, and the resulting angles and velocities after impact are observed and analyzed.

By recording the collisions with asmartphone camerafrom above, students can performframe-by-frame video analysisto measure speed and direction before and after the collision. This hands-on activity highlights thevector nature of momentum, and helps distinguish betweenelasticandinelastic collisions, while introducing concepts likeangle of deflectionandimpulse.

References: [1] “Collisions of coins,”http://practicalphysics.org/collisions-coins.html

Energy

Bouncing Ball Energy Loss (MECH-15)

Sensors Used: Camera (high-speed video)
What’s Measured: Bounce heights; energy loss per bounce; comparison of initial and rebound heights to quantify inelastic energy dissipation

Description
Drop a bouncy ball from a known height onto a hard surface and use either ahigh-speed videoto track its motion. Measure the height of each successive bounce to calculate how much energy is lost after each impact.

This experiment illustrates how energy is dissipated duringinelastic collisionsand provides a simple way to exploreenergy conservation,gravitational potential energy, andreal-world losses due to heat and deformation.

Measuring Projectile Motion and Energy Using a Rubber Gun and Smartphone Sensors (MECH-16)

Sensors Used: Camera (slow-motion video), Thermal camera (optional), Accelerometer (optional)
What’s Measured: Projectile velocity, flight time, impact energy (qualitative via thermal imaging), energy loss from inelastic collision, deviation from ideal projectile motion

Description
This experiment exploresprojectile dynamics and energy transferby launching a rubber pellet from a toy gun and tracking its motion with smartphone sensors. Usingslow-motion video, students can estimate the projectile’s initial velocity and flight time. If the phone supports it (e.g., theCAT S60), athermal cameracan capture the heat signature at the point of impact, offering qualitative insight into where and how energy is deposited.

Students can further measureacceleration, estimatekinetic energy, and evaluateenergy lossfrominelastic collisionsby comparing predicted and observed motion. For advanced analysis, air friction can be approximated by tracking deviations from ideal motion over distance.

References: [1] “Speed of a rifle pellet - momentum method,”http://practicalphysics.org/speed-rifle-pellet-momentum.html

Angular Momentum

Circular Motion with a Smartphone: Horizontal vs Vertical Rotation (MECH-17)

Sensors Used: Accelerometer, Gyroscope, Rotation Vector Sensor (optional), Camera (optional)
What’s Measured: Radial (centripetal) acceleration, angular velocity, effects of gravity on circular motion in different orientations (horizontal vs vertical)

Description
In this experiment, students investigatecentripetal acceleration and angular velocityusing a smartphone placed securely inside ashopping bag or pouch. By swinging the bag in a circular path, eitherhorizontally(parallel to the ground) orvertically(like a Ferris wheel), students can observe how orientation relative to gravity affects the measured motion.

The phone’saccelerometer and gyroscoperecord how radial acceleration and angular velocity change with speed and radius. Students can compare sensor readings from both orientations and relate them to the formula:a = v² / r

This hands-on activity offers a vivid demonstration ofnon-inertial reference frames.

Possible extensions for deeper analysis include using therotation vector sensorto track angular velocity more directly. Students can also film the motion from above or the side to correlate visual rotation with sensor data.

References: [1] “Whirling forces,” https://www.thenakedscientists.com/get-naked/experiments/whirling-forces

Measuring Centripetal Acceleration in a Salad Spinner (MECH-18)

Sensors Used: Accelerometer, Gyroscope, Camera (optional for external recording), FFT tools (optional)
What’s Measured: Centripetal acceleration, rotation speed, angular acceleration, and rotational frequency

Description
In this experiment, students investigatecentripetal accelerationusing a smartphone placed inside a salad spinner — a common kitchen device used to dry lettuce. As the spinner rotates, the phone’saccelerometer and gyroscoperecord how radial forces change with time.

Most salad spinners rotate between200-600 RPM(roughly 3-10 rotations per second), and a smartphone’s100 Hz sensor rateprovides enough resolution to track these changes. Students can extractrotation speed,angular acceleration, and observe howcentripetal accelerationdepends on radius and velocity using the formula:a = v² / r.

For advanced exploration, a second phone can be used to record the motion from the outside, or performFFT analysison the acceleration data to extract rotation frequency with higher precision.

References: [1] “An experimental test of F = mv² / r,”http://practicalphysics.org/experimental-test-f-mv%C2%B2r.html

Exploring Friction and Energy Dissipation with Euler’s Disk (MECH-19)

Sensors Used: Camera (optional, for slow-motion or time-lapse video)
What’s Measured: Duration of spin; precession rate over time; energy loss due to friction and air resistance; effect of mass and surface conditions on motion

Description
In this experiment, students analyze the motion of a spinning disk — known asEuler’s Disk— to investigatefrictionandenergy dissipation. When spun on a smooth, slightly concave surface, the disk both spins and rolls, with itsprecession rate increasingas it slows down. Just before stopping, it emits a rapid, rising whir — a striking illustration of energy loss in action.

This phenomenon helps reveal the effects ofrolling frictionandair resistanceon kinetic energy dissipation. By varying themass of the disk, theangle of the surface, or itssmoothness, students can explore how these factors influence the duration and dynamics of the motion.

References: [1] “Gravity-Defying Coin Takes 2 Minutes to Tip Over—Euler’s Disk,”https://www.youtube.com/watch?v=48BL2s3gdgc

Momentum of Inertia

Rolling Races: Moment of Inertia in Action (MECH-20)

Sensors Used: Camera (slow-motion video)
What’s Measured: Time to reach bottom; acceleration; comparative motion of objects with different moments of inertia; energy conversion between potential, translational, and rotational forms

Description
In this experiment, students investigate howmass distributionaffects rotational motion by racing different objects — such as afull and empty bottle— down an inclined ramp. Despite having the same shape, their differing mass distributions result in differentmoments of inertia, which influence how quickly they accelerate.

Usingslow-motion video, students can analyze which object reaches the bottom first and relate the outcome to differences in rotational inertia. The experiment demonstrates how torque and inertia affect rolling motion and helps connect real-world observations to the conservation of energy — including the conversion betweengravitational potential,translational, androtationalenergy.

References: [1] “Racing Jam Jars,”https://www.thenakedscientists.com/get-naked/experiments/racing-jam-jars

Measuring the Moment of Inertia of a Smartphone with a Torsion Pendulum (MECH-21)

Sensors Used: Gyroscope
What’s Measured: Oscillation period; changes in rotational inertia; moment of inertia (I) based on period and torsional constant; effect of mass distribution on rotational motion

Description
In this experiment, a smartphone is suspended horizontally from a thin wire, allowing it tooscillate in torsional motion. When gently twisted and released, the phone rotates back and forth about a vertical axis. Using the phone’sgyroscope, students can measure theoscillation period (T). With a known or estimatedtorsional constant (κ)of the suspension wire, themoment of inertia (I)can be calculated using the formula:** I = (κ · T²) / (4π²)

This experiment demonstrates how a smartphone can be used to measure a diagonal component of its owninertia tensor, linking classical mechanics to modern digital sensors.

As an extension, known masses (e.g. coins or washers) can be attached at varying distances from the center of rotation. Students observe how theperiod increaseswith greater rotational inertia, confirming the dependence of motion on mass distribution.

In a variation, the smartphone is suspended beneath a tray holding a fixed amount ofmodeling clayshaped into different geometries (e.g. sphere, rod, disc). Because the mass is constant, changes in oscillation period reflect differences inrotational inertiabetween shapes. This setup allows students to experimentally compare theoretical formulas for different mass distributions.

For an added challenge, students can attempt toreverse-engineer the shapeof an unknown clay distribution based on measured oscillation periods.

Demonstrating the Intermediate Axis Theorem Using a Smartphone (MECH-22)

Sensors Used: Gyroscope, Camera (high-speed video for visual analysis, optional)
What’s Measured: Rotational stability vs. instability across three principal axes; relative rotational behavior due to differing moments of inertia (I₁, I₂, I₃)

Description
In this experiment, students explore therotational stabilityof a rigid body — using a smartphone as a stand-in gyroscope. By tossing or spinning the smartphone gently (e.g., on a soft surface like a bed), students rotate it around each of its three principal axes one at a time. They will observe that rotation isstablearound two axes — the one with thesmallestand the one with thelargestmoment of inertia — butunstablearound the intermediate axis. This is a physical realization of theTennis Racket Theorem, where objects exhibit flipping or tumbling motion when spun around their “middle” axis. Students can analyze rotation using the smartphone’sgyroscope sensoror high-speed video, and compare the stability of each spin.

By spinning the smartphone about all 3 axes, you use the gyroscope data to observe stability/instability. It also tells you the relative values ofI1,I2,I3, but not their magnitudes.

Advanced Concepts

Demonstrating the Coriolis Effect with a Spinning Globe and Fluid Droplets (MECH-23)

Sensors Used: None required (visual demonstration); optional: Camera for recording motion
What’s Measured: Deflection of a fluid droplet due to rotation; qualitative demonstration of motion in a rotating reference frame (Coriolis effect)

Description
In this experiment, students use aspinning globeand adroplet of liquid(such as water or olive oil) to visualize theCoriolis effect— the apparent deflection of moving objects due to Earth’s rotation. A small droplet is placed near the top of the globe and allowed to flow downward as the globe rotates slowly.

Instead of moving in a straight path toward the equator, the droplet curves sideways, demonstrating how motion is deflected in arotating reference frame. This effect models the behavior of large-scale phenomena on Earth, such asatmospheric windsandocean currents.

Olive oilworks particularly well due to its higher viscosity, which slows the motion and makes the deflection easier to observe. This simple yet powerful demonstration provides an intuitive introduction to the physics of rotating systems and geophysical motion.

Building a Simple Balance to Measure Weight (MECH-24)

Sensors Used: Camera (optional), Angle-measurement app (optional)
What’s Measured: Mass comparison via torque balance; equilibrium angle; displacement from level to detect imbalance

Description
In this experiment, students build a basic balance to measure unknown weights bycomparing masses. A lightweight beam — such as a ruler or wooden strip — is suspended at its midpoint to act as a fulcrum. Small containers or trays are attached to each end to hold objects.

By placing aknown mass(e.g., coins, washers, or bolts) on one side and anunknown objecton the other, students adjust the balance untilequilibriumis reached. This allows them to determine the unknown mass throughdirect comparison.

The activity introduces core physics concepts such astorque,center of mass,rotational equilibrium, andmeasurement accuracy. For more precise analysis, students can use asmartphone cameraorangle-measurement appto detect slight imbalances or track tilt changes quantitatively.

References: [1] “Simple balance,”http://practicalphysics.org/simple-balance-1.html

Measuring Mass Using a Smartphone’s Internal Vibrator and Accelerometer (MECH-25)

Sensors Used: Accelerometer, Internal vibration motor
What’s Measured: Vibration amplitude as a function of added mass; indirect measurement of inertial mass through change in oscillation response

Description
In this experiment, students explore how adding mass to a vibrating smartphone changes its motion — providing an indirect method forestimating massbased on measured acceleration. The smartphone is placed on a flat surface and set to vibrate continuously (using a test app or alarm), while its internalaccelerometermeasures vibration amplitude. As different small masses of similar shape are placed on the smartphone, the system’s inertia increases,reducing the measured vibration amplitude. To ensure reliable contact and reduce slippage (due to the slick glass surface), a thinsheet of rubber or saran wrapis placed between the phone and the added mass. By calibrating with known weights, students can construct a simple curve linking vibration amplitude to mass, turning the phone into a primitivedigital mass sensor.

Bonus Insight: This experiment specifically measuresinertial mass— the property that resists acceleration under an applied force — not gravitational mass. It’s an elegant way to illustrate Newton’s second law in action.Students can be challenged to consider:What kind of setup would measure gravitational mass instead? And how could one test the equivalence of inertial and gravitational mass, asEötvösonce did?

Capstone / Research Extension

Measuring Gravitational Attraction with a Torsion Balance (MECH-26)

In this advanced project, students attempt to detect the gravitational attraction between masses using a modern reinterpretation of the historic 1798 Cavendish experiment. A lightweight horizontal beam, approximately 30 to 50 cm long, is suspended from the ceiling by a thin torsion wire—such as fishing line, fine copper, or tungsten. Equal 1 kg masses are attached to each end of the beam, forming a delicate balance that can twist freely about its axis.

To amplify the minuscule rotational motion, a small mirror is mounted at the center of the beam or on one of the test masses. A laser pointer aimed at the mirror reflects a beam onto a distant wall or screen, where even the slightest angular movement becomes visible as a shifting dot. Near one of the test masses, a 10 kg source mass is carefully positioned to induce a gravitational attraction. If the system is sufficiently isolated from disturbances, the gravitational force should cause the suspended apparatus to twist, displacing the laser dot over time.

Students can measure the angular deflection by tracking the movement of the laser spot or analyzing video recordings. By reversing the position of the source mass, they can test for directional consistency. If conditions are favorable and noise is minimized, students may even attempt to estimate the gravitational constant,G, based on observed displacement and the known properties of the system.

This experiment introduces the physics of Newtonian gravity at an unusually small scale. The torque produced by gravitational attraction, the restoring force of the torsion wire, and the resulting angular deflection are all connected through fundamental relationships:F = G·m₁·m₂ / r²τ = F · (L/2)θ = τ / κ

Here,Lis the length of the beam, andκis the torsional constant of the wire. The experiment brings these equations to life and offers a hands-on encounter with precision measurement, experimental design, and the challenges of detecting weak forces in the presence of environmental noise.

Beyond the physics, this project echoes the foundational techniques that underpin modern gravitational experiments—from torsion balances to LIGO. It encourages critical thinking, careful observation, and deep engagement with the experimental process.

Because the gravitational force involved is extremely weak, the apparatus must be protected from air currents, electrostatic effects, and mechanical vibration. Even small asymmetries in mass placement or beam construction can distort results. Students must also allow for long settling times—often several minutes to an hour—and work in a stable environment with minimal temperature and light fluctuations.

Historical Inquiry

Eötvös, Einstein, and the Equivalence Principle (MECH-27)

As students experiment with a torsion balance to explore gravitational attraction or compare inertial and gravitational mass, they’re retracing steps that have shaped the very foundations of modern physics. This is not just an experiment — it’s a doorway into one of the deepest ideas in science: theequivalence principle.

To connect this hands-on exploration with the history of physics, students can investigate how ideas evolved fromNewton’s law of gravitytoEinstein’s theory of General Relativity, stopping along the way to meet figures likeLoránd Eötvös, whose pioneering work in the late 19th century provided some of the most sensitive tests of gravitational and inertial mass ever conducted.

Now that you’ve explored the difference between gravitational and inertial mass, you can trace the historical path that connects your experiment to one of the most profound insights in physics. Use ChatGPT to guide your investigation. To begin, consider asking:

Who was Loránd Eötvös, and what was his contribution to physics?
What was the Eötvös experiment, and how did it test the equivalence of gravitational and inertial mass?
What is Einstein’s “happiest thought,” and how is it connected to the Eötvös experiment?
Why is the equivalence principle central to General Relativity?
What modern experiments continue to test the equivalence principle today?