AP Physics C: Mechanics FRQ Room

Air Resistance and Projectile Motion

In an experiment, two projectiles with different surface textures (one smooth and one rough) are lau

Analysis of a Velocity-Time Graph

A velocity vs. time graph for an object moving in one dimension displays a linear increase in veloci

Analyzing Motion with a Nonlinear Acceleration Function

A particle moves along the x-axis with an acceleration given by $$a(t)=4\cos(t)$$ (m/s²). It has an

Average vs. Instantaneous Quantities

A particle’s displacement is given by the integral function $$x(t)= \int_0^t e^{-\tau} \cos(\tau)\,d

Circular Motion: Centripetal Acceleration from Tangential Speed Function

An object moves in a circular path of constant radius $$R = 3.0\,m$$. Its tangential speed varies wi

Comparative Analysis of Kinematic Equations

A researcher claims that the kinematic equation $$s=ut+\frac{1}{2}at^2$$ is universally valid for al

Determining Launch Angle from Experimental Data

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

Determining Motion from a Sine Position Function

An object's position is given by $$x(t)= 3*\sin(t) + t$$ (meters). Analyze its motion using calculus

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

Differential Equation of Motion Under Gravity and Drag

A particle of mass $$m$$ is falling under gravity and experiences a drag force proportional to its v

Displacement-Time Graph Analysis for Non-Uniform Motion

A displacement vs. time graph for an object moving in one dimension is given by the function $$s(t)=

Distance vs. Displacement Analysis in One-Dimensional Motion

An experiment recorded the motion of a car along a straight road where its distance traveled and dis

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

Effect of Initial Velocity on Displacement

A student investigates how altering the initial velocity of a cart affects its displacement on a lev

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

FRQ 1: Constant Acceleration Experiment

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

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 4: Vector Addition and Displacement Analysis

A researcher studies an object moving along a straight path where its motion includes reversals in d

FRQ 5: Calculus-Based Displacement Calculation

An object has a velocity given by the function $$v(t) = 3*t^2 - 2*t + 5$$ (with t in seconds and v i

FRQ 5: Derivation of Motion Equations from Calculus

A researcher aims to derive the standard kinematic equations using calculus for an object moving wit

FRQ 7: Projectile Trajectory Analysis

A projectile is fired from ground level with an initial speed of 50 m/s at an angle of 40° above the

FRQ 8: Circular Motion Kinematics (MEDIUM)

An object moves in uniform circular motion with its position given by $$\vec{r}(t)=(R\cos(\omega*t),

FRQ 8: Projectile Motion – Targeting a Moving Object

A researcher is tasked with designing a projectile launch system that accurately targets an object l

FRQ 8: Vector Addition in Two-Dimensional Motion

An object moves in a plane following these displacements in sequence: 4 m east, 3 m north, 5 m west,

FRQ 9: Non-Uniform Acceleration: Parabolic Motion

A researcher observes a car whose acceleration is not constant but given by the function $$ a(t) = 2

FRQ 12: Parametric Representation of Projectile Motion

A projectile’s motion is given by the parametric equations: $$x(t) = 5*t$$ and $$y(t) = 4*t - 2*t^2$

FRQ 14: Work and Energy in Kinematics – Rolling Ball on an Incline

A researcher is studying a ball rolling down an inclined plane with friction. In addition to the kin

FRQ 15: Investigating Uniformly Accelerated Motion Using Integrals

A researcher records the acceleration of an object with a sensor, finding that the acceleration vari

FRQ 17: Analyzing Motion from a Cubic Position Function

An object’s position is given by $$x(t)= 2*t^3 - 9*t^2 + 4*t$$ (in meters, with time in seconds). An

Impulse and Momentum with a Time-Dependent Force

A baseball (mass m = 0.145 kg) is struck by a bat. The force exerted by the bat is given by $$F(t)=

Kinematics with Calculus: Non-Uniform Acceleration

An object moves along the x-axis under a non-uniform acceleration given by $$a(t) = 4*t - 2$$ m/s²,

Motion on an Inclined Plane with Friction

Design an experiment to measure the acceleration of an object sliding down an inclined plane with fr

Piecewise Motion Analysis

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

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

Rotational Motion: Angular Kinematics

A disk initially at rest undergoes constant angular acceleration $$\alpha = 2\,rad/s²$$. (a) Derive

Uniformly Accelerated Free Fall Analysis

In a free fall experiment, a rock is dropped from a height of 80 m. The following table presents mea

Uniformly Accelerated Motion: Derivation and Application

A car accelerates uniformly from rest. Its velocity as a function of time can be expressed as $$v(t)

Variable Acceleration Analysis Using Calculus

Design an experiment to investigate the motion of an object under a time-varying force that produces

Vector Decomposition in Projectile Motion

A projectile is launched from the ground with an initial speed of 40 m/s at an angle of 50° above th

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

Verification of Uniformly Accelerated Motion

A student conducts an experiment on a frictionless track using a cart. The student hypothesizes that

Analysis of Elastic and Inelastic Collisions

Consider two scenarios involving collisions between two identical 2 kg masses. In Scenario 1, the ma

Analysis of Force and Velocity Data

An object is subjected to a variable force recorded by a sensor, given by $$F(t)=100-5*t$$ (in newto

Analysis of Mechanical Advantage and Work in a Lever System

A lever is used to lift a 500 N weight. The operator applies a force that varies with the angle, giv

Assessing Energy Conversion in a Pendulum Experiment

A researcher conducts an experiment with a simple pendulum of length L = 2 m and a bob of mass 0.5 k

Calculus‐Based Work Calculation with Constant Force

A constant force of 20 N acts along the direction of displacement over a distance of 3 m. Use calcul

Comparing Work–Energy Analysis Across Different Reference Levels

A researcher examines the impact of choosing different reference levels for potential energy calcula

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

Conservation of Mechanical Energy in a Pendulum

A simple pendulum consists of a bob attached to a massless string of length 2 m. The bob is pulled a

Damped Oscillations and Energy Dissipation in a Mass-Spring System

A damped harmonic oscillator with mass m = 2 kg, spring constant k = 50 N/m, and damping coefficient

Deriving and Applying the Work–Energy Theorem

Starting from Newton's second law, $$F = m*a$$, derive the work–energy theorem by relating force, di

Efficiency Analysis of a Mechanical System

A motor lifts a 100 kg mass by raising it 10 m in 20 seconds, using an electrical energy input of 15

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

Elastic Potential Energy in a Spring

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.1 m. Analyze the energy store

Energy Conservation in a Pendulum

A simple pendulum of length 2 m and mass 0.5 kg is released from an initial angle of 30° with respec

Energy Dissipation due to Friction

A 10 kg block is pushed along a horizontal surface with a coefficient of kinetic friction $$\mu = 0.

Energy Dissipation in an Oscillatory System

Consider a mass-spring oscillator with mass 1 kg and spring constant $$ k = 100 \;N/m $$, oscillatin

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

Friction‐Influenced Kinetic Energy Loss Experiment

A 1 kg block is pushed along a horizontal rough surface with a coefficient of kinetic friction μ = 0

FRQ 1: Analysis of Work at an Angle

A media report claims that when a constant force is applied at an angle to the displacement, the wor

FRQ 2: Work-Energy Theorem in Lifting

A news article claims that the work done in lifting an object is independent of the velocity at whic

FRQ 3: Kinetic Energy Measurement in Free Fall

A researcher presents data claiming that objects dropped from rest convert all gravitational potenti

FRQ 4: Potential Energy Curve Analysis for a Diatomic Molecule

A scientific paper provides the potential energy function for a diatomic molecule as $$U(x) = U_0 \l

FRQ 11: Deriving Force from a Potential Energy Function

A study posits that the force acting on a particle can be obtained via $$F(x) = - \frac{dU}{dx}$$. E

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 18: Work–Energy Analysis of a Decelerating Elevator

An elevator with a mass of 1200 kg decelerates uniformly as it approaches a floor. A motion analysis

Investigating Power Output in a Mechanical System

A researcher measures the power output of a machine that exerts a constant force while moving an obj

Non-Uniform Gravitational Field Work-Energy Calculation

An object of mass $$m = 1000 \;\text{kg}$$ is lifted from the Earth's surface (taken as $$x=0 \;\tex

Particle Dynamics in a Variable Force Field

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 12 - 2*x$$ (in newton

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

Power Output Fluctuations in a Jogger

A 70 kg jogger runs along a track with an instantaneous velocity given by $$v(t) = 2 + 0.5\,t$$ (in

Relationship Between Force and Potential Energy

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

Roller Coaster Energy Analysis

A roller coaster car of mass 500 kg is released from rest at a height of 50 m and descends along a t

Rotational Dynamics and Work-Energy in a Disk

A solid disk of mass 10 kg and radius 0.5 m starts from rest. A constant torque of 5 N·m is applied

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

Runner's Power Output Analysis

In a track experiment, a runner’s power output is calculated using the formula $$P = m*a*v$$ obtaine

Work and Energy in a Pulley System

A researcher investigates a two-mass system connected by a massless, frictionless pulley. Mass m1 =

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 on an Inclined Plane

A 3 kg block slides down a fixed inclined plane that makes an angle of 30° with the horizontal. The

Work-Energy Theorem in a Non-Uniform Gravitational Field

A particle of mass 1 kg is raised from the surface of a planet where the gravitational acceleration

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, and Power in Circular Motion

A car of mass $$m = 1000 \;\text{kg}$$ is moving on a circular track of radius $$R = 50 \;\text{m}$$

Analyzing a Force-Time Graph: Impulse and Momentum

A hockey puck of mass 0.15 kg is struck by a hockey stick. The force exerted on the puck during the

Angular Impulse and Rotation

A uniform disk (mass = 4 kg, radius = 0.5 m) rotates initially at 20 rad/s. A constant tangential fo

Astronaut Momentum Conservation

An astronaut with a total mass of 89 kg is floating in space near her shuttle. To reorient herself,

Ballistic Pendulum Analysis

A bullet with mass $$0.02$$ kg is fired horizontally into a pendulum bob of mass $$0.98$$ kg suspend

Center of Mass Acceleration under Variable Force

Two blocks with masses $$m_1 = 3\,\text{kg}$$ and $$m_2 = 2\,\text{kg}$$ are connected by a light ro

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 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 a Ladder with Varying Linear Density

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

Circular Motion: Banked Curve Analysis

A car of mass 1200 kg negotiates a banked curve of radius 50 m with no friction required. The curve

Complex Rotational and Translational Collision Involving Center of Mass

A uniform rod of length $$2$$ m and mass $$4$$ kg is pivoted frictionlessly about its center. A smal

Conservation of Linear Momentum in a Glider Collision

On a frictionless air track, two gliders collide. The experimental data below list the masses and ve

Dynamics of a Falling Object with Air Resistance

An object of mass 0.1 kg is dropped from a height and experiences air resistance modeled as $$F_{air

Explosive Fragmentation: Momentum Transfer

A stationary object with a mass of 12 kg explodes into three fragments in a frictionless environment

FRQ 1: Center of Mass of a Non-Uniform Rod

Consider a rod of length $$L = 1.2 \ m$$ with a linear mass density given by $$\lambda(x) = 2 + 3*x$

FRQ 2: Center of Mass of a Composite Lamina

Consider a composite lamina composed of two rectangular regions in the xy-plane. Region A is 0.8 m b

Glider Collision on an Air Track

Two gliders on a frictionless air track collide and stick together. Glider A (mass = $$0.50\,\text{k

Impulse and Angular Momentum in a Collision

A 0.2 kg ball traveling at 5 m/s collides with a thin rod (mass = 2 kg, length = 1.5 m) pivoted abou

Impulse and Kinetic Energy from a Time-Dependent Force on a Car

A 1200 kg car, initially at rest, is subjected to a time-dependent force given by $$F(t)=4000 - 500*

Impulse from a Collision with a Wall

A ball of mass $$0.2\,kg$$ strikes a rigid wall and rebounds. During the collision, it experiences a

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

Inelastic Collision Energy Loss Analysis

Two carts on a frictionless track undergo a completely inelastic collision. Cart A has a mass of $$1

Inelastic Collision of a Pendulum Bob with a Block

A pendulum bob of mass 2 kg is released from rest from a 30° angle from the vertical with a pendulum

Inelastic Collision on a Frictionless Surface

Two gliders on a frictionless air track collide and stick together. Glider A (mass = 2 kg) moves rig

Inelastic Collision: Bullet-Block Interaction

A 0.02-kg bullet is fired at 400 m/s into a stationary 2-kg wooden block on a frictionless surface.

Momentum Analysis in an Asteroid Breakup

An asteroid of total mass 1000 kg fragments into two pieces in deep space. Fragment A (mass unknown)

Momentum Analysis of a Variable Mass Rocket

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

Momentum Analysis of a Variable-Density Moving Rod

A rod of length $$L=1.5\,m$$ has a linear density function $$\lambda(x)=4+3*x$$ (in kg/m) and is mov

Momentum and Angular Momentum in a Rotational Breakup

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

Motion of the Center of Mass Under an External Force

A system consists of two particles with masses 2 kg and 3 kg located at x = 0 m and x = 4 m, respect

Multi-object Collision Dynamics

Three carts on a frictionless track collide and stick together. The carts have masses and initial ve

Nonuniform Rod: Total Mass and Center of Mass

A rod of length $$1.0$$ m has a linear density given by $$\lambda(x) = 10 + 6*x$$ (kg/m), where $$x$

Oblique Collision of Ice Pucks

Two ice pucks are sliding on a frictionless ice surface. Puck A (mass = 0.2 kg) is moving with a vel

Rocket Propulsion Momentum Problem

A model rocket with an initial mass of $$2\,kg$$ (including fuel) ejects $$0.5\,kg$$ of fuel instant

Rocket Propulsion: Variable Mass System

A rocket with an initial mass of 500 kg (including fuel) expels gas at a constant exhaust velocity o

Rotational Dynamics of a Composite Object

A composite object consists of two connected rods: Rod A is 1 m long and has uniform density, while

Rotational Impulse and Angular Momentum

A rigid disc of mass 4 kg and radius 0.5 m rotates about its central axis with an initial angular sp

Stability Analysis: Center of Mass vs. Center of Gravity

A uniform rectangular block of mass 10 kg with dimensions 0.5 m × 0.3 m × 0.2 m rests on a flat surf

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 $

Acceleration of a Rotating Rigid Body with Frictional Torque

A disk with moment of inertia $$I=2\text{ kg\cdot m}^2$$ experiences a frictional torque proportiona

Analysis of Angular Displacement in a Rotating Disk

In this experiment, several dots are marked along the radius of a rotating disk. The students record

Angular Impulse Analysis

A flywheel is subjected to a time-dependent torque given by $$\tau(t) = 50 * e^{-2*t}$$ N*m for $$t

Angular Impulse and Change in Angular Momentum

Design an experiment to measure the angular impulse delivered to a rotating object and its resulting

Angular Impulse and Change in Angular Momentum

A stationary flywheel is subjected to a constant torque $$\tau$$ for a time interval $$\Delta t$$.

Angular Momentum Conservation in Explosive Separation

A symmetric rotating disk of mass $$M$$ and radius $$R$$ is spinning with an angular velocity $$\ome

Angular Momentum Conservation on a Merry-Go-Round

A child of mass m = 30 kg stands on the edge of a merry-go-round, modeled as a disk with mass M = 10

Angular Momentum Conservation on a Merry-Go-Round

A child of mass $$m = 30\,kg$$ stands at the edge of a merry-go-round that is modeled as a uniform d

Angular Momentum Conservation: Merry-Go-Round with a Moving Child

A child of mass $$m = 30 \text{ kg}$$ stands on a merry-go-round modeled as a solid disk of mass $$M

Angular Momentum in a Variable Moment of Inertia System

A researcher examines the dynamics of a rotating system whose moment of inertia changes with time du

Angular Momentum Transfer in Coupled Rotating Disks

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

Application of the Parallel Axis Theorem

An object with mass 5 kg has a moment of inertia about its center of mass $$I_{cm} = 0.2\,kg\,m^2$$.

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

Calculus-Based Torque Distribution in a Non-uniform Rod

A student attempts to measure the net torque on a non-uniform rod whose mass distribution varies alo

Centripetal Force and Angular Velocity Measurement

Design an experiment to measure the centripetal force acting on an object in circular motion and rel

Comparative Calculations for a Composite System

Consider a system of three beads, each of mass $$m$$, arranged along a rod of negligible mass and le

Comparative Study of Angular Kinematics at Different Radii

In a lab experiment, students measure the angular displacement and corresponding linear displacement

Composite Body Rotation

A composite object is formed by welding a solid disk to a thin rod. The disk has mass $$M$$ and radi

Conservation of Angular Momentum in Rotational Collisions

Two disks (Disk A and Disk B) rotate independently and are then brought into contact, eventually rot

Conveyor Belt Dynamics Driven by a Rotating Drum

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

Correlation Between Torque and Rotational Energy via Calculus

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

Critical Analysis of Torque in Mechanical Systems

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

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

Discrete Mass Distribution and Moment of Inertia

A set of three small beads, each of identical mass $$m$$, is arranged along a light rod of length $$

Dynamic Equilibrium in Rotational Motion

Design an experiment to investigate the conditions for rotational equilibrium in a lever system. You

Energy Dissipation Due to Friction in a Spinning Disk

A disk is spun up to a high angular velocity and then allowed to slow down due to friction. The expe

Equilibrium Analysis in a Rotating Beam System

In a lab experiment, a student investigates the equilibrium of a beam pivoted at its center with var

Experimental Data: Angular Velocity vs Time Analysis

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

FRQ 4: Rotational Kinematics of a Disk

A disk starts from rest and rotates with a constant angular acceleration. After t = 4.00 s, the disk

FRQ 11: Impact of Mass Distribution on Angular Acceleration

Two wheels have identical mass M and radius R. Wheel A has all its mass concentrated at the rim (\(I

FRQ 15: Angular Momentum in an Inelastic Collision on a Rotating Platform

A figure skater stands on a frictionless rotating platform with a moment of inertia of \(10.00\,kg\c

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

Investigation of Torque on a Rotating Pulley

In an experiment, a student applied a constant force of $$F = 40\,N$$ at varying distances (moment a

Lever and Torque Computations

This problem involves calculating torque in a lever system. A diagram is provided below.

Lever Torque Application

A mechanic uses a long lever to lift a heavy object. A force of $$F = 100 \text{ N}$$ is applied at

Parallel Axis Theorem Application in Complex Systems

A composite object consists of a uniform rod of mass $$M = 4\,kg$$ and length $$L = 3\,m$$, and an a

Rotational Dynamics in a Non-Inertial Frame

In a rotating frame (such as on a merry-go-round), fictitious forces arise. Consider a situation whe

Rotational Inertia of a Hollow Cylinder vs. a Solid Cylinder

Compare the rotational inertias of a solid cylinder and a thin-walled hollow cylinder, both of mass

Rotational Inertia of a Non-Uniform Disk

A disk of radius R has a surface mass density given by $$\sigma(r)= \sigma_0 \left(1+\frac{r}{R}\rig

Simulation Analysis of Rotational Motion with Non-uniform Mass Distribution

A simulation of a rotating flexible system shows that the moment of inertia, $$I$$ (in kg m^2), chan

Stability Analysis of a Block on a Rotating Platform (Tipping Experiment)

A block is placed on a rotating platform, and the conditions under which the block tips are investig

Torque and the Right-Hand Rule Verification Experiment

Design an experiment to verify the direction of the torque vector as predicted by the right-hand rul

Torque from a Distributed Load

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

Torque Measurement and Analysis

A recent experimental study claims that the relationship between force and torque is strictly linear

Torque Measurement and Angular Acceleration Experiment

In this experiment, you will investigate the relationship between applied force, moment arm, and the

Torque on a Uniform Rod with Distributed Force

A researcher is studying the effect of a distributed force along a uniform rod of length $$L$$ pivot

Using Experimental Data to Evaluate Conservation of Angular Momentum

An experimental setup involves a rotating platform where the moment of inertia and angular velocity

Calculus Application in SHM: Derivatives and Acceleration

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

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 of Horizontal vs Vertical Oscillations

Two identical mass-spring systems have a mass of $$m = 0.5\,\text{kg}$$ and a spring constant of $$k

Comparative Analysis: Spring-Mass System vs. Pendulum Oscillations

A researcher compares the oscillatory behavior of a horizontal spring-mass system with that of a sim

Composite Oscillator: Two Springs in Series

A block with mass $$m = 1.0\,kg$$ is attached to two springs connected in series, with spring consta

Coupled Oscillations in a Two-Mass Spring System

Consider two masses, $$m_1$$ and $$m_2$$, connected by a spring with spring constant $$k$$. Addition

Coupled Oscillators Investigation

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

Data Analysis of Damped Oscillations

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

Dependence of Maximum Speed on Amplitude

For a spring-mass oscillator undergoing simple harmonic motion, analyze how the maximum speed $$v_{m

Derivation of Total Mechanical Energy Conservation in SHM

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

Determination of Angular Frequency from Displacement Data

Displacement measurements for a spring-mass oscillator are given by the equation $$y = A\sin(\omega

Determining Spring Constant from Experimental Data

An experiment on a spring produced the following data relating displacement $$x$$ (in meters) to for

Determining Spring Constant from Force-Displacement Data

In a laboratory experiment, the force exerted by a spring is measured for various displacements. The

Determining the Phase Constant from Experimental Data

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

Determining the Spring Constant from Oscillation Data

A student conducts an experiment to determine the spring constant $$k$$ of a spring by measuring the

Energy Exchange in Coupled Oscillators

Two identical masses \(m\) are connected by identical springs and allowed to oscillate on a friction

Estimating Spring Constant from Kinetic Energy Measurements

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

Evaluating the Impact of Initial Conditions on SHM Motion

An educational resource asserts that 'the initial displacement and velocity of a mass-spring system

Fourier Analysis of Oscillation Data

In an advanced experiment, the oscillation of a spring-mass system is recorded and analyzed using Fo

Horizontal Mass-Spring Oscillator Analysis

A laboratory experiment involves a block of mass $$m = 0.50\,kg$$ attached to a horizontal spring of

Impact of Spring Constant Variation on Oscillatory Motion

A researcher studies how varying the spring constant affects the oscillatory motion of a block attac

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

Investigating Damping Effects in a Spring-Mass Oscillator

In real oscillatory systems, damping forces affect the motion of the oscillator. Consider a spring-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

Measuring the Spring Constant: An Experimental Investigation

A student performs an experiment to determine the spring constant of a coil spring. The following da

Nonlinear Restoring Force: Effects on the Period of Oscillations

A system is characterized by a restoring force that is not perfectly linear: $$F = -k\,x - \alpha\,x

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 Period and Data Analysis

Explore the period of a simple pendulum and compare experimental data with theoretical predictions.

Pendulum Period Measurement Experiment

A group of students measure the period of a simple pendulum by timing multiple oscillations using a

Period and Frequency Determination from Time Measurements

A block oscillates on a spring. It takes 0.25 s for the block to move from its maximum displacement

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 =

Period of a Physical Pendulum: A Calculus Approach

A physical pendulum consists of a uniform thin rod of length $$L$$ and mass $$m$$, pivoted at one en

Phase Shift Determination in SHM

In an experiment to determine the phase shift of a simple harmonic oscillator, a block attached to a

Restoring Force in a Non-Ideal Pendulum

For a pendulum with a bob of mass $$m$$ swinging through large angles, the exact restoring force is

Sinusoidal Description and Phase Constant in SHM

A spring-mass oscillator has an amplitude $$A = 0.04 \; m$$ and a frequency $$f = 5.0 \; Hz$$. The d

Time-Derivative Analysis of Displacement in SHM

An experiment aims to determine the velocity of a mass undergoing simple harmonic motion by calculat

Vertical Oscillator with Offset Equilibrium

A vertical mass-spring system has a mass of $$m = 1.0\,kg$$ attached to a spring with force constant

Vertical Spring-Block Oscillations

A researcher investigates the vertical oscillations of a block attached to a spring hanging from a f

Vertical Spring-Mass Oscillator Dynamics

A block of mass $$m = 1.5 \; kg$$ is attached to the end of a vertical spring with force constant $$

Analysis of Low Earth Orbit Satellite Decay

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

Analysis of Orbital Transfer Maneuvers Using Calculus

A spacecraft is initially in a circular orbit of radius $$ r_1 $$ and is to be transferred to a circ

Analyzing a Two-Body Gravitational Interaction Using Calculus

Consider two objects of masses $$m_1$$ and $$m_2$$ that are initially at rest and start moving towar

Analyzing Tidal Forces in a Two-Body System

Explain the origin of tidal forces in a gravitational two-body system and derive their expression us

Calculating Gravitational Potential in a Non-Uniform Planet

A researcher investigates the gravitational potential inside a planet with a radially varying densit

Calculating the Gravitational Field from a Spherical Mass Distribution

Consider a planet with a spherically symmetric density profile given by $$ \rho(r) = \rho_0 \left(1

Calculus in Orbital Motion: Area Sweep in an Elliptical Orbit

Kepler's Second Law implies that the rate of area sweep (dA/dt) is constant for an orbiting body. In

Center of Mass of the Sun-Earth System

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

Comparative Analysis of Gravitational Forces

Using the data provided, compare the gravitational forces between various pairs of celestial bodies.

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

Derivation of Kepler's Second Law from Angular Momentum Conservation

Using the principle of conservation of angular momentum, derive Kepler's Second Law which states tha

Determining the Center of Mass in a Celestial System

In studying the two-body problem, such as the Sun-planet system, the center of mass (or barycenter)

Determining the L1 Lagrange Point

In a star-planet system, an object is positioned along the line connecting the two bodies at the L1

Dynamics of Comet Orbits

A comet follows a highly elliptical orbit around the Sun. Analyze its speed variation along its path

Effective Gravitational Field on an Irregular Asteroid

An irregularly shaped asteroid has a non-uniform density distribution. To determine the effective gr

Energy Balance at Apoapsis and Periapsis

Analyze the provided energy data for a planet at apoapsis and periapsis in its orbit. Discuss how co

Escape Velocity Derivation

The escape velocity is the minimum speed required for an object to escape from the gravitational fie

Examining Relativistic Corrections to Newtonian Gravity

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

Free-Fall Measurement on a Curved Incline

An experiment is designed to measure gravitational acceleration by allowing a small ball to roll dow

FRQ 18: Non-Uniform Circular Motion in a Varying Gravitational Field

An object in orbit around a planet experiences non-uniform circular motion due to variations in the

FRQ 19: Relativistic Corrections and Perihelion Precession

General relativity provides corrections to Newtonian gravity that can explain the observed perihelio

FRQ 20: Determining the Mass of a Central Body from Satellite Orbits

A satellite is observed to orbit a celestial body with a period $$T$$ and at a radius $$r$$. Using t

Gravitational Field Modeling for Extended Bodies

Compare the gravitational field produced by an extended, spherically symmetric body to that of a poi

Gravitational Lensing: Deflection of Light

Using a Newtonian approximation, a light ray passes near a massive object with mass $$M$$ at a close

Gravitational Parameter in Exoplanetary Systems

Using the provided exoplanetary data, analyze the consistency of the gravitational parameter (derive

Gravitational Slingshot Maneuver

A spacecraft uses a gravitational slingshot maneuver around a planet to gain additional speed. Answe

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

Investigating Tidal Forces and Differential Gravity Effects

Consider a moon orbiting a planet, where tidal forces arise due to the variation in gravitational fo

Kepler's Laws and Orbital Dynamics

A researcher investigates several near-circular planetary orbits around a distant star. Observationa

Kepler's Third Law and Satellite Orbits

Using the provided data for satellite orbits, analyze Kepler’s Third Law. Determine the relationship

Laboratory Test of Newton's Law of Gravitation using a Torsion Balance

Newton's Law of Gravitation can be tested in a laboratory setting using sensitive apparatus such as

Modeling Tidal Forces with Calculus

Tidal forces arise due to the gradient in gravitational force across an object. These forces can cau

Newton vs. Einstein: Conceptual Analysis of Gravity

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

Orbital Dynamics and Energy Conservation

Examine the dynamics of a satellite in a circular orbit around the Earth by using energy conservatio

Orbital Motion of a Satellite

A satellite of mass m is in a stable circular orbit around a planet of mass M at a distance r from t

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

Orbital Simulation Ignoring Relativistic Effects

A simulation models the orbit of a fast-moving object near a massive body using Newton's law of grav

Planetary Orbits and Energy Considerations

Consider a planet in a highly eccentric orbit. The total orbital energy (kinetic plus potential) is

Planetary Orbits and Kepler's Laws

Consider a planet orbiting a star under the influence of gravity. The orbit is elliptical with the s

Simulating Satellite Orbital Decay and Atmospheric Drag

An experimental simulation is set up to study the orbital decay of a satellite due to atmospheric dr

Variation of Gravitational Force with Distance

Consider the gravitational force given by $$F(r) = \frac{G m_1 m_2}{r^2}$$. Answer the following par