AP Physics C: Mechanics FRQ Room

Analysis of Air Resistance Effects on Free Fall

In an experiment on free fall, an object is dropped and its velocity is measured over time. The calc

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

Analysis of Experimental Data Table

An experiment on an air track records the displacement of a cart at various times. The data is shown

Designing a Kinematics Lab Experiment

An engineer is tasked with measuring the constant acceleration of a cart on a frictionless track usi

Determination of Maximum Height in Projectile Motion

An experiment was conducted to determine the maximum height reached by a projectile using a motion s

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

Displacement Calculation from a Velocity-Time Graph

The velocity of an object is depicted by the following graph. Answer the subsequent questions based

Effect of Initial Velocity on Displacement

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

Evaluating an Experimental Claim on Presumed Uniform Acceleration

A media report claims that a series of experiments have shown that objects in free fall experience a

Evaluating Non-Uniform Acceleration from Experimental Data

A student records the following velocity data for an object undergoing non-uniform acceleration:

Free Fall Analysis with Terminal Velocity Consideration

A rock is dropped from a cliff 80 meters high. A researcher claims that the measured fall time of th

FRQ 1: Constant Acceleration Experiment

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

FRQ 1: One‐Dimensional Constant Acceleration

An object moves along a straight line with constant acceleration. Its initial velocity is 5 m/s and

FRQ 3: Displacement Data Analysis from a Position-Time Table

The table below provides the position (in meters) of an object at various times (in seconds): | Tim

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 6: Motion on an Inclined Plane

A researcher studies the motion of a block sliding down an inclined plane with friction. The block i

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 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 11: Modeling Motion with Differential Equations

A researcher is modeling the motion of a ski jumper. The dynamics of the jumper's flight can be expr

FRQ 13: Analyzing a Two-Dimensional Collision with Projectiles

A researcher conducts an experiment with two projectiles launched simultaneously from different posi

FRQ 13: Comparative Analysis of Two Free Fall Experiments

The following data summarizes two experiments where objects were dropped from different heights: |

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: Variable Acceleration – An Object Under Changing Forces

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

FRQ 20: Real-World Application – Car Braking Analysis

A car traveling at 30 m/s begins braking uniformly until it comes to a complete stop. Sensors record

Instantaneous vs. Average Velocity

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

Kinematics with Resistive Forces

Design an experiment to study the motion of an object falling under gravity while experiencing a dra

Lab Investigation: Effects of Launch Angle on Projectile Range

In a controlled laboratory experiment, a student launches a projectile with a fixed initial speed of

Motion on an Inclined Plane with Friction

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

Multi-Phase Rocket Motion Analysis

A rocket is launched vertically. Its engines provide a constant acceleration for 10 seconds. After e

One-Dimensional Uniform Acceleration Analysis

An object moves along a straight line with its position given by $$x(t) = -2*t^3 + 5*t^2 - 3*t + 1$$

Photogate Timer in Free Fall

A student uses a photogate timer to record the free fall of an object dropped from a height of 1.5 m

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

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/

Relative Motion: Two Trains on Parallel Tracks

Two trains move on parallel tracks. Train A has its position given by $$x_A(t)=25t$$ and Train B by

Rotational Kinematics of a Spinning Disk

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

Slope Analysis in a Velocity-Time Graph

A physics lab recorded an object’s velocity over time using an electronic sensor, and the resulting

Terminal Velocity Experiment

An experiment involves dropping objects of varying shapes from a tall building to study terminal vel

Time vs. Position Data Analysis: Initial Conditions Overlooked

A student conducted an experiment to study an object’s motion by recording its position over time us

Two-Dimensional Motion with Vector Decomposition

An object moves in the plane and its position is given by the vector function $$\vec{r}(t)= \langle

Uniform Acceleration in One Dimension

An object moves along a straight line with constant acceleration. Its motion is described by the pos

Uniformly Accelerated Motion on a Track

Design an experiment to test the hypothesis that in uniformly accelerated motion, the displacement i

Variable Acceleration and Integration

An object moves along a line with a time-dependent acceleration given by $$a(t)=6*t - 2$$ (in m/s²).

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

Analysis of Fall Dynamics with Air Resistance

An object with a mass of 0.5 kg is dropped from a height of 50 m. Air resistance is modeled as a dra

Analysis of Kinetic Energy Dissipation in Inelastic Collisions

A researcher examines a perfectly inelastic collision. Object A (mass 2 kg) moving at 4 m/s collides

Ballistic Kinetic Energy with Air Resistance

A ball is thrown upward within a controlled chamber while sensors record its velocity as a function

Calculus-Based Examination of a Spring System

A spring with a spring constant $$k = 200\,N/m$$ is compressed by a distance x and then released. An

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

Composite System: Roller Coaster Energy Analysis

A roller coaster car of mass 600 kg travels along a frictionless track. The following table provides

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

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

Derivation of the Work-Energy Theorem

Using calculus, derive the work–energy theorem starting from the definition of work and Newton's sec

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 Dissipation in an Oscillatory System

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

FRQ 3: Kinetic Energy Measurement in Free Fall

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

FRQ 13: Energy Loss Analysis in a Bouncing Ball

A 0.5-kg ball is dropped and allowed to bounce on a hard surface. The maximum height reached after e

FRQ 19: Analysis of Force–Time Data in a Crash Test

During a crash test, the force experienced by a dummy is recorded as a function of time. The force–t

FRQ 20: Evaluating Efficiency in a Conveyor Belt System

In a conveyor belt system used for transporting goods, the work input and the corresponding measured

Multi‐Phase Cart Energy Experiment

A small cart is sent along a track that includes a flat section, an inclined ramp, and a loop-the-lo

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

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 Variation in a Machine

An object is being moved along a straight path by a constant force of 10 N. The displacement as a fu

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 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 Motion Work–Energy Experiment

In a rotational experiment, a disc is accelerated by a motor that applies a measured torque over a s

Tidal Energy Extraction Analysis

A tidal turbine is installed in a coastal area where water flow speed varies cyclically. The force e

Time-Varying Velocity and Instantaneous Power Measurement

A vehicle follows a displacement function given by $$x(t)= 4*t^3 - 2*t^2 + 3*t$$ (in meters) while a

Variable Force and Velocity: Power and Work Analysis

A machine applies a time-dependent force given by $$F(t)=50+10\,t$$ (N) while the displacement of an

Work by Non-Conservative Forces in a Loop

A block experiences a variable nonconservative force along a path given by $$ F(x)= 20 * \sin(x) $$

Work Done by a Variable Gravitational Force

An object of mass 2 kg moves from a height of 50 m to 30 m above the Earth's surface. The gravitatio

Work Done by Non‐Conservative Forces with Variable Friction

A 10-kg block slides along a horizontal surface. The coefficient of kinetic friction varies with pos

Work Done in a Non-uniform Gravitational Field

An object of mass 500 kg is moved radially outward from the Earth. Assume the Earth’s mass is $$M =

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 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 Inelastic Collisions

A 3-kg cart moving at 4 m/s collides inelastically with a stationary 2-kg cart on a frictionless tra

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

Analysis of an Oblique Collision

Two ice skaters are initially at rest on a frictionless ice surface. They push off each other; skate

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 Mass of a Non-uniform Rod

A rod of length $$L = 2\,m$$ has a linear density given by $$\lambda(x) = 5 + 3*x$$ (in kg/m), where

Center of Mass of a Non‐Uniform Rod

A non‐uniform rod of length $$L = 1$$ m has a linear density that is modeled by $$\lambda(x) = 5 + 2

Center of Mass of a Nonuniform Circular Disk

A circular disk of radius $$R$$ has a surface mass density that varies with radial distance $$r$$ ac

Center of Mass of a Rectangular Plate with Variable Density

A thin rectangular plate extends from $$x=0$$ to $$x=2$$ m and from $$y=0$$ to $$y=3$$ m. Its surfac

Center of Mass of a Variable-Density Rod

Consider a thin rod of length $$L=2.0$$ m with a linear density given by $$\lambda(x)=2+3*x$$ (kg/m)

Central Force and Center-of-Mass Motion in a Binary Star System

A binary star system consists of Star A (mass $$2\times10^{30}\,kg$$) and Star B (mass $$3\times10^{

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

FRQ 10: Collision with Rotational Motion

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

FRQ 13: Critical Analysis: Momentum Experiment

A research study investigating momentum transfer in vehicle collisions reports that the measured mom

FRQ 16: Momentum Conservation in a Multi-Particle System

Three particles are aligned along the x-axis with masses $$m_1 = 1 \ kg$$, $$m_2 = 2 \ kg$$, and $$m

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 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 Center of Mass in a Soccer Kick

A soccer ball (mass $$0.45\,kg$$) initially at rest is kicked, resulting in a launch speed of $$25\,

Impulse and Momentum in Ball Kicking

In an experiment, a soccer player kicks a 0.4 kg ball. A force sensor records the force exerted by t

Impulse Calculation from a Force-Time Graph

A force acting on an object is depicted by a force-versus-time graph over the interval from t = 0 s

Impulse Delivered by a Decreasing Force from a Water Jet

A water jet impinging on a stationary nozzle exerts a time-varying force given by $$F(t)=50 - 10*t$$

Impulse Delivered by a Time-Dependent Damping Force

A 3 kg block on a frictionless surface experiences a damping force that varies with time, given by $

Impulse from Force Sensor Data

In a collision experiment, a force sensor attached to a small car records the force applied during i

Inelastic Collision with Time-Dependent Force

Two gliders on a frictionless air track collide inelastically. The red glider of mass $$0.5\,kg$$ is

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 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 Center of Mass under a Time-Dependent Force

A system with a total mass of 4.0 kg, initially at rest, is subjected to a time-dependent external f

Motion of Center of Mass Under External Force

Consider a system with total mass $$M=5\,kg$$. An external force acts on the system given by $$F_{ex

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-Dimensional Inelastic Collision Analysis

Two particles undergo a perfectly inelastic collision. Particle A (mass = 2 kg) moves with velocity

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

Off-Center Collision and Angular Momentum

A small ball (mass $$0.5\,kg$$) moving at $$4\,m/s$$ strikes a uniform rod (mass $$3\,kg$$, length $

Oscillations: Simple Pendulum Analysis

For a simple pendulum of length 1.5 m undergoing small oscillations, the angular displacement is giv

Projectile Explosion and Center of Mass Motion

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

Rotational Impulse in a Spinning Disc Experiment

In an experiment to measure angular impulse, a student applies a variable torque to a spinning disc

Spring-Loaded Collision with Impulsive Force

A 0.5 kg ball moving horizontally at $$8$$ m/s collides with a spring-mounted barrier that exerts a

Stability Analysis Using Center of Mass on a Pivoted Beam

A uniform beam of length 2.0 m and mass 10 kg is pivoted at one end. A 20 kg mass is suspended from

Stability of a Suspended Mobile

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

Angular Impulse and Change in Angular Momentum

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

Angular Kinematics on a Rotating Platform

A rotating platform starts from rest and accelerates with a constant angular acceleration $$\alpha$$

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 Changes in a Skater's Spin

A figure skater initially spins with a moment of inertia $$I_i$$ and angular velocity $$\omega_i$$.

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

Assessment of Rotational Kinematics Equations

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

Calculus Based Determination of Moment of Inertia for a Non-uniform Rod

A rod of length $$L = 2\,m$$ has a linearly varying density given by $$\lambda(x) = \lambda_0 \,(1 +

Calculus-Based Determination of Angular Displacement

A rotating object's angular velocity is recorded as a function of time, and a graph of angular veloc

Composite Body Rotation

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

Computational Modeling of a Spinning Disk with Variable Torque

A spinning disk with a constant moment of inertia of $$I = 0.3\,kg\,m^2$$ is subjected to a time-var

Conservation of Angular Momentum in Explosive Separation

A rotating disk of mass $$M = 5.0 \text{ kg}$$ and radius $$R = 0.5 \text{ m}$$, spinning at $$\omeg

Coupled Rotational and Translational Dynamics in a Rolling Sphere

A sphere is allowed to roll down a curved track without slipping. The experiment examines the coupli

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

Driven Rotational Pendulum with Variable Torque

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

Effect of Friction on Rotational Motion

Design an experiment to quantify the torque losses due to friction in a rotating apparatus. Your goa

Energy Considerations in a Rotating Pendulum

A physical pendulum consisting of a rigid body with moment of inertia $$I$$ rotates about a pivot. T

Engine Torque Measurement Analysis

A mechanical engineer is analyzing the torque output of a car engine. The engine uses a lever arm at

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 2: Rotational Inertia of a Composite System

A thin uniform rod of length L = 2.00 m and mass M = 5.00 kg has two small beads, each of mass m = 1

FRQ 3: Application of the Parallel Axis Theorem

A solid disk has a mass M = 10.00 kg and a radius R = 0.50 m. Its moment of inertia about its centra

FRQ 17: Moment of Inertia of a Non-Uniform Rod

A rod of length L = 2.00 m has a density that varies with position according to $$\rho(x) = \rho_0 *

Investigating the Big Five Equations for Rotational Motion

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

Investigating the Parallel Axis Theorem

A researcher examines the effect of changing the axis of rotation on the moment of inertia of a rigi

Investigation of Gyroscopic Precession

Design an experiment to study gyroscopic precession. You will use a spinning wheel mounted on a gimb

Mass Redistribution and Kinetic Energy in Rotating Systems

In a rotating system, a person on a rotating platform moves closer to the axis, reducing the system’

Moment of Inertia of a Hollow Cylinder with Thickness

Derive an expression for the moment of inertia of a hollow cylinder with inner radius $$R_1$$, outer

Net Torque and Angular Acceleration Calculation

A disc with a moment of inertia of 0.5 kg m^2 is acted upon by two tangential forces: 10 N at a radi

Non-Uniform Angular Velocity: Integration and Differentiation

A rotating disk has an angular velocity given by $$\omega(t)=3*t^2 - 2*t + 1$$ (in rad/s), where t i

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

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 Impact and Energy Dissipation in Collisions

Two disks rotating about the same fixed axis are brought into contact and stick together. Disk A has

Rotational Inertia of a Composite Bead System

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

Rotational Kinematics on a Spinning Disk

A rotating disk's angular displacement is measured by the function $$\theta(t) = 0.2\,t^2 + 2\,t$$ (

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 its Direction: Vector Analysis

A force of magnitude $$F = 25 \text{ N}$$ is applied at an angle of $$30^\circ$$ above the horizonta

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

Torque Measurement and Angular Acceleration Experiment

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

Torsion Pendulum Method for Irregular Objects' Inertia

This experiment uses a torsion pendulum to determine the moment of inertia of irregular objects. The

Using Experimental Data to Evaluate Conservation of Angular Momentum

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

Work Done by Torque and Rotational Kinetic Energy

An engine applies a constant torque to a flywheel, causing it to rotate from rest through an angular

Amplitude Dependence in a Nonlinear Oscillator

Consider an oscillator whose restoring force is not perfectly linear but is given by: $$F = -k * x

Analyzing the Half-Cycle Method in Oscillation Experiments

A media report asserts that 'timing just half a cycle of a pendulum is sufficient to determine its f

Calculus Application in SHM: Derivatives and Acceleration

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

Comparative Analysis: Spring-Mass vs. Pendulum Oscillators

An experiment compares the oscillatory behavior of a spring-mass system and a simple pendulum. Answe

Comparative Period Analysis: Mass-Spring Oscillator vs. Simple Pendulum

A researcher is comparing the oscillatory behavior of a horizontal mass-spring system and a simple p

Conservation of Mechanical Energy in SHM

A frictionless spring-mass oscillator conserves mechanical energy during its motion. Demonstrate thi

Derivation of SHM Equations Using Calculus

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

Deriving the General Solution of SHM

Derive and analyze the general solution for simple harmonic motion from the governing differential e

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 Spring Constant Using SHM Data

An experiment on a mass-spring oscillator provides the following data for different masses and their

Determining the Phase Constant from Experimental Data

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

Effect of Mass Variation on SHM

A block attached to a spring oscillates, and the period of oscillation is given by $$T = 2\pi\sqrt{\

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 Conservation in a Mass–Spring Oscillator

Examine energy conservation in a mass–spring system and its experimental verification.

Energy Conservation in Vertical Oscillators

A media claim states that 'in a vertical spring-mass system, mechanical energy is always conserved r

Energy Transformation in SHM

A block of mass $$m = 0.2 \; kg$$ oscillates on a horizontal spring with a force constant of $$k = 1

Energy Transformations in SHM

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

Experimental Analysis of SHM Data

The displacement data for a mass-spring system were recorded during a laboratory experiment. The fol

Forced Oscillations and Resonance

An oscillator is driven by an external force and is modeled by the equation $$m\ddot{x} + kx = F_0 \

Forced Oscillations and Resonance

A mass-spring system is driven by an external force of the form \(F(t) = F_0 \cos(\omega_d t)\) and

Fourier Analysis of Oscillation Data

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

Fourier Analysis of Oscillatory Motion

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

FRQ 5: Period of a Simple Pendulum

An ideal simple pendulum has a length of $$L = 1.0\ m$$ and swings with a maximum angular displaceme

FRQ 6: Sinusoidal Description of SHM

A simple harmonic oscillator has an amplitude of $$A = 3.0\ cm$$ and a frequency of $$f = 4.0\ Hz$$.

FRQ 8: Energy Exchanges in SHM

A graph of elastic potential energy vs. displacement for a spring is provided. Use calculus to deriv

FRQ 9: Damped Oscillatory Motion Analysis

An oscillator experiencing damping shows a decrease in amplitude over successive cycles. Analyze the

FRQ 15: Determination of the Phase Constant

An oscillator with amplitude $$A = 0.05\ m$$ and angular frequency $$\omega = 8\ rad/s$$ is observed

FRQ 17: Determination of Damping Coefficient

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

FRQ 17: Pendulum Nonlinear Effects

Consider a simple pendulum with length $$L = 1\ m$$. While for small angular displacements the perio

FRQ 19: Vertical Oscillator Dynamics

A mass is attached to a vertical spring. When displaced by a distance $$y$$ from its equilibrium pos

FRQ11: Work Done in Stretching a Spring – Integral Calculus Approach

A spring has a force constant of $$k = 150\,N/m$$. The work done in stretching the spring from its e

FRQ14: Oscillations on an Inclined Plane

A block is attached to a spring and placed on a frictionless inclined plane that makes an angle of $

FRQ15: Nonlinear Behavior of a Large-Angle Pendulum

A simple pendulum of length \(L\) does not exactly follow simple harmonic motion when the amplitude

FRQ18: Effect of Mass Variation on the Oscillation Period

The period \(T\) of a mass-spring oscillator is given by the formula: $$T = 2\pi\sqrt{\frac{m}{k}}.

Hooke's Law Force Calculation

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

Horizontal Mass-Spring Oscillator Analysis

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

Integration Approach to SHM: From Acceleration to Displacement

A mass undergoing simple harmonic motion has an acceleration given by $$a(t) = -\omega^2 * A * \sin(

Integration of Variable Force to Derive Potential Energy

A non-linear spring exerts a force given by $$F(x)= - k * x - \alpha * x^3$$, where $$k = 200 \; N/m

Modeling Nonlinearities in Pendulum Motion

While the small-angle approximation leads to simple harmonic motion for a pendulum, larger angles in

Nonlinear Restoring Force: Beyond Hooke's Law

Some real-world springs do not obey Hooke's law for large displacements. Consider a spring that exer

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

A mass on a spring is observed to take $$0.20\,s$$ to move from its maximum displacement on one side

Period Estimation Using Calculus in Simple Pendulum Experiments

An experimental study reports that integrating the motion equations of a simple pendulum leads to pe

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 Analysis in Driven Oscillators

Consider an experiment where a driven oscillator is used to measure phase shifts as the driving freq

Phase Space Analysis of SHM

A student plots the phase space diagram (velocity vs. displacement) of a simple harmonic oscillator

Sinusoidal Oscillator and Phase Constant

A mass attached to a spring oscillates horizontally on a frictionless surface, and its displacement

Stress Testing of Oscillatory Limits

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

Vertical Mass-Spring Oscillator: Equilibrium and Oscillations

A 2.0-kg mass is attached to the end of a vertical spring with a force constant $$k = 250\,N/m$$. Th

Vertical Spring–Block Oscillator Dynamics

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

Analysis of Low Earth Orbit Satellite Decay

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

Analysis of Tidal Forces Acting on an Orbiting Satellite

A researcher studies the tidal forces acting on a satellite orbiting a massive planet. Due to the fi

Analyzing Multi-body Interactions in a Three-Body Problem

Consider a simplified three-body system consisting of two stars, each of mass M, and a planet of mas

Analyzing Three-Body Gravitational Interactions

Consider a system comprising a star, a planet, and a moon. In such a three-body system, gravitationa

Areal Velocity and Angular Momentum in Planetary Motion

A planet of mass $$m$$ orbits a star while conserving its angular momentum $$L$$. Answer the followi

Assessment of Newton's Second Law Along a Gravitational Incline

A small cart is placed on a frictionless inclined plane that makes an angle $$\theta$$ with the hori

Calculating Gravitational Potential in a Non-Uniform Planet

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

Calculus Derivation of Kepler's Second Law

Kepler's Second Law states that a line joining a planet and the Sun sweeps out equal areas during eq

Calculus in Determining Work Against Gravity over Altitude Change

A spacecraft gradually moves from an initial orbital radius r₁ to a higher radius r₂. The work done

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 Analysis in Two-Body System

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

Comparison of Gravitational and Centripetal Forces

For a satellite in a stable circular orbit, investigate the balance between gravitational and centri

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

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

Designing a Cavendish Experiment to Measure the Gravitational Constant

A student plans to design a version of the Cavendish experiment to measure the gravitational constan

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

Eccentricity and Elliptical Orbits

Discuss the role of eccentricity in elliptical orbits and derive the orbit equation in polar coordin

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 Angular Motion

A planet follows an elliptical orbit around its star. Investigate how variations in orbital distance

Energy Analysis in Multi-Body Systems

Consider a system of three bodies interacting gravitationally. Derive the expression for the total g

Energy Conversion in an Asteroid's Elliptical Orbit

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

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

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 16: Graphical Analysis of Gravitational Potential Energy vs. Height

The graph provided shows experimental data for gravitational potential energy (in joules) versus hei

Gravitational Energy in a Binary Star System

Binary star systems are bound by gravity. The total mechanical energy of such a system includes kine

Gravitational Field Modeling for Extended Bodies

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

Gravitational Field of a Uniform Ring

A researcher is investigating the gravitational field created by a thin uniform ring of mass $$M$$ a

Gravitational Field Produced by a Thin Uniform Disk

A researcher calculates the gravitational field produced by a thin circular disk of total mass $$M$$

Gravitational Parameter in Exoplanetary Systems

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

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

Modeling Orbital Decay with Differential Equations

A satellite in orbit experiences a drag force proportional to its velocity, leading to orbital decay

Non-uniform Gravitational Fields in Planetary Interiors

Investigate how gravitational acceleration varies within a planet assuming it has a uniform density.

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 Dynamics and Energy Conservation

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

Orbital Energy Analysis in Elliptical Orbits

The total mechanical energy of a satellite in an elliptical orbit is the sum of its kinetic and grav

Orbital Perturbation due to Radial Impulse

A satellite in a circular orbit of radius R receives a small radial impulse, altering its orbit into

Orbital Speed Variation in Elliptical Orbits

Analyze the provided data on orbital speeds at different points in an elliptical orbit. Discuss how

Orbital Transfer Trajectories and Hohmann Transfers

A spacecraft in low Earth orbit (LEO) is planning a Hohmann transfer to reach a geosynchronous orbit

Predicting Orbital Decay Due to Atmospheric Drag

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

Satellite Orbit Simulation: Finite Burn and Hohmann Transfer Error

A research team develops a computer simulation to model a satellite's orbital transfer using a Hohma

Tidal Forces and their Impact on Orbital Dynamics

A moon orbits a planet and experiences tidal forces. Analyze how these forces are derived and their

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

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