AP Physics C: Mechanics FRQ Room

Calculating Displacement via Integration of a Velocity Function

An object moves in one dimension with its velocity described by $$v(t)=4*t-2$$ m/s. Determine the di

Calculus-Based Analysis of Varying Acceleration

An object moves with a velocity function given by $$v(t)=3*t^2 - 12*t + 5$$. A table below shows cal

Centripetal Acceleration in Circular Motion

Design an experiment to measure the centripetal acceleration of an object in circular motion and det

Comparing Theoretical and Experimental Data in Uniform Acceleration

An experiment measures the velocity of an object under uniform acceleration, and the following table

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°

Deriving Velocity and Acceleration from a Position Function

Consider an object moving along a straight line with its position given by $$x(t)=\sin(t)$$, where x

Designing a Trajectory for a Manufacturing Robot

A robot in a manufacturing plant moves along a straight track with a piecewise position function: Fo

Designing an Experiment: Motion on an Inclined Air Track

You are asked to design an experiment to determine the coefficient of kinetic friction on an incline

Determination of Acceleration Due to Gravity

A student drops a small metal ball from a 45 m high platform and records its height over time using

Determining Velocity from a Position Function with Differentiation Error

An experiment recorded the position of a particle moving along a straight line, modeled by the funct

Determining Zero Acceleration from a Non-linear Position Function

An object's position is given by the function $$x(t)=t^4-8*t^2$$. Determine at what time the object'

Distance vs. Displacement Analysis in One-Dimensional Motion

An object moves along a straight path and its motion is described by the velocity function $$v(t) =

Experimental Determination of g

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

Free-Fall Motion Analysis

A rock is dropped (from rest) from the top of an 80 m cliff. Assume that the acceleration due to gra

FRQ 3: Graphical Analysis of Velocity-Time Data

A researcher collects velocity vs. time data from an object undergoing several phases of motion: acc

FRQ 10: Comparative Analysis of Two Cars with Different Acceleration Profiles

A researcher compares the motion of two cars starting from rest. Car A accelerates at a constant rat

FRQ 13: Average Speed vs. Average Velocity Analysis (EASY)

An object's position along the x-axis is given by $$x(t)=t^2-4*t+3$$ (in m) for $$0 \le t \le 5\,s$$

FRQ 14: Analysis of a Parabolic Projectile Trajectory (MEDIUM)

The vertical motion of a projectile is described by the equation $$h(t)=-4.9*t^2+20*t+5$$ (in m). (a

FRQ 16: Integration of a Decaying Velocity Function (HARD)

An object has a velocity function given by $$v(t)=4*e^{-t}-2$$ (in m/s) for $$t\ge0$$, and its initi

FRQ 19: Comparative Kinematics – Two Launch Angles

Two objects are launched from the same point with the same initial speed of 40 m/s, but at different

FRQ 20: Evaluating Data Uncertainty from a Velocity-Time Graph (EXTREME)

A velocity vs. time graph obtained from an experiment includes error bars on each data point. The be

Graphical Analysis of Distance and Displacement

A student records the position of a runner during a 10-second race, resulting in the following table

Impact Analysis: Collision Avoidance

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

Impulse and Variable Force Analysis

Design an experiment to measure the impulse delivered to a ball by a bat, given that the force varie

Instantaneous vs. Average Velocity

An object’s position is given by the function $$x(t)=t^4 - 4*t^2$$ (x in meters).

Motion Along a Curved Track

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

Motion with Time-Varying Acceleration (Drag Force Approximation)

An object in free fall experiences a time-dependent acceleration due to air resistance approximated

Newton's Second Law and Force Measurement on a Cart

Design an experiment using a cart on a frictionless track with a pulley system to verify Newton's se

Optimization of Projectile Range

A projectile is launched from ground level with a fixed initial speed of 50 m/s. Use calculus to det

Parametric Trajectory Analysis

A particle moves in the plane with its position given by: $$x(t)=2t^2$$ and $$y(t)=20t - 4.9t^2$$, w

Piecewise Motion Analysis

An object moves along a straight line with acceleration defined piecewise as follows: for $$0 \le t

Polynomial Position Function Analysis

A particle’s position along the x-axis is given by the function $$x(t)=t^3 - 6*t^2 + 9*t$$ (in meter

Projectile Launch from an Elevated Platform

A ball is launched from a platform 10 meters above the ground with an initial speed of 30 m/s at an

Projectile Motion Analysis

An object is launched with an initial speed of $$60\,m/s$$ at an angle of $$30^\circ$$ above the hor

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 with Air Resistance

Design an experiment to study the effect of launch angle on the horizontal range of a projectile in

Projectile Motion: Maximum Height and Range

An object is launched from ground level at an angle of 30° above the horizontal with an initial spee

Relative Displacement in Different Frames

A particle moves along the x-axis and its position is described by $$x(t)=7*t-2*t^2$$ (in meters). T

Rotational Kinematics of a Spinning Disk

Design an experiment to measure the angular acceleration of a spinning disk using a photogate sensor

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

Time-Dependent Acceleration Analysis

A particle experiences a time-dependent acceleration given by $$a(t)=2t\,e^{-t}\,m/s^2$$. At time \(

Time-Dependent Force and Work-Energy Theorem

A particle of mass m moves along a straight line under a time-dependent force $$F(t)= 100\,e^{-t}$$

Uniformly Accelerated Motion With Non-Zero Initial Velocity

An object moves along a straight path with an initial velocity of $$u=5\,m/s$$ and a constant accele

Uniformly Accelerated Motion: Incorrect Baseline Velocity

A physics lab experiment tracked the displacement of a moving object using a laser sensor, measuring

Vector Addition in Two-Dimensional Motion

An object is launched with a horizontal velocity of $$30\,m/s$$ and a vertical velocity of $$40\,m/s

Vector Displacement and Total Distance

An object moves along a straight line in two phases. First, it moves 10 m to the east, then it moves

Analysis of a Potential Energy Curve

An object of mass 3 kg moves along the x-axis in a potential energy field given by $$U(x) = (x-1)(x-

Analysis of Potential Energy Curves

Consider the provided graph representing the potential energy function $$U(x)$$ for a diatomic molec

Calculating Work on an Inclined Plane with Variable Force

A 6 kg box is pushed up a frictionless incline that makes an angle of 30° with the horizontal. The a

Calculus-based Integration of Work over a Variable Force

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 5*x$$ N. The particle

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

Conservation of Energy in a Roller Coaster

A 500 kg roller coaster car is released from rest at the top of a hill 50 m above the bottom of a di

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 Collision and Energy Transfer

Two blocks, A (2 kg) and B (3 kg), slide without friction on a horizontal surface. Initially, block

Elastic Potential Energy and Block Dynamics

A 2 kg block is placed on a frictionless horizontal surface and compresses a spring by 0.1 m. The sp

Energy Analysis in Circular Motion

A 2 kg object moves in a horizontal circular path of radius 5 m. It is subject to a tangential force

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 Dissipation in an Oscillatory System

An oscillatory system with damping has its energy decay described by $$E(t) = E_0*e^{-\gamma*t}$$.

Energy Loss in a Ball with Air Resistance

A ball with a mass of 0.5 kg is thrown vertically upward with an initial speed of 20 m/s. During its

Energy Loss in an Inelastic Collision

A 2 kg object moving at 4 m/s collides and sticks to a 3 kg object initially at rest.

Evaluation of Elastic Potential Energy in a Spring-Mass System

A researcher studies energy storage in a spring–mass system. A spring with a spring constant $$k = 2

FRQ 3: Kinetic Energy Change in a Car's Acceleration

A 1200-kg car accelerates along a straight, level road. Measurements of the car's speed at various d

FRQ 7: Roller Coaster Energy Conversion and Energy Loss Analysis

A roller coaster car is reported to convert all its gravitational potential energy into kinetic ener

FRQ 8: Investigation of Variable Power Output in a Pulley System

A pulley system is used to tow a load with a constant force of 100 N. A sensor records the instantan

FRQ 9: Interpreting Drop Test Kinetic and Potential Energy Data

A study provides experimental data for a 3 kg ball dropped from various heights, with the measured s

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

Instantaneous and Average Power in a Variable Force System

A block is subjected to a variable force and its velocity varies with time. The force acting on the

Instantaneous and Average Power of a Rocket Engine

A rocket engine produces a time-dependent force given by $$F(t) = 1000 + 200 * t$$ (N) for t in the

Kinetic Energy Gain in a Roller Coaster Ride

A roller coaster car of mass 500 kg is released from rest at the top of a frictionless hill at a hei

Loop-the-Loop Roller Coaster: Work-Energy and Normal Force Analysis

A roller coaster car of mass 300 kg enters a vertical loop with a radius of 10 m.

Optimization of Work in a System with Resistive Force

A particle of mass 2 kg moves along the x-axis under a constant driving force of 10 N and a resistiv

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

Potential Energy Curve of a Diatomic Molecule

The interatomic potential energy of a diatomic molecule can be approximated by the function $$U(r) =

Power and Energy Efficiency in a Conveyor Belt Experiment

A researcher investigates the energy efficiency of a conveyor belt system in a manufacturing facilit

Power Output in a Variable Force Scenario

A force acting on an object causes work to be done such that the work as a function of time is given

Power Output in Elevator Lifting

A motor lifts an elevator of mass 500 kg at a constant speed of 2.5 m/s over a vertical displacement

Relationship Between Force and Potential Energy

For a conservative force, the relationship between force and potential energy is given by $$F(x) = -

Rolling Motion on an Incline: Combined Energy Analysis

A solid sphere with mass 3 kg and radius 0.2 m rolls without slipping down an inclined plane of heig

Rotational Energy Transfer in a Spinning Disc

A disc of mass 2 kg and radius 0.5 m has a moment of inertia given by $$I = \frac{1}{2} m R^2$$ and

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

Rotational Work and Energy Experiment

A uniform disk with a moment of inertia $$I = 0.5\,kg\cdot m^2$$ is subjected to a variable torque d

Sliding Block on an Incline with Friction

A 5-kg block is released from rest at the top of an incline that is 10 m high. The incline is 20 m l

Spectroscopic Potential Energy Curve Analysis

A spectroscopic experiment is conducted to infer the potential energy curve of a diatomic molecule f

Spring Energy Experiment: Measuring Nonlinear Work

A spring exhibits a force-displacement relationship given by $$F(x) = k*x + a*x^3$$, where $$k = 50\

Wind Tunnel Analysis of Mechanical Energy Extraction

In a wind tunnel experiment, a miniature wind turbine is tested. The force exerted by the wind on th

Work Done on a Variable Inclined Plane

An object of mass $$m = 2 \;\text{kg}$$ is moved along an inclined plane whose angle of inclination

Work with a Variable Force on a Straight Path

A particle experiences a variable force along the x-axis given by $$F(x)= 10 + 3*x \; (\text{N})$$.

Work-Energy Analysis in Collisions

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

Work-Energy Principle in a Frictional System

A 5 kg block is moving upward along a 10-degree incline of length 5 m with an initial speed of 7 m/s

Work-Energy Theorem in a Rotational System

A solid disk with moment of inertia $$I = 0.5 \;\text{kg·m}^2$$ is subjected to a variable torque gi

Billiard Ball Collision and Impulse Analysis

In a game of billiards, a moving ball (mass $$0.17\,kg$$, speed $$2\,m/s$$) collides with a stationa

Car Collision Analysis

Two cars collide head-on and come to a complete stop. Car A has a mass of $$1200\,kg$$ and an initia

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 Measurement Using a Suspended Rod

In this experiment, students attempt to determine the center of mass of a non-uniform rod by suspend

Center of Mass of a Non-uniform Circular Disk

A thin circular disk of radius $$R$$ has a surface mass density given by $$\sigma(\theta)= k\,(1+\co

Center of Mass of a Variable Density Disk

A circular disk of radius R = 0.5 m has a surface mass density that varies with the radial distance

Center-of-Mass Motion Under an External Force

Consider a system composed of two masses, $$m_1=2.0\,kg$$ and $$m_2=3.0\,kg$$, connected by a light,

Center-of-Mass of a Ladder with Varying Linear Density

A ladder of length $$L=2.0\,m$$ has a vertical linear mass density given by $$\lambda(y)=3.0(1+y)$$

Combined Translational and Rotational Motion Analysis

A uniform rod of length $$2\,m$$ and mass $$4\,kg$$ is pivoted at one end. Initially at rest, an imp

Damped Harmonic Oscillator Analysis

A mass-spring system subject to damping has its displacement described by the function $$x(t)=0.2\,e

Elastic and Inelastic Collision Analysis

Two balls collide head-on. The red ball (mass $$0.5\,kg$$, velocity $$+4\,m/s$$) and the green ball

Elastic Collision Analysis

Two balls undergo an elastic head-on collision. Ball A has a mass of $$0.6\,\text{kg}$$ and an initi

Elastic Collision of Gliders

Two gliders undergo an elastic collision on a frictionless air track. Glider A (mass = 1.5 kg) is mo

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 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 9: Rocket Propulsion and Momentum Conservation

A rocket in space initially has a mass of $$500 \ kg$$ and is traveling at $$20 \ m/s$$. It ejects $

FRQ 10: Collision with Rotational Motion

A uniform disk of mass $$2 \ kg$$ and radius $$0.5 \ m$$ is rolling without slipping at $$4 \ m/s$$

Glider Collision on a Frictionless Air Track

Two gliders on an air track are involved in a head-on elastic collision. The data for the gliders is

Impulse from Force-Time Graph

A soccer ball (mass = 0.43 kg) is kicked, and the force exerted by the kicker’s foot varies with tim

Impulse in a Rebounding Ball

A 0.5-kg ball is dropped from a height of 2 m onto a rigid surface. It rebounds with a speed of 1.2

Impulse on Coupled Freight Cars

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

Impulsive Collision Involving Rotation

A 0.5 kg ball traveling at $$8\,m/s$$ horizontally strikes the end of a thin uniform rod of length $

Momentum Change under a Non-Constant Force

A 1.2-kg block on a frictionless surface is subjected to a time-varying force given by $$F(t) = 12*s

Motion of the Center of Mass with External Force

Consider two blocks (masses 3 kg and 5 kg) connected by a light spring on a frictionless surface. In

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

Multiple Particle Center of Mass in Two Dimensions

Four particles in a plane have the following properties: Particle A (2 kg) at (0, 0), Particle B (3

Nonuniform Rod Center of Mass

Consider a rod of length $$L = 1.0\,m$$ whose linear density is given by $$\lambda(x)=6+4*x$$ (in kg

Projectile Center-of-Mass Trajectory

A projectile is launched and its trajectory is recorded with emphasis on the motion of its center of

Projectile Explosion and Center of Mass Motion

A 5 kg projectile is launched vertically upward. At its highest point, it explodes into two fragment

Projectile Motion with Air Resistance Approximation

A 0.2 kg projectile is launched with an initial speed of 15 m/s at an angle of 40° above the horizon

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)

Structural Stability: Crane Center of Mass Analysis

An engineer is analyzing a crane consisting of a 500-kg horizontal beam whose center of mass is at 5

Tethered Satellites: Center of Mass and Thruster Impulse

Two satellites are connected by a 10-m long tether in space. Satellite A has a mass of 800 kg and Sa

Three-Body Collision on a Frictionless Table

Three particles with masses 1 kg, 2 kg, and 3 kg are initially placed along the x-axis at x = 0 m, 4

Two-Dimensional Collision Analysis

Two gliders on a frictionless air track collide in a two-dimensional plane. Glider A has a mass of $

Two-Stage Collision in Coupled Carts

Two carts on a frictionless track undergo a two-stage event. Initially, cart A (mass $$2\,kg$$, velo

Analysis of a Variable Moment of Inertia System

A rotating disk has a moment of inertia that decreases over time as its arms retract, following the

Analyzing Variable Torque and Angular Acceleration

A rotor with constant moment of inertia $$I = 4 \text{ kg\cdot m}^2$$ is subjected to a time-varying

Angular Kinematics of a Rotating Disk

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

Angular Momentum and Torque in Circular Motion

A particle of mass m moves in a circle of radius r with an angular velocity given by $$\omega(t)= 3t

Angular Momentum Conservation in Rotational Collisions

In this experiment, two disks with different moments of inertia and angular velocities are coupled t

Angular Momentum in Explosive Separation

A spinning disk with an initial angular velocity $$\omega_i$$ undergoes an explosion that breaks it

Angular Momentum Transfer in Colliding Rotational Bodies

A student studies collisions between two rotating bodies—a spinning disk and a smaller rotating whee

Angular Momentum Transfer in Coupled Rotating Disks

In an experiment, two disks are coupled so that they eventually rotate together without any external

Applying the Parallel Axis Theorem to a Composite Object

A composite object has been tested to determine its moment of inertia about different axes. The foll

Calculation of Rotational Inertia for Composite System

A uniform disk of mass $$M = 2\,kg$$ and radius $$R = 0.5\,m$$ has two small beads, each of mass $$m

Calculus Derivation of the Moment of Inertia for a Uniform Disk

Derive the moment of inertia for a uniform solid disk of mass M and radius R about its central axis

Calculus in Determining the Moment of Inertia of a Continuous Object

A researcher is investigating how non-uniform mass distribution affects the moment of inertia of a t

Comparative Angular Momentum in Different Systems

Compare the application of conservation of angular momentum in two systems: a spinning mechanical wh

Comparative Dynamics of Rotational vs. Translational Motion in Rolling Objects

In a complex investigation, an object is rolled down an incline and both its angular and linear acce

Composite Rotational and Translational Dynamics in Rolling Motion

A hoop of mass m and radius R rolls down an inclined plane. Initially, it slides and rolls, so that

Conveyor Belt Dynamics Driven by a Rotating Drum

A rotating drum of radius R drives a conveyor belt without slipping. Derive the relationship between

Critical Analysis of Torque in Mechanical Systems

A media report on engine performance claims that a 10% increase in the applied force always results

Cylinder Rolling Down an Incline

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an incline of angle $$\t

Derivation of Angular Kinematics Equations

A set of experiments records the angular displacement of a rotating wheel over time, yielding the fo

Designing a Rotational System with Specified Kinetic Energy

A researcher is tasked with designing a rotational system that must store a specified amount of kine

Determining Angular Acceleration from Time-Resolved Measurements

A researcher measures the angular velocity of a rotating wheel at several time intervals. The follow

Differentiating Between Contact and Rolling Without Slip

An experiment is conducted to analyze the motions of objects rolling down an inclined plane. Both th

Dynamic Stability of a Spinning Object

A gyroscope (spinning top) has a moment of inertia $$I=0.1\text{ kg\cdot m}^2$$ and spins with an an

Dynamics of a Rotating Rod with Sliding Masses

In this advanced experiment, masses are allowed to slide along a horizontal rod attached to a pivot.

Dynamics of Coupled Rotational Systems

Two disks are coupled by a belt: Disk A has a moment of inertia $$I_A = 0.8 \text{ kg m}^2$$ and ini

Energy Conservation in Combined Rotational and Translational Motion

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

Experimental Data: Angular Velocity vs Time Analysis

An experiment records the angular velocity of a rotating object over time. The provided graph shows

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

Graphical Analysis of Rotational Kinematics

A graph of angular velocity $$\omega$$ (in rad/s) versus time $$t$$ (in s) for a rotating wheel is p

Impact of Changing Radius on Rotational Motion

A rotating disk experiences a change in its effective radius from $$R_1$$ to $$R_2$$ due to deformat

Impulse and Angular Momentum: Impact on a Rotating Disk

A solid disk of mass $$M = 2.0 \text{ kg}$$ and radius $$R = 0.25 \text{ m}$$ is spinning with an in

Non-uniform Mass Distribution Effects on Rotational Inertia

Consider a rod of length $$L$$ whose linear mass density varies as $$\lambda(x) = \lambda_0 * (1 + x

Parallel Axis Theorem in Compound Systems

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

Parallel Axis Theorem: Composite Body Moment of Inertia

Consider a composite object consisting of a solid disk (mass $$M = 4.0 \text{ kg}$$, radius $$R = 0.

Rolling Motion of a Sphere on an Incline

A solid sphere of mass M and radius R rolls without slipping down an inclined plane of height h star

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 Inertia of a Composite Bead System

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

Rotational Kinetic Energy Storage in a Flywheel

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

Static Equilibrium of a Beam

A uniform beam of length 4 m and weight 200 N is supported at one end by a wall and held horizontal

Time-Varying Torque and Angular Acceleration

A researcher is exploring the effects of a time-varying torque on the rotational motion of a rigid b

Torque and Equilibrium: Balancing a Non-Uniform Beam

A beam of length $$L$$ has a non-uniform mass distribution such that its center of mass is located a

Torque from a Distributed Load

A uniform beam of length $$L$$ has a constant linear weight density $$w$$ (in N/m).

Torque in a Multi-force System: Seesaw Equilibrium

A seesaw of length $$L = 3.0 \text{ m}$$ is balanced on a fulcrum located 1.0 m from the left end. T

Amplitude Decay in Damped Oscillations

A damped oscillator has its displacement described by the function $$y(t)=A_0*e^{-\frac{b}{2*m}t}*\c

Analysis of Maximum Kinetic Energy in a Spring-Mass Oscillator

For a spring-mass oscillator with spring constant $$k$$ and amplitude $$A$$, the elastic potential e

Anharmonic Effects in a Pendulum

A simple pendulum of length $$L = 0.8 \; m$$ is released from an initial angle of $$15^\circ$$. For

Calculating Damped SHM Energy Loss

A student records the amplitude decay of a damped oscillator and calculates the energy using $$U=\fr

Calculus Application in SHM: Derivatives and Acceleration

Given the position function of an oscillator, apply calculus to derive its velocity and acceleration

Calculus-Based Prediction of Maximum Speed

A photogate system is set up to record the speed of a mass-spring oscillator. Answer the following p

Calculus-Derived Velocity and Acceleration in SHM

For a simple harmonic oscillator described by $$x(t)=A\sin(\omega t+\phi)$$, determine its velocity

Comparative Analysis of Horizontal and Vertical SHM Systems

A researcher compares the oscillations of a mass attached to a spring in horizontal and vertical con

Comparative Analysis: Energy Methods vs. Force Methods in SHM

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

Critical Analysis of Frequency Measurement Techniques in SHM

A newspaper article claims that 'simply timing a few cycles of an oscillator provides an exact measu

Data Analysis and Calculus Estimation in SHM

Using discrete experimental data for a harmonic oscillator, estimate derivatives and analyze acceler

Data Analysis from a Virtual SHM Experiment

A virtual experiment on simple harmonic motion produces the following data for the displacement of a

Data Analysis of Damped Oscillations

A damped oscillator is described by the function: $$y(t) = 0.1 * e^{-\frac{b * t}{2m}} * \cos(\omeg

Derivation of Total Mechanical Energy Conservation in SHM

For a block-spring system undergoing simple harmonic motion, demonstrate that the total mechanical e

Deriving Equations for a Damped Harmonic Oscillator

An experiment is designed to study the effects of damping in a spring-mass oscillator. This version

Determination of Spring Constant from Oscillation Data

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

Determination of the Spring Constant from Experimental Force-Displacement Data

In an experiment to determine the spring constant, a series of force and displacement measurements w

Determining the Phase Constant from Experimental Data

An experiment measuring the displacement of a simple harmonic oscillator produced the following data

Elastic Energy and Maximum Speed Calculation

Consider a block attached to a horizontal spring with a spring constant of $$k = 100\,N/m$$. The blo

Energy Conservation in a Spring Oscillator

A block of mass $$m = 0.2\,kg$$ oscillates horizontally on a frictionless surface attached to a spri

Energy Conservation Verification Using Calculus

A mass-spring system oscillates with a displacement given by $$x(t) = 0.06 \cos(15t)$$. (a) Derive t

Energy Exchange in Oscillatory Systems

A new research article claims that 'the maximum speed of a block on a spring is invariant with respe

Energy Transformations in SHM

Consider a block attached to a spring undergoing simple harmonic motion with displacement $$x(t)=A\s

Estimating Spring Constant from Kinetic Energy Measurements

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

Fourier Analysis of Oscillatory Motion

In an advanced experiment, students record the motion of a nonlinear oscillator and attempt to decom

FRQ 7: Differentiation of SHM to Obtain Velocity and Acceleration

Consider an oscillator described by $$y = A \sin(\omega t + \phi_0)$$. A set of experimental velocit

FRQ 14: Work Done by a Spring via Integration

Using calculus, derive an expression for the work done in stretching a spring from its natural lengt

FRQ1: Hooke’s Law in a Horizontal Spring-Mass System

A horizontal spring with force constant $$k = 250\,N/m$$ is fixed at one end. A block attached to th

FRQ6: Calculus Derivation of Velocity and Acceleration in SHM

For a mass undergoing simple harmonic motion described by the displacement function $$x(t)= A\sin(\o

FRQ9: Energy Exchanges in a Mass-Spring Oscillator

In a frictionless mass-spring oscillator the energy continuously oscillates between kinetic and pote

FRQ12: Phase Shift and Time Translation in SHM

An oscillator is described by the equation $$x(t) = A\sin(\omega t+\phi_0)$$. Answer the following:

FRQ20: Energy Dissipation in Damped Pendulum Oscillations

A damped pendulum oscillates with small angles such that its motion is approximately described by $

Graphical Analysis of SHM Experimental Data

A block attached to a horizontal spring oscillates, and its displacement $$x(t)$$ (in meters) is rec

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

Modeling Amplitude Reduction Due to Non-Conservative Forces

In a real oscillatory system, non-conservative forces (like friction) result in an energy loss per c

Momentum and Impulse Analysis in Oscillatory Motion

A block of mass $$m = 0.4 \; kg$$ oscillates on a spring, and its displacement is given by $$x(t)=0

Non-conservative Forces in Oscillating Systems

In an experiment with a spring-mass oscillator, students study the effect of friction on the oscilla

Oscillation Frequency's Dependence on Mass and Spring Constant

A research claim suggests that 'doubling the mass of an oscillating system will always decrease the

Pendulum Dynamics Beyond the Small-Angle Approximation

Investigate the dynamics of a pendulum when the small-angle approximation begins to break down.

Period and Frequency of a Vertical Oscillator

A block of mass $$m = 1.5 \; kg$$ is suspended from a vertical spring with a force constant of $$k =

Resonance in Forced Oscillations

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

Sinusoidal Description and Phase Shift in SHM

A block attached to a spring oscillates while a marker records its position on paper over time. This

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

Vertical Spring-Block Oscillator: Equilibrium and Oscillations

A block of mass $$m = 1.80\,kg$$ is attached to a vertical spring with force constant $$k = 400\,N/m

Vertical Spring–Block Oscillator Dynamics

Investigate the dynamics of a block oscillating vertically on a spring.

Calculus Derivation of Gravitational Potential Energy

Derive the expression for gravitational potential energy using calculus and compare your result to e

Calculus-based Derivation of Gravitational Force Variation

The gravitational force between two point masses is given by $$ F(r) = -G * \frac{m_1 * m_2}{r^2} $$

Center of Mass Analysis in Two-Body System

For a star-planet system, the barycenter determines the common center of mass around which both bodi

Comparative Analysis of Orbital Periods for Different Planets

Two planets orbit a star. (a) Derive the relationship $$\frac{T^2}{a^3} = \text{constant}$$ for thes

Derivation of Equations of Motion in a Gravitational Field Using Lagrangian Mechanics

A researcher analyzes the motion of a particle of mass $$m$$ moving radially under the influence of

Derivation of Escape Velocity from Earth's Surface Using Calculus

Using the principle of energy conservation and calculus, derive the expression for the escape veloci

Derivation of Escape Velocity Using Calculus

Using the concept of gravitational potential energy, derive the escape velocity from a planet of mas

Derivation of the Virial Theorem for Gravitational Systems

Derive the virial theorem for a gravitationally bound system and apply it to a hypothetical star clu

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 Orbital Speed in a Circular Orbit

A satellite is in a near-circular orbit around a planet. Its orbital speed can be determined by equa

Effects of Stellar Mass Variation in Binary Systems

In a binary star system, one star gradually loses mass. Analyze the impact on the orbital parameters

Energy Comparisons in Circular and Elliptical Orbits

Compare the total mechanical energy of a satellite in a circular orbit with that in an elliptical or

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 an Asteroid's Elliptical Orbit

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

Examining Relativistic Corrections to Newtonian Gravity

In strong gravitational fields, relativistic corrections become significant compared to Newtonian pr

Experimental Design for Measuring Gravitational Constants

Design an experiment using a torsion balance to measure the gravitational constant $$G$$.

FRQ 9: Kepler’s Second Law – Area Sweep Rate

Kepler’s Second Law states that a line connecting a planet to its star sweeps out equal areas in equ

FRQ 13: Orbital Transfer Maneuver – Hohmann Transfer

A spacecraft in a circular orbit of radius $$r_1$$ wishes to transfer to a higher circular orbit of

FRQ 14: Work Done in Changing Orbital Radius

The work done against gravity in changing the orbital radius of an object is computed by integrating

Gravitational Energy Trade-offs in a Multi-Body System

Examine the experimental data provided for gravitational potential energies between different pairs

Gravitational Interaction between Two Bodies

Consider two masses m1 and m2 separated by a distance r in deep space. Investigate the gravitational

Gravitational Potential Energy Measurement on a Ramp

In a laboratory experiment, a block is released down a long ramp to measure the conversion of gravit

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

Mass Determination using Orbital Motion and Kepler's Laws

A planet orbits a star in a nearly circular orbit with period $$T$$ and orbital radius $$r$$. (a) De

Newton vs. Einstein: Conceptual Analysis of Gravity

Compare and contrast Newton's Law of Gravitation with Einstein's theory of General Relativity. Answe

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

Orbital Decay Due to Atmospheric Drag

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

Orbital Period and Semimajor Axis Relationship Using Kepler's Third Law

A researcher collects observational data for various moons orbiting a giant planet. The table below

Orbital Period Determination Using Kepler's Third Law

Kepler’s Third Law relates the period T of an orbiting body to the semimajor axis a of its orbit. In

Perturbation Analysis in Elliptical Orbits

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

Perturbation in Orbital Motion

A small asteroid deviates from a perfect elliptical orbit due to a time-dependent perturbative force

Tidal Heating and Energy Dissipation

Tidal forces in planetary systems can lead to energy dissipation in satellites, resulting in tidal h