AP Physics C: Mechanics FRQ Room

Acceleration Calculation by Differentiating a Position Function

In an experiment, the position of an object was modeled by the function $$x(t)= 3*t^4 - 2*t^2 + t$$.

Analysis of a Velocity vs. Time Graph in a Lab Experiment

In an experiment on an air track, a cart's velocity was recorded and is depicted in the following gr

Analysis of Air Resistance on a Falling Object

An object falling under gravity experiences a net acceleration modeled by $$a(t)= 9.8 - 0.5*t$$ (m/s

Analyzing a Two-Dimensional Collision

Two objects undergo a collision in a plane. Object A (mass m_A) experiences a force during the colli

Analyzing Circular Motion: Speed and Acceleration

A particle moves with a constant speed of 15 m/s along a circular path of radius 10 m.

Analyzing Two-Dimensional Motion Using a High-Speed Camera

In an experiment on projectile motion, a high-speed camera was used to record the two-dimensional mo

Application of the Big Five Kinematic Equations

An experiment provides the following data for an object in motion: initial velocity $$u=5$$ m/s, tim

Block on an Inclined Plane: Kinematic Analysis

A block of mass m is released from rest at the top of a frictionless inclined plane of length L = 10

Conservation of Energy in Projectile Motion

A projectile is launched from the ground with an initial speed $$v_0 = 50$$ m/s at an angle of $$45°

Decoupling Horizontal and Vertical Motions in Projectile Motion

A projectile is launched from the ground, and its position is recorded over time. The following tabl

Determining Acceleration Due to Gravity from Free Fall

A student conducted an experiment by dropping a metal ball from a height of 45 m. A digital timer co

Determining Instantaneous Acceleration from a Displacement Graph

An experiment recorded the displacement of an object as a function of time using a high-precision se

Determining Launch Angle from Experimental Data

A projectile is launched with an unknown initial speed and angle. In an experiment, the total flight

Dynamic Cart on Air Track: Misinterpretation of Frictionless Environment

In an experiment, a dynamic cart was set in motion on an air track to study uniformly accelerated mo

Experimental Determination of g

In a free-fall experiment, a student fits the data to obtain the position function $$h(t)= 4.9*t^2$$

Experimental Evaluation of Vector Addition in Two Dimensions

In a laboratory, two displacement vectors are measured. The following table provides their magnitude

Free Fall from a Cliff with Calculus Insights

A rock is dropped from an 80-meter cliff. Assuming the only acceleration is due to gravity (with $$g

Free Fall Kinematics

A rock is dropped from the top of a 100-meter tall building (neglect air resistance).

FRQ 1: Constant Acceleration Experiment

A researcher is investigating the motion of a cart along a frictionless track. The cart, starting fr

FRQ 6: Motion with Non-Uniform Acceleration

An object experiences a time-dependent acceleration given by $$a(t) = 4*t - 2$$ (in m/s²) and an ini

FRQ 7: Effects of Air Resistance in Free Fall

A researcher is examining the motion of an object in free fall where air resistance is not negligibl

FRQ 18: Motion of a Robot – Programming and Modeling Its Movement

A robotics research laboratory programs a robot to move on a flat plane. The robot's position is mod

FRQ 19: Analysis of Motion on an Inclined Plane (MEDIUM)

An object slides down a frictionless inclined plane that makes an angle $$\theta$$ with the horizont

FRQ 19: Variable Acceleration – An Object Under Changing Forces

A researcher studies an object whose acceleration varies with time according to the function $$ a(t)

Impact Analysis: Collision Avoidance

Two vehicles are moving along a straight road. Vehicle A travels with constant velocity described by

Investigation of Constant Acceleration in a Car

In an experiment, a motion sensor was set up along a straight track to measure the displacement of a

Kinematic Analysis of a Cyclist

A cyclist starts from rest and accelerates uniformly at 1.5 m/s² for 10 seconds, then rides at a con

Motion Along a Curved Track

A roller coaster car moves along a curved track. Its displacement along the track is given by $$s(t)

Piecewise Defined Acceleration

A particle moves along a frictionless surface with a piecewise defined acceleration given by: For $

Projectile Motion and Calculus Analysis

A ball is thrown from a platform. Its horizontal and vertical positions are given by $$x(t)=20*t$$ a

Projectile Motion Revisited: Maximum Height and Impact Velocity

An object is projected vertically upward from ground level with an initial speed of $$50\,m/s$$. Ass

Projectile Motion using Calculus

A projectile is launched from ground level at an angle of 30° with an initial speed of 40 m/s (negle

Projectile Motion: Maximum Height and Range

A projectile is launched from ground level with an initial speed of $$v_0 = 60$$ m/s at an angle of

Relative Motion: Meeting of Two Objects

Two objects move along a straight line. Object A starts at x = 0 m with a constant velocity of 10 m/

Simple Harmonic Motion in a Spring-Mass System

Design an experiment to investigate simple harmonic motion (SHM) using a spring-mass system. Describ

Simultaneous Measurement of Velocity and Acceleration

In an experiment, separate sensors were used to simultaneously measure both the velocity and acceler

Terminal Velocity in Free Fall

Design an experiment to determine the terminal velocity of an object in free fall within a fluid med

Variable Acceleration Analysis

An object experiences variable acceleration described by $$a(t) = 2*t$$ (in m/s²).

Vector Addition in Two-Dimensional Projectile Motion

Design an experiment to test the principles of vector addition by analyzing the two-dimensional moti

Comparative Analysis of Constant and Variable Force Work

Two experiments are performed on a 3 kg object. In Experiment 1, the object is moved under a constan

Comparative Analysis of Constant vs. Variable Gravitational Work

An object of mass $$m= 10 \;\text{kg}$$ is lifted from the ground (assume $$x=0$$) to a height of $$

Compound Machine Energy Analysis Experiment

A compound machine consisting of a lever and a pulley is used to lift a load, and the work input is

Determining Maximum Height using Energy Conservation

A researcher launches a ball of mass 0.06 kg vertically upward with an initial speed of 50 m/s in a

Elastic Potential Energy in a Spring System

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.25 m from its equilibrium pos

Energy Conservation in a Pendulum

A simple pendulum has a length $$L = 2 \text{ m}$$ and is released from rest at an initial angle of

Energy Loss in a Damped Pendulum

A pendulum with a length of 1.5 m and a bob of 0.8 kg experiences damping such that its amplitude de

Energy Transformation in a Roller Coaster

A roller coaster car of mass $$m = 500 \;\text{kg}$$ starts from rest at a height $$H = 50 \;\text{m

Evaluating Work Done on an Object in Rotational Motion

A researcher examines the work done on a rotating disc by a variable torque. The applied torque is d

Experiment on Electric Motor Power Output

Design an experiment to measure the power output of an electric motor used in a small robotic car.

Free‐Fall Impact Energy Experiment

In this experiment, a small cart is dropped from a known height and allowed to free-fall until it im

FRQ 8: Pendulum Energy Transformations with Damping

An experimental study on a pendulum claims that its mechanical energy is conserved, assuming only gr

FRQ 10: Conservation of Energy in a Pendulum Experiment

A simple pendulum with a length of 2.0 m is released from an angle of 30° with respect to the vertic

FRQ 11: Analysis of a Bungee Jump – Variable Net Force

A bungee jumper with a mass of 70 kg experiences a net force that changes as a function of vertical

FRQ 12: Analysis of Cyclist Power Output During a Ride

A cyclist’s performance is monitored by a power meter during a ride. The measured power output over

FRQ 12: Quantifying the Work Done by Friction

An experimental report claims that the negative work done by friction is constant regardless of the

FRQ 19: Equilibrium Points from a Nonlinear Potential Energy Function

A report presents the nonlinear potential energy function $$U(x) = (x - 2)^2 - (2 * x - 3)^3$$ and c

Gravitational Potential Energy Conversion

A 10 kg object is dropped from a height of 15 m in a vacuum. Using conservation of energy methods, a

Impulse and Work in a Collision

A 1 kg particle is subject to a time-varying force during a collision given by $$F(t)=100-20\,t$$ (N

Nonlinear Spring Energy Experiment

In an experiment with a nonlinear spring, a mass is attached and the force is measured as a function

Numerical Integration of Work in a Variable Force Field

A researcher studies the work done on a particle moving along the x-axis under the influence of a va

Oscillatory Motion Energy Exchange Experiment

A mass attached to a vertical spring oscillates up and down, and a sensor records its displacement a

Potential Energy Curve Analysis

A particle moves in a one-dimensional potential given by $$U(x) = (x-2)^2 - (2x-3)^3$$. A graph of t

Potential Energy Curve Analysis

An object has a potential energy given by $$ U(x) = (x-2)^2 - (2*x-3)^3 $$, where U is in joules and

Power and Energy in High-Speed Systems: Rocket Launch Analysis

A researcher is analyzing the power requirements of a rocket engine during a launch phase. A rocket

Projectile Launch: Energy and Air Resistance Considerations

A 0.2 kg projectile is launched vertically upward in a vacuum with an initial speed of 30 m/s.

Rotational Kinetic Energy in a Rolling Object

A solid sphere of mass 3 kg and radius 0.2 m rolls without slipping down an incline from a height of

Solar Energy Mechanical Conversion Experiment

In a lab setup, a solar panel powers an electric motor that lifts a 10 kg mass vertically by 3 m at

Variable Mass Rocket Energy Calculation

A rocket burns fuel at a constant rate so that its mass decreases with time according to $$m(t)= M_0

Work-Energy Analysis in Collisions

Two blocks with masses 2 kg and 3 kg undergo collisions under different circumstances. In an elastic

Work-Energy Theorem in a Variable Force Field

A particle of mass 2 kg is acted on by a position-dependent force given by $$F(x) = 10 - 2*x$$ N, wh

Work–Energy Experiment in Varying Potential Fields

A researcher studies the motion of an object of mass 4 kg in a potential energy field described by $

Work–Energy Theorem Verification in Projectile Motion

A projectile of mass 0.2 kg is launched horizontally from a platform 4 m high. High-speed cameras me

Angular Momentum Change in a Disc–Rod Collision Experiment

In a rotational collision experiment, a spinning disc collides with a stationary rod. Motion sensors

Angular Momentum Transfer in a Collision

A 0.5 kg ball moving horizontally at 10 m/s strikes a stationary 0.3 kg rod (length 2 m) pivoted at

Balancing a Composite System's Center of Mass

A thin uniform rod of length $$3$$ m (mass $$1$$ kg) has two point masses attached to it: a $$2$$ kg

Calculating Center of Mass Acceleration

A system of mass 10 kg experiences a net external force described by $$F(t)=20+5*t$$ N, where t is i

Calculus-Based Analysis of a Variable Density Disk

A thin disk of radius $$R = 0.5$$ m has a surface mass density that varies with radius as $$\sigma(r

Center of Gravity vs. Center of Mass in a Non-Uniform Rod

A vertical rod of length 1.0 m has a linear density given by $$\lambda(x)=5+2*x$$ (kg/m), where x is

Center of Gravity vs. Center of Mass in a Tilted Rod

A uniform rod is supported on one end by a fulcrum. In an experiment, additional weights were placed

Center of Mass in a Coupled Mass-Spring System

Two masses (m1 = 1.5 kg and m2 = 1.5 kg) are connected by a light spring on a frictionless track. In

Center of Mass of a Composite Three-Dimensional Object

A uniform cube (mass $$4$$ kg, side length $$0.5$$ m) and a uniform sphere (mass $$2$$ kg, radius $$

Center of Mass of a Lamina with Nonuniform Density

A thin, triangular lamina has vertices at (0,0), (4,0), and (0,3). Its surface mass density is given

Center of Mass of a System of Particles in 3D Space

Three point masses are located in 3D space with the following properties: - Mass 1: 2 kg at coordin

Center of Mass of an Irregular Lamina in Polar Coordinates

Consider a lamina occupying the region defined in polar coordinates by $$0 \le r \le 2(1+\cos(θ))$$

Composite Object: Rod with Attached Sphere

A uniform rod of length 1 m and mass 2 kg has a small sphere of mass 1 kg attached to its right end.

Determination of Collision Time from Impulse Data

In a crash-test experiment, the force on a car during impact is modeled by the equation $$F(t) = 100

Evaluating Energy Dissipation in an Inelastic Collision

Two vehicles collide and stick together in an inelastic collision. The experimental data below provi

Experimental Design: COM Independence in Collisions

Design an experiment to test the hypothesis that "the motion of the center of mass (COM) of a system

Explosive Separation in a Multi‐Stage Rocket

A multi‐stage rocket undergoes an explosive separation. Experimentally, Stage 1 (mass = 2000 kg) is

Explosive Separation of Particle System

A 10 kg object at rest explodes into three fragments with masses 3 kg, 4 kg, and 3 kg. After the exp

FRQ 14: Derivation of the Continuous Center of Mass Formula

Consider a one-dimensional object with a continuous mass distribution described by the density funct

FRQ 18: Critical Evaluation: Inelastic Collision Study

A published study on vehicle collisions claims that experimental momentum measurements in inelastic

Glancing Collision of Billiard Balls

Two billiard balls, each of mass 0.17 kg, undergo an elastic glancing collision. Initially, Ball 1 m

Impulse Analysis with Error Bars

In an experiment, a variable force acting on a cart is measured and described by the equation $$F(t)

Impulse and Momentum Change for a Hockey Puck

A 0.1 kg hockey puck initially has a momentum of 0.5 kg·m/s. It then receives an impulse that increa

Impulse and Momentum under a Variable Force

A 5 kg cart on a frictionless track is initially at rest. A variable force given by $$F(t)=10*t$$ (N

Impulse and Work: Discerning Differences

A particle of mass 3 kg is subjected to a force that varies with position according to $$F(x)=12*x^2

Impulse from a Time-Varying Force with Graph Stimulus

A force sensor records the force applied to a hockey puck as a function of time while a player strik

Impulse in a Variable Gravitational Field

An object of mass 2 kg is thrown vertically upward from the ground with an initial velocity of 50 m/

Impulse on a Rolling Soccer Ball with Piecewise Force

A soccer ball of mass $$0.43\,kg$$ is rolling and experiences friction that acts in two stages: a co

Impulse on Coupled Freight Cars

Two freight cars are coupled and moving on a frictionless track at an initial speed of $$10\,m/s$$.

Inelastic Collision: Two Blocks on a Frictionless Surface

Block A (mass = 5 kg, initial velocity = 2 m/s) and Block B (mass = 3 kg, initially at rest) collide

Meteor Impact: Conservation of Momentum and Energy Dissipation

A meteor with a mass of 5000 kg is traveling at 20 km/s (20000 m/s) and impacts the Earth, breaking

Momentum Analysis of a Variable Mass Rocket

A rocket in space, free from external gravitational forces, expels fuel continuously. Its mass is gi

Momentum and Angular Momentum in a Rotational Breakup

A rotating disk in space breaks apart into two fragments. Experimental measurements record both the

Multi-Stage Rocket Propulsion using Momentum Conservation

A rocket with an initial total mass of 1000 kg expels propellant at a constant exhaust velocity of $

Multiple Collisions in a Figure Skating Routine

In a choreographed figure skating routine, two skaters push off from each other. Skater A has a mass

Rocket Propulsion with Variable Mass

A rocket has an initial mass of $$M_0 = 50$$ kg (including fuel) and ejects fuel such that its mass

Rolling Cylinder on an Incline

A uniform solid cylinder (mass = 4 kg, radius = 0.5 m) rolls without slipping down a 30° incline. An

Rotational Dynamics Using Center of Mass

A uniform rod of length $$L=2.0\;m$$ and mass M rotates about a frictionless pivot at one end. (a)

Satellite Debris: Center of Mass and Impulse Effects

In Earth orbit, three pieces of debris are observed. Their properties are recorded in the following

Stability of a Suspended Mobile

A suspended mobile consists of three masses hanging from strings attached to a horizontal bar. The m

Two-Dimensional Collision and Momentum Conservation

Two ice skaters push off each other on a frictionless surface. Skater A (mass $$60\,kg$$) moves with

Two-Dimensional Collision of Ice Skaters

Two ice skaters initially at rest push off from each other on frictionless ice. Skater A (50 kg) mov

Variable Density Rod: Mass and Center of Mass Calculation

A thin rod of length $$L = 2$$ m has a linear density given by $$\lambda(x) = 2 + 3 * x$$ (kg/m), wh

Angular Kinematics of a Rotating Disk

Consider a rotating disk for which you want to measure angular displacement, angular velocity, and a

Angular Momentum Conservation in a Spinning System

Design an experiment to verify the conservation of angular momentum using a rotating platform and mo

Assessment of Rotational Kinematics Equations

Experimental data for a rotating disk include measurements of angular displacement, angular velocity

Calculus-Based Analysis of Angular Impulse

In rotational dynamics, angular impulse is defined as the integral of torque over a time interval, w

Correlation Between Torque and Rotational Energy via Calculus

A student designs an experiment to investigate the relationship between applied torque and rotationa

Coupled Rotational Dynamics of Two Disks

Two disks with different masses and radii are mounted on a common, frictionless axle. Disk A has mas

Cylinder Rolling on an Incline

A solid cylinder of mass M rolls without slipping down an inclined plane of height h and length L (w

Derivation of the Moment of Inertia for a Thin Rod

A uniform thin rod of length $$L$$ and mass $$M$$ rotates about an axis perpendicular to the rod thr

Driven Rotational Pendulum with Variable Torque

A rotational pendulum is subject to a driving torque given by $$\tau(\theta) = \tau_0 \sin(\theta)$$

Energy Conservation in Combined Rotational and Translational Motion

A sphere is made to roll down an incline without slipping, converting gravitational potential energy

Energy Conservation in Rotational Motion

A hollow sphere of mass $$m = 5\,kg$$ and radius $$R = 0.3\,m$$ rolls without slipping down an incli

Experimental Measurement of Rotational Inertia Using Oscillations

A researcher is designing an experiment to measure the moment of inertia of various objects using an

FRQ 1: Torque Analysis on a Wrench

A mechanic uses a wrench of length L = 0.30 m to loosen a bolt. The mechanic applies a force of F =

FRQ 9: Experimental Determination of Moment of Inertia

A student performs an experiment to determine the moment of inertia of a uniform disk by measuring i

FRQ 13: Dynamics of a Variable Torque System

A rod with moment of inertia \(I = 4.00\,kg\cdot m^2\) is subjected to an angular-dependent torque g

FRQ 16: Composite Rotational Inertia via Integration

A thin rod of length L = 3.00 m has a linear mass density that varies along its length according to

Graphical Analysis of Angular Motion

A rotating system has been monitored and the angular velocity (in rad/s) recorded over time (in seco

Impact of Mass Distribution on Rotational Kinetic Energy

This experiment investigates how different mass distributions affect the rotational kinetic energy o

Impact of Off-Center Mass in Rotational Dynamics

A student attaches a small mass to a rotating disk at a point away from the center to study its effe

Investigating Non-uniform Density Effects on Moment of Inertia

Design an experiment to determine the moment of inertia for an object with a non-uniform mass distri

Investigating the Big Five Equations for Rotational Motion

A researcher is verifying the 'Big Five' equations of rotational kinematics under constant angular a

Parallel Axis Theorem in Compound Systems

A composite system consists of a uniform disk of mass $$M$$ and radius $$R$$ and a point mass $$m$$

Relation Between Linear and Angular Velocity on a Rotating Disk

In an experiment, a rotating disk is used to measure the linear speed of points located at different

Rolling Cylinder Down an Incline

A solid cylinder rolls without slipping down an incline. A set of measurements were made at differen

Rolling Motion Dynamics Down an Incline

A solid cylinder with mass $$M = 3\,kg$$ and radius $$R = 0.2\,m$$ rolls without slipping down an in

Rolling Motion Energy Analysis on an Inclined Plane

A cylinder is allowed to roll down an inclined plane without slipping, converting gravitational pote

Rolling Motion with Transition from Slipping to Pure Rolling

A solid sphere of mass m and radius R is initially sliding and rolling down an inclined plane with a

Rotational Dynamics of a Gyroscope

A gyroscope with a moment of inertia of $$I = 0.05\,kg\,m^2$$ is spinning with a spin angular veloci

Rotational Energy Distribution in a Compound System

A compound system consists of a uniform disk (mass M and radius R) and an attached thin rod (mass m

Rotational Impulse and Change in Angular Momentum

A flywheel initially at rest receives a constant torque impulse over a brief time interval.

Rotational Inertia of a Composite Bead System

A researcher is investigating the effect of discrete mass distribution on the rotational inertia of

Rotational Kinematics from Angular Velocity Graph

A rotating object's angular velocity increases linearly with time. The graph provided shows that $$\

Rotational Kinetic Energy Storage in a Flywheel

An engineer is tasked with designing a flywheel energy storage system. The flywheel is required to s

Time-Resolved Analysis of Angular Acceleration

A wheel's angular velocity is given by $$\omega(t)= 3t^2 + 2t\,rad/s$$. Calculate its angular accele

Time-varying Angular Acceleration in a Rotational System

A disk experiences an angular acceleration described by $$\alpha(t)=5\sin(2t) \text{ rad/s}^2$$.

Torque and Rotational Inertia in Engine Mechanisms

You are exploring the behavior of torque in a small engine. Design an experiment to measure the engi

Torque and Rotational Inertia: Uniform Rod

A uniform rod of length $$L = 2.0 \text{ m}$$ is pivoted about one end. A force of $$F = 10 \text{ N

Verification of the Parallel Axis Theorem

A solid disk of mass M = 4 kg and radius R = 0.5 m is considered. The moment of inertia about its ce

Analysis of SHM Under Driving Force

A researcher studies a damped, driven harmonic oscillator subjected to an external sinusoidal force

Calculus Approach to Maximum Velocity in SHM

Consider an oscillator whose displacement is given by the sinusoidal function $$y(t) = A \sin(\omega

Comparative Analysis: Energy Methods vs. Force Methods in SHM

In analyzing simple harmonic motion (SHM), two common approaches are the energy conservation method

Comparison of Horizontal and Vertical Oscillations

Compare a mass-spring oscillator on a frictionless horizontal surface with a vertical spring-block s

Coupled Oscillators Investigation

A researcher investigates two masses, $$m_1$$ and $$m_2$$, connected in series by two identical spri

Coupled Oscillators: Two Springs in Parallel

A block is attached to two springs arranged in parallel with spring constants $$k_1 = 250\,\text{N/m

Damped Oscillation: Logarithmic Decrement Analysis

A spring-mass oscillator exhibits damped oscillations, and the amplitudes of successive cycles are r

Damped Oscillations: Determining the Damping Coefficient

A mass-spring system oscillates vertically but in a medium that exerts a damping force proportional

Data Analysis of Oscillatory Motion with Damping Effects

A student report claims that 'damped oscillations follow the exact same simple harmonic law as undam

Derivation of SHM Equations Using Calculus

Starting with Newton’s second law and Hooke’s law for a mass-spring system, derive the differential

Determination of Spring Constant from Oscillation Data

A researcher collects oscillation data for different masses attached to a spring. The data is summar

Determining Phase Shift in Sinusoidal SHM

A simple harmonic oscillator follows the equation $$y = A * \sin(\omega * t + \phi_0)$$ with amplitu

Differentiation in SHM: Velocity and Acceleration

The position of an oscillator is given by the function $$y(t)=0.05 * \sin(10*t+0.3)$$ (with $$y$$ in

Driven Oscillations and Resonant Response

Consider a mass-spring system that is subjected to a periodic driving force given by $$F(t)=F_0*\cos

Elastic Potential Energy and Maximum Speed Calculation

A block of mass $$m = 0.10\,\text{kg}$$ is attached to a spring with a spring constant of $$k = 200\

Energy Analysis and Instantaneous Power in SHM

A block of mass $$m = 0.1\,kg$$ is attached to a spring with force constant $$k = 800\,N/m$$ and osc

Energy Loss Analysis in a Spring Oscillator

In a laboratory experiment, the amplitude of a mass-spring oscillator is observed to decrease expone

Energy Transformations in a Spring Oscillator

A block attached to a horizontal spring oscillates with amplitude $$A$$. Consider the energy transfo

Equation of Motion for a Simple Pendulum Beyond Small Angle Approximation

A researcher examines the motion of a simple pendulum without relying on the small-angle approximati

Error Analysis in SHM Measurements

A student conducting an experiment on a mass-spring oscillator records the following period measurem

Estimating Spring Constant from Kinetic Energy Measurements

A spring-mass oscillator's maximum kinetic energy is measured during its motion. Using energy conser

Evaluating Damped Oscillatory Motion Effects

A mass-spring oscillator with mass $$m = 0.5 \; kg$$, spring constant $$k = 100 \; N/m$$, and dampin

Evaluating Experimental Uncertainties in SHM Measurements

Accurate measurement of the oscillation period is crucial in SHM experiments. In this context, uncer

Experimental Determination of Spring Constant

In a lab experiment, students measure the displacement of a spring under various applied forces. The

FRQ 3: Determining Period and Frequency

An oscillating block moves from its position of maximum stretch to its maximum compression in $$0.25

FRQ 13: Determining Angular Frequency from Oscillation Data

An oscillator’s motion is described by $$y = A \sin(\omega t + \phi_0)$$. A set of position measurem

FRQ 16: Maximum Speed in SHM via Energy Methods

Experimental data for a mass–spring oscillator, including amplitude, mass, spring constant, and meas

FRQ 17: Determination of Damping Coefficient

An oscillatory system exhibits damping, with its amplitude decreasing over successive cycles. Analyz

FRQ 18: Pendulum Motion Beyond the Small Angle Approximation

A simple pendulum is tested at various amplitudes, including larger angles where the small angle app

FRQ2: Maximum Speed of a Spring Oscillator via Energy Conservation

Consider a mass attached to a spring oscillating on a frictionless surface. The spring has a force c

FRQ3: Kinematics of SHM – Period and Frequency

A block-spring oscillator is observed to take 0.4 s to travel from its maximum displacement in one d

FRQ16: Resonance in a Driven, Damped Oscillator

A damped oscillator is subjected to an external periodic driving force of the form $$F_d \cos(\omega

Graphical Analysis of Oscillatory Data

A researcher records and graphs the displacement of a mass-spring oscillator as a function of time.

Hooke's Law and Spring Force Calculation

Consider a spring that obeys Hooke’s law, $$F = -k * x$$, where k is the spring constant and x is th

Hooke's Law Force Calculation

A spring at its natural length is stretched by a displacement $$x$$. (a) Derive the expression that

Impact of Varying Spring Constants on Oscillatory Behavior

Two identical blocks of mass $$m = 0.2 \; kg$$ are attached to two different springs with spring con

Integral Calculation of Work Done in SHM

An experiment is devised to measure the work done on a spring during a complete compression-extensio

Lagrangian Mechanics of the Simple Harmonic Oscillator

A researcher employs Lagrangian mechanics to analyze a mass-spring oscillator. Consider a mass $$m$$

Mass Dependence in Oscillatory Motion

A spring with a force constant of $$k = 300 \; N/m$$ is used in two experiments. In the first, a blo

Momentum Transfer in a Spring-Mass Collision

A block of mass $$m = 0.25\,kg$$ moving at a speed of $$v = 2.0\,m/s$$ collides with the free end of

Nonlinear Pendulum Oscillations and Error Analysis

A pendulum with a length of $$L = 2.0\,m$$ is released from an initial angle of 45° (approximately 0

Pendulum Oscillations for Large Angles

For a simple pendulum with length \(L\) oscillating with a maximum angle \(\theta_{\text{max}}\) tha

Phase Shift Determination in SHM

A researcher studies a mass-spring oscillator and observes that at time $$t = 0$$ the block is at it

Resonance in Forced Oscillations

A researcher sets up a forced oscillation experiment with a mass-spring system subject to a sinusoid

Sinusoidal Analysis of SHM with Phase Shift

Examine a sinusoidally described simple harmonic oscillator with a phase shift.

Small-Angle Pendulum Experiment

In a physics lab, a small pendulum of length $$L = 0.80\,m$$ is used to study simple harmonic motion

Spring Force and Elastic Potential Energy

A spring with a force constant $$k = 300\,N/m$$ and a natural length of 0.50 m is stretched to a len

Spring Force and Energy Analysis

A spring with a force constant $$k = 350 \; N/m$$ and a natural (unstretched) length of 20 cm is str

Stress Testing of Oscillatory Limits

In an advanced experiment, a student increases the amplitude of oscillation for a spring–mass system

Time-Dependent Analysis of Oscillatory Motion

An oscillator's displacement is given by the function $$x(t)=0.03 * \cos(12*t)$$ (with $$x$$ in mete

Torsional Oscillator as a Rotational Analogy

A disk with a moment of inertia \(I=0.05\,\text{kg}\cdot\text{m}^2\) is suspended by a wire that pro

Vertical Oscillator in a Gravitational Field

A block of mass $$m = 2.0 \;\text{kg}$$ is attached to a vertical spring with force constant $$k = 4

Analysis of Low Earth Orbit Satellite Decay

A low Earth orbit (LEO) satellite experiences gradual orbital decay due to atmospheric drag. Analyze

Application of Kepler's Third Law

A planet orbits its star in an almost circular orbit with radius a. Use Kepler's Third Law to analyz

Application of Kepler's Third Law in the Solar System

A table below provides the semi-major axis and orbital period for several planets. Use this data to

Average Orbital Energy and Angular Momentum

For an orbiting satellite, the specific orbital energy and angular momentum are key conserved quanti

Barycenter Determination in a Sun-Planet Analog with Magnetic Models

A lab experiment simulates the Sun-Earth system using scaled models equipped with magnetic component

Calculus Analysis of Gravitational Potential Energy

Gravitational potential energy (GPE) plays a crucial role in systems influenced by gravity. Answer t

Calculus in Gravitational Work: Integration of Inverse Square Force

Using Newton's Law of Gravitation, the force on an object is given by $$F(r) = -\frac{G * M * m}{r^2

Center of Mass in the Sun-Earth System

Using the provided data for the Sun-Earth system, analyze the location of the barycenter. Use the ex

Center of Mass of the Sun-Earth System

Consider the Sun-Earth system where the mass of the Earth is $$m = 5.98 \times 10^{24} \text{ kg}$$,

Comparing Circular and Elliptical Orbits in a Lab Simulation

Different orbital shapes exhibit differing energy distributions. Answer the following: (a) A simula

Derivation and Calculation of Escape Velocity

A researcher is tasked with determining the escape velocity $$v_{esc}$$ from a planet using energy c

Derivation of Gravitational Potential Energy

Starting from Newton's law of gravitation given by $$F(r) = -G * \frac{M * m}{r^2}$$, derive the exp

Deriving Gravitational Force from Gravitational Potential Energy

In a region where the gravitational potential energy between two masses is given by $$U(r) = -\frac{

Designing a Modern Cavendish Experiment

A researcher designs an experiment modeled after the Cavendish torsion balance to determine the grav

Determination of Gravitational Constant Using a Compound Pendulum

A compound pendulum experiment is conducted to determine the gravitational constant (G) by measuring

Determination of Gravitational Parameter (GM) from Satellite Orbits

An observational study is undertaken to determine the gravitational parameter (GM) of a planet by an

Determining Planetary Mass from Satellite Orbital Data

Satellite orbital data can be used to determine the mass of the planet they orbit. Answer the follow

Elliptical Orbit Dynamics: Speed Variation Analysis

For a planet or satellite in an elliptical orbit, the speed varies along the orbit due to conservati

Elliptical Orbits and Eccentricity Calculation

An object is in an elliptical orbit around a star. (a) Define the eccentricity $$e$$ in relation to

Energy Conservation in Gravitational Fields: Application to Roller Coaster Dynamics

Although gravitational potential energy is most famously applied in celestial mechanics, the concept

Energy Conversion in a Gravitational Slingshot Maneuver

A spacecraft performing a gravitational slingshot maneuver around a planet converts gravitational po

Energy Conversion in an Asteroid's Elliptical Orbit

A computer simulation is conducted to study the energy conversion in an asteroid's elliptical orbit

Energy Dissipation in Orbital Decay

A satellite experiences a tangential drag force given by $$F_{drag} = -b * v$$, where $$b$$ is a con

Escape Velocity Derivation

A spacecraft of mass m is located on the surface of a planet with mass M and radius R. Using energy

FRQ 15: Gravitational Anomalies and Their Effects on Orbits

A satellite experiences a small perturbation in the gravitational potential due to a local mass anom

FRQ 17: Tidal Forces and Differential Gravity

An extended object in a gravitational field experiences differential gravitational forces (tidal for

Gravitational Potential Energy Differences in Multi-Body Systems

Consider a test mass moving in the gravitational field of two bodies, $$M_1$$ and $$M_2$$. Answer th

Gravitational Potential via Integration in a Varying Density Sphere

A computational experiment is conducted to calculate the gravitational potential inside a spherical

Gravitational Slingshot Maneuver

A spacecraft uses a gravitational slingshot maneuver to gain speed by passing near a planet. (a) Des

Gravitational Slingshot Maneuver Analysis

Analyze the gravitational slingshot maneuver used by spacecraft to gain speed during a flyby of a pl

Gravity Assist in Three-Body Dynamics

In a gravitational slingshot (gravity assist) maneuver, a spacecraft can change its velocity by inte

Impact of Relativistic Effects on Orbital Motion

Discuss how relativistic effects modify the orbital motion of a planet when it orbits close to a ver

Inferring Mass Distribution of a Galaxy through Orbital Dynamics

The rotation curves of galaxies can reveal information about their mass distribution and the possibl

Newton's Law in Binary Star Systems

Using the provided data for a binary star system, analyze Newton's Law of Gravitation to determine t

Optimization of Orbital Maneuvers in Multi-Stage Rockets

A multi-stage rocket performs orbital maneuvers to transition from a low Earth orbit to a higher geo

Orbital Decay due to Atmospheric Drag

A satellite in low Earth orbit experiences atmospheric drag, which can be modeled as a force $$F_d =

Orbital Decay Due to Atmospheric Drag

A satellite in low Earth orbit experiences atmospheric drag, resulting in a gradual decrease of its

Perturbation Analysis in Elliptical Orbits

An elliptical orbit is subject to a small, constant perturbing force that causes a slow change in th

Predicting Orbital Decay Due to Atmospheric Drag

A low Earth orbit satellite experiences orbital decay due to atmospheric drag. Assume that the drag

Role of Eccentricity in Orbital Dynamics

Orbital eccentricity (e) quantifies how much an orbit deviates from a circle. (a) Define orbital ec

Satellite Maneuver Simulation with Finite Burn Dynamics

An experimental simulation aims to predict a satellite's trajectory during a maneuver that involves

Speed Variation in Elliptical Orbits via Angular Momentum Conservation

In an elliptical orbit, a satellite’s speed varies along its path. Using the conservation of angular

Torsion Balance Gravitational Force Measurement

A research group performs an experiment using a torsion balance to measure the gravitational attract

Variation of Gravitational Force with Distance

Newton’s Law of Gravitation indicates that the gravitational force between two masses varies inverse

Work Done in a Variable Gravitational Field

An object of mass $$m$$ is moved radially from a distance $$r_1$$ to $$r_2$$ in the gravitational fi