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 vs. Time Graph in a Lab Experiment

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

Analysis of a Velocity-Vs-Time Graph

An object’s velocity is recorded along a straight path. The velocity-time graph indicates that the o

Analyzing a Parabolic Trajectory

A projectile is launched with an initial speed of 50 m/s at an angle of 45°. Analyze its trajectory

Analyzing Circular Motion: Speed and Acceleration

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

Average vs. Instantaneous Quantities

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

Combined Translational and Rotational Motion Experiment

Design an experiment to study an object that exhibits both translational and rotational motion as it

Decoupling Horizontal and Vertical Motions in Projectile Motion

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

Determining Instantaneous Rates from Discrete Data

A sensor records the position of a moving particle at various times. The recorded data is shown in t

Displacement Calculation from a Velocity-Time Graph

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

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)=

Free Fall from a Cliff with Calculus Insights

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

Free Fall under Gravity

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

FRQ 2: Distance vs. Displacement in Variable Motion (MEDIUM)

An object moves along the x-axis with a velocity given by $$v(t)=3*t-6$$ (in m/s). (a) Determine the

FRQ 2: Projectile Motion – Launch Experiment

A researcher uses a projectile launcher to study the flight of a ball launched at an angle. The ball

FRQ 3: Projectile Motion with Calculus Analysis (MEDIUM)

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

FRQ 4: Projectile Motion – Maximum Height and Range

A projectile is launched from the ground with an initial speed of 60 m/s at an angle of 30° above th

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 6: Relative Motion in Two Dimensions (HARD)

In the xy-plane, two particles have position functions given by: $$\vec{r}_A(t)=(2*t, t^2)$$ and $$\

FRQ 9: Piecewise Acceleration Motion (HARD)

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

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 13: Analyzing a Two-Dimensional Collision with Projectiles

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

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 16: Exploring the Relationships in the Big Five Kinematic Equations

A researcher conducts a series of experiments to test the Big Five kinematic equations, in which eac

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

Impulse and Momentum with a Variable Force

A cart of mass $$m = 5.0\,kg$$ is subjected to a time-dependent force described by $$F(t)=5*t²$$ (in

Impulse and Variable Force Analysis

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

Inferring Acceleration from Velocity Data Using Calculus

The following table shows the time and corresponding velocity for an object moving in one dimension,

Investigation of Constant Acceleration in a Car

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

Kinematics in a SmartLab Setup: Integration Error

In a SmartLab experiment, sensors measured both the acceleration and displacement of an object movin

Motion on an Inclined Plane

A student investigates the motion of a block sliding down a 30° inclined plane initially in a fricti

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

Non-linear Position Function Analysis

A particle moves along the x-axis with a non-linear, decaying oscillatory position given by $$x(t)=e

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

Predicting Motion in a Resistive Medium

An object in free fall experiences a time-dependent acceleration given by $$a(t)=g\left(1-\frac{t}{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

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 Timing Error

In an outdoor lab experiment, a projectile launcher was used to fire a ball at a 45° angle relative

Rotational Dynamics: Variable Torque

A solid disk with moment of inertia I is subject to a time-dependent torque given by $$\tau(t)= 5*t$

Rotational Motion: Angular Kinematics

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

Simultaneous Measurement of Velocity and Acceleration

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

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

Time-Dependent Acceleration Analysis

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

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

Variable Acceleration Analysis

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

Variable Acceleration: Analyzing a Non-Uniformly Accelerated Motion

An object moves along a straight path with an acceleration given by $$a(t)=\frac{12}{(t+2)^2}$$ 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

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

Bouncing Ball Energy Loss Experiment

A ball is dropped from a known height and allowed to bounce repeatedly. The experiment calculates en

Collision and Energy Loss Analysis

Two objects collide inelastically and stick together. Object A has mass 3 kg moving at 4 m/s, and Ob

Comparing Work–Energy Analysis Across Different Reference Levels

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

Conservation of Mechanical Energy in a Pendulum

A pendulum bob of mass 1.2 kg and length 2 m is released from a 60° angle relative to the vertical.

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 Hooke’s Law

A spring with a spring constant of $$ k = 200 \;N/m $$ is compressed by 0.1 m from its equilibrium p

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 on a Frictional Ramp with Calculus Approach

A 2-kg block slides down a straight inclined ramp from a height of $$h = 2 \;\text{m}$$. The ramp is

Energy Dissipation in a Bouncing Ball

A ball of mass 0.2 kg is dropped from a height of 2 m and rebounds to a height of 1.5 m. Assume that

Energy in a Spring–Mass System

A mass is attached to a spring with a spring constant of $$k = 200\,N/m$$. The spring is compressed

Energy Loss Due to Position-Dependent Friction

A block of mass $$m = 5 \;\text{kg}$$ slides along a horizontal surface. The coefficient of kinetic

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

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 1: Vertical Lifting Experiment – Work Calculation

A lab student lifts a 1.5 kg mass vertically at constant velocity and records the force applied alon

FRQ 4: Conservation of Mechanical Energy in a Roller Coaster

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

FRQ 5: Analysis of Work and Potential Energy in a Spring–Mass System

A mass–spring system is tested in a laboratory. A spring attached to a mass is stretched from its eq

FRQ 7: Energy Loss Due to Friction on a Sliding Object

An object is sliding along a horizontal surface and loses kinetic energy due to friction. A sensor r

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 11: Analysis of a Bungee Jump – Variable Net Force

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

FRQ 12: Quantifying the Work Done by Friction

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

FRQ 14: Calculus Analysis of Work Done on an Object with a Time-Varying Force

An object with a mass of 3 kg is subjected to a force that varies with time according to the equatio

FRQ 16: Work and Energy Transformation in a Compound Machine

A 10-kg block is pushed up a frictionless ramp using a compound pulley system. A force sensor record

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

Instantaneous Power in a Variable Force Scenario

An object is subjected to a time-dependent force given by $$F(t)=5*t$$ N, and its displacement is de

Kinetic Energy and Work-Energy Theorem Application

A block of mass 2 kg undergoes net work in two different scenarios. Use the work–energy theorem to f

Oscillations in a Mass-Spring System

A 1 kg mass is attached to a spring with a spring constant of 100 N/m. The mass is displaced 0.1 m f

Potential Energy Curves and Equilibrium Analysis

An object of mass 4 kg has a potential energy function given by $$U(x) = (x - 2)^2 - (2\,x - 3)^3$$.

Power Output Measurement in an Elevator Experiment

A 500 kg model elevator is accelerated from rest to a speed of 2 m/s over a distance of 3 m under th

Projectile Energy Analysis with Air Resistance Correction

A 0.5 kg ball is thrown vertically upward with an initial speed of 20 m/s. Air resistance does negat

Pulley System Work–Energy Verification

A two-mass pulley system is used to verify the work–energy theorem. Velocities of both masses are re

Rocket Engine Power Output Under Variable Thrust

A rocket engine produces a time-varying thrust described by $$T(t) = 5000 + 1000*sin(t)$$ (in newton

Roller Coaster Energy Transformation Experiment

A 250 kg roller coaster car is released from rest at the top of a hill of 20 m height. The car then

Spring with Nonlinear Force: Elastic Potential Energy via Integration

A nonlinear spring exerts a restoring force given by $$F(x)= k*x + \alpha*x^3$$, where $$k = 200 \;\

Variable Force Robotic Arm Power Experiment

In this experiment, a robotic arm exerts a time-varying force on an object as it moves along a horiz

Variable Force Work Calculation and Kinetic Energy Analysis

Consider an object moving along the x-axis under the influence of a variable force given by $$F(x) =

Variable Gravitational Acceleration Over a Mountain

A hiker lifts a 10 kg backpack up a mountain where the gravitational acceleration decreases linearly

Work Done along a Curved Path Under Variable Force

A particle moves along a curve defined by $$ y = x^2 $$ in the xy-plane. It is subjected to a force

Work Done by a Time-Dependent Force

A 4 kg object on a frictionless surface is subjected to a horizontal force given by $$ F(t) = 10 * t

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 on an Object by a Central Force

An object moves under a central attractive force given by $$F(r)= -\frac{A}{r^2}$$ with $$A = 50 \;\

Work-Energy Analysis in Collisions

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

Astronaut Momentum Conservation

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

Center of Mass for Discrete Particles

Consider a system of three particles in the xy-plane with the following properties: • Particle A: m

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 Shift in an Internal Explosion

Three masses are arranged on a frictionless horizontal surface: m1 = 1 kg at (0,0), m2 = 2 kg at (1,

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 Colliding Carts

Two carts on a frictionless track collide. Cart A (mass = 2 kg) moves to the right with a speed of 3

Data Analysis: Momentum from Experimental Graphs

In an experiment, a cart of mass $$2\,kg$$ undergoes a collision event. The following data were reco

Derivation of the Rocket Equation Using Momentum Conservation

A rocket moving in space expels mass continuously. Assume that during an infinitesimal time interval

Dynamics of Center of Mass under a Time-Varying External Force

A system consists of two blocks with masses of 3 kg and 5 kg. A time-varying external force given by

Elastic Collision Analysis

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

Explosive Separation and Momentum Conservation

An object of total mass $$M$$ at rest explodes into two fragments. One fragment has mass $$m$$ and i

FRQ 4: Impulse from a Time-Dependent Force

A 0.8 kg ball experiences a force given by $$F(t)=10*\sin((\pi*t)/2)$$ (N) for $$0 \le t \le 2\ s$$.

FRQ 17: Impulse from a Functional Force

A particle experiences a force given by $$F(t) = 4*t$$ (N) over the time interval $$0 \le t \le 5\ s

Impulse and Kinetic Energy Loss in a Perfectly Inelastic Collision with a Spring

A 0.7 kg ball moving at $$6\,m/s$$ collides inelastically with a 2.3 kg block at rest. After collisi

Impulse Calculation from Force-Time Graph

A particle is subjected to a time-varying force represented by the graph provided. Using calculus, d

Impulse Delivered by a Variable Force on a Soccer Ball

A soccer ball of mass 0.45 kg is struck by a kick that applies a variable force over a brief time in

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

Integrated Analysis of Momentum and Center of Mass in a Multi-Stage Experiment

In a complex experiment, a projectile traveling at 10 m/s with a mass of 5.0 kg breaks apart mid-fli

Momentum and Angular Momentum in a Rotational Breakup

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

Momentum and Energy in Elastic Collisions

Two gliders on a frictionless air track undergo an elastic head-on collision. Glider A (mass $$0.8\,

Motion of the Center of Mass under External Force

Consider a two-particle system consisting of masses $$m_1 = 4\,kg$$ and $$m_2 = 6\,kg$$. An external

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

Multi-Stage Rocket Propulsion using Momentum Conservation

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

Oblique Collision of Two Billiard Balls

Two billiard balls, each of mass $$0.17\,\text{kg}$$, undergo an oblique collision on a frictionless

Projectile and Cart Collision: Trajectory Prediction

A 0.2 kg projectile is launched horizontally at 10 m/s from a 20 m high platform. At the same moment

Rebound Velocity from a Time-Dependent Impact Force

A ball with mass $$m=0.5\,kg$$ is dropped from a height of $$10\,m$$ and approaches the ground with

Recoil Dynamics in a Firearm Event

A 5.0 kg rifle fires a 0.025 kg bullet horizontally with a speed of 400 m/s. Experimental measuremen

Rocket Propulsion and the Tsiolkovsky Rocket Equation

A rocket expels mass continuously and can be modeled as a variable mass system. Starting with a mass

Rocket Propulsion and Variable Mass System

A rocket has an initial mass of $$500$$ kg (including fuel) and expels gas with a constant relative

Rocket Propulsion with Variable Mass

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

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

Analysis of Rolling Motion on an Incline

Consider a solid cylinder of mass $$M$$ and radius $$R$$ that rolls without slipping down an incline

Angular Kinematics from Experimental Data

A laboratory experiment measures the angular velocity $$\omega$$ of a rotating object as a function

Angular Kinematics: Modeling a Rotating Spring System

A disk is attached to a torsional spring that exerts a restoring torque given by $$T = -k \times \th

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

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 +

Comparative Analysis of Rotational and Translational Dynamics

A rolling object on a rough surface exhibits both translational and rotational motion. Its total kin

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

Coupled Rotational Dynamics of Two Disks

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

Derivation of Angular Kinematics Equations

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

Determining Moment of Inertia of Irregular Objects

Design an experiment to determine the moment of inertia of an irregularly shaped object using a pend

Determining the Moment of Inertia of a Non-Uniform Rod

A non-uniform rod of length $$L = 2\,m$$ is analyzed to determine its moment of inertia about one en

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.

Energy Conversion in a Rolling Cylinder Experiment

A cylinder rolls without slipping down an inclined plane. The experiment examines how gravitational

Energy Transfer in Rolling Objects

Design an experiment to study the energy conversion in a rolling object down an incline, by measurin

Equilibrium Analysis in Rotational Systems

A seesaw, modeled as a uniform beam 4 m long pivoted exactly at its center, is in rotational equilib

FRQ 12: Combined Translational and Rotational Motion with Slipping

A disk of mass M = 2.00 kg and radius R = 0.30 m is released on a 30° inclined plane with a kinetic

Inelastic Collision of Rotating Disks

Two disks mounted on a common frictionless axle collide and stick together. Disk A has a moment of i

Investigation of Angular Acceleration from Experimental Data

In an experiment, the angular displacement (in radians) of a rotating object was recorded at various

Lever Torque Application

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

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’

Parallel Axis Theorem in Compound Systems

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

Physical Pendulum with Offset Mass Distribution

A physical pendulum is constructed from a rigid body of mass $$M$$ with its center of mass located a

Rolling Motion Down an Inclined Plane

A solid cylinder of mass m and radius R rolls without slipping down an inclined plane of height h, s

Rolling Motion Energy Conversion Experiment

A researcher investigates the energy conversion in rolling motion without slipping. A solid cylinder

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 Energy Distribution in a Compound System

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

Rotational Equilibrium Analysis of a Beam

A beam is in static equilibrium under the influence of several forces applied at different distances

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 Measurement with a Disk and Pendulum

In this experiment a flat disk is mounted as a pendulum with its pivot offset from its center. The o

Rotational Inertia of a Composite Bead System

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

Rotational Inertia of a Uniform Rod

A uniform, thin rod of length $$L$$ and mass $$m$$ rotates about an axis perpendicular to the rod th

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

Torque and Angular Acceleration Relationship

An experiment measures the response of a rotating object to different applied torques. A graph is pl

Torque and Rotational Inertia in Engine Mechanisms

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

Torque Equilibrium in a Beam

A horizontal beam is pivoted at one end. On the beam, a force F1 = 100 N is applied 0.8 m from the p

Torsion Pendulum Method for Irregular Objects' Inertia

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

Torsional Oscillator Analysis

A torsional pendulum consists of a disk suspended by a wire with torsion constant $$k$$. The system

Verification of the Parallel Axis Theorem

Students test the parallel axis theorem by measuring the moment of inertia of a uniform rod about se

Wrench Torque Analysis

A mechanic uses a wrench to loosen a bolt. Consider a wrench of length L = 0.3 m. In part (a), the m

Amplitude and Maximum Speed Relationship in SHM

A mass-spring oscillator undergoes simple harmonic motion with amplitude $$A$$ and angular frequency

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

Calculus-Based Derivation of Oscillator Velocity and Acceleration

For an oscillator described by $$y = A\sin(\omega t + \phi_0)$$, derive its velocity and acceleratio

Comparative Analysis of Horizontal and Vertical Oscillators

Students set up two oscillatory systems: one horizontal spring–block oscillator and one vertical spr

Comparative Analysis: Spring-Mass vs. Pendulum Oscillators

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

Comparative Dynamics of Mass-Spring and Pendulum Oscillators

Analyze the motion of two oscillatory systems: a mass-spring oscillator and a simple pendulum (using

Complex SHM: Superposition of Two Harmonic Motions

A block experiences two independent harmonic motions such that its displacement is given by $$x(t)=

Conservation of Mechanical Energy in SHM

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

Coupled Oscillators Investigation

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

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

Damped Oscillation: Logarithmic Decrement Analysis

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

Damped Oscillations in a Spring System

Consider a lightly damped oscillator described by the displacement function $$x(t)=A e^{-\frac{b}{2m

Damped Oscillations: Amplitude Decay Analysis

A mass-spring oscillator with $$m = 0.2\,kg$$ and damping constant $$b = 0.1\,kg/s$$ experiences dam

Deriving the Equation of Motion Using Calculus

An undamped harmonic oscillator is governed by the differential equation $$\ddot{x} + \omega^2 x = 0

Driven Oscillations and Resonance in a Spring Oscillator

A mass-spring oscillator is subjected to an external sinusoidal driving force given by $$F_d(t)=F_0\

Driven Oscillations and Resonant Response

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

Driven Oscillator and Resonance

A forced mass-spring-damper system is subject to an external driving force given by $$F(t) = F_0\sin

Effect of Amplitude on the Period of an Oscillator

An experiment is conducted to investigate if the period of a spring-mass oscillator depends on the a

Effects of Spring Constant Variation on Oscillatory Motion

A spring-mass system oscillates with motion given by $$y(t)=A*\cos(\omega*t)$$ where $$\omega=\sqrt{

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 Mass–Spring Oscillator

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

Energy Conversion in a Spring-Mass Oscillator

Consider an experiment to investigate energy conversion in a spring-mass oscillator. In this experim

Energy Loss Analysis in a Spring Oscillator

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

Energy Transformations in a Mass-Spring System

A researcher investigates energy transformations in a mass-spring oscillator. The system consists of

FRQ 9: Effect of Spring Constant on Frequency

For a mass-spring oscillator, the frequency is given by $$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$. An

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:

Graphical Analysis of Oscillatory Data

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

Hooke's Law Force Calculation

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

Integral Calculation of Work Done in SHM

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

Interpretation of a Lab Setup Diagram for a Spring-Mass Oscillator

Examine the provided schematic diagram of a spring-mass oscillator experimental setup. (a) Describe

Lagrangian Mechanics of the Simple Harmonic Oscillator

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

Mass Variation and Frequency in SHM

Consider a spring oscillator with a constant spring constant of $$k = 200\,N/m$$. The frequency of o

Mechanical Energy in SHM

A researcher attaches a block of mass $$m = 0.05 \; kg$$ to a horizontal spring with force constant

Nonlinear Pendulum Oscillations and Error Analysis

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

Pendulum Motion and the Small Angle Approximation

A simple pendulum of length $$L$$ oscillates with small angular displacements. Analyze its motion us

Pendulum on a Rotating Platform

A simple pendulum of length $$L$$ is mounted on a platform that rotates with constant angular speed

Pendulum Oscillations: Small Angle Approximation

A simple pendulum of length $$L=1.5\,\text{m}$$ is set into small-amplitude oscillations. (a) Derive

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 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 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

Resonance and Energy Amplification in Oscillatory Systems

In a driven, damped oscillator, the amplitude as a function of the driving frequency is given by $$

SHM with Phase Shift: Initial Conditions Analysis

An oscillator is described by the equation: $$y(t) = A * \sin(2\pi * f * t + \phi)$$ In a particul

Sinusoidal Motion: Phase Constant Determination

An oscillator’s motion is described by the equation $$y = A \sin(\omega t + \phi_0)$$ with an amplit

Sinusoidal SHM with Phase Shift

An oscillator undergoes simple harmonic motion described by the equation $$y(t) = A\sin(\omega t + \

Spring-Block Oscillator: Phase Angle and Motion Description

A block attached to a horizontal spring oscillates without friction. The motion of the block is desc

Torsional Oscillator as a Rotational Analogy

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

Vertical Oscillations: Energy and Force Analysis

Consider a block attached to a vertical spring. Analyze the system from both the force and energy pe

Vertical Oscillator in a Gravitational Field

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

Analysis of Gravitational Anomalies: Local Variations in g

Local measurements of gravitational acceleration $$g$$ exhibit small variations due to underlying de

Application of Kepler's Third Law

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

Barycenter of the Sun-Earth System

A researcher investigates the center of mass (barycenter) of the Sun-Earth system. Given that the ma

Calculus in Gravitational Work: Integration of Inverse Square Force

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

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

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 Determination in the Sun-Earth System

A researcher is calculating the barycenter (center of mass) for the Sun-Earth system using a one-dim

Center of Mass in the Sun-Earth System

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

Derivation of Escape Velocity Using Calculus

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

Derivation of Orbital Period in Binary Star Systems

A researcher studies a binary star system in which two stars of masses $$m_1$$ and $$m_2$$ orbit the

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 Eccentricity from Observational Data

Astronomers collect data of a planet's distance from its star at various times and wish to determine

Elliptical Orbit Simulation Error in Barycenter Consideration

A computer simulation is developed to mimic the elliptical orbits of planets using Newton's law of g

Elliptical Orbits and Eccentricity Calculation

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

Energy Dissipation in Orbital Decay

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

Experimental Design for Measuring Gravitational Constants

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

FRQ 10: Gravitational Interactions in a Three-Body System

Consider a simplified system with three masses, $$m_1$$, $$m_2$$, and $$m_3$$, located at fixed posi

FRQ 11: Time-Dependent Gravitational Force in Radial Motion

A spaceship travels radially away from a planet under the influence of gravity. Consider the gravita

FRQ 17: Tidal Forces and Differential Gravity

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

Gravitational Force Calculation Between Celestial Bodies

Consider two celestial bodies with masses $$m_1$$ and $$m_2$$ separated by a distance $$r$$. Newton'

Gravitational Parameter in Exoplanetary Systems

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

Gravitational Potential Energy Change for a Satellite

A satellite is moved from a lower orbit to a higher orbit around a planet. Gravitational potential e

Gravitational Potential Energy Measurement on a Roller Coaster

An amusement park ride features a roller coaster that reaches a maximum height of $$50 \;m$$ above t

Gravitational Slingshot Maneuver

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

Impact of Mass Loss on a Comet's Orbit

A comet loses mass due to sublimation as it approaches the Sun. This variable mass affects its orbit

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

Investigating Orbital Eccentricity Effects

Orbital eccentricity affects the dynamics of a planet's motion. Answer the following: (a) A graph i

Mathematical Modeling of Tidal Forces

Using the provided data on tidal forces measured at different distances, analyze how the tidal force

Modeling Orbital Decay due to Atmospheric Drag

A low Earth orbit satellite experiences orbital decay primarily due to atmospheric drag. The drag fo

Optimization of Orbital Maneuvers in Multi-Stage Rockets

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

Orbital Decay due to Atmospheric Drag

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

Orbital Simulation Ignoring Relativistic Effects

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

Perturbation Analysis of Satellite Orbits

Study the graph showing small deviations from a nominal circular orbit of a satellite. Analyze the p

Tidal Heating and Energy Dissipation

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

Work Done by Gravitational Force in Radial Motion

A spacecraft of mass $$m$$ moves radially under the gravitational influence of a mass $$M$$. Answer

Work Done in a Variable Gravitational Field

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