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 Motion from a Position Function

A particle moving along a line has its position described by $$x(t)=t^4 - 8t^2 + 16$$ (in meters) wh

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 Two-Dimensional Motion Using a High-Speed Camera

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

Block on an Inclined Plane: Kinematic Analysis

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

Combined Translational and Rotational Motion Experiment

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

Comparing Theoretical and Experimental Data in Uniform Acceleration

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

Conservation of Momentum in Collisions

Design an experiment using an air track to test the conservation of momentum in elastic collisions.

Critical Evaluation of Experimental Kinematics Data

A researcher claims that the motion of a falling object is characterized by uniform acceleration. Th

Determining Instantaneous Acceleration from a Displacement Graph

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

Displacement and Critical Points for a Time-Dependent Position Function

A particle moves along the x-axis such that its position is given by $$x(t)=4t^2 - t^3$$, where t is

Experimental Evaluation of Vector Addition in Two Dimensions

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

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: One‐Dimensional Constant Acceleration

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

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 4: Vector Addition and Displacement Analysis

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

FRQ 4: Velocity-Time Graph Analysis (EASY)

A velocity versus time graph for an object shows the following behavior: • From $$t=0$$ to $$t=2\,s$

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 10: Experimental Analysis of Free Fall

Below is experimental data from free fall tests for objects dropped from various heights: | Height

FRQ 13: Analyzing a Two-Dimensional Collision with Projectiles

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

FRQ 14: Differentiation of a Position Function

An object’s position is given by the function $$x(t)= t^3 - 6*t^2 + 9*t$$ (with x in meters and t in

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 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

FRQ 18: Determining Launch Parameters from Projectile Data (EXTREME)

A projectile’s path was experimentally measured, and the following graph (provided) shows its parabo

FRQ 20: Experimental Design – Determining g with a Free Fall Apparatus

In a physics laboratory, researchers design an experiment using a free fall apparatus to measure the

Gravitational Effects in a Non-Uniform Field

Design an experiment to measure the variation of gravitational acceleration with altitude. Provide a

Impact Analysis: Collision Avoidance

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

Integrating an Acceleration Function to Determine Motion

An object's acceleration is described by $$a(t)=6*t-5$$, with initial velocity $$v(0)=2$$ m/s and in

Investigation of Variable Friction in Curvilinear Motion

Design an experiment to study the motion of an object along a curved path where friction varies with

Motion Analysis Using Integrals

An object moves along a straight line with an acceleration given by $$a(t)=6-2*t$$ (m/s²) for $$0\le

Motion of a Bus on a Curved Track

A bus moves along a curved track with its acceleration given by $$a(t)= 0.2*t + 0.5*\sin(t)$$ (m/s²)

Motion with Air Resistance: Approximating Terminal Velocity

A small sphere falling through a medium experiences air resistance proportional to its velocity. Its

Optimization of Projectile Range

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

Oscillatory Motion: Mass-Spring System

A 1.0-kg mass is attached to a spring with a spring constant of $$k = 16\, N/m$$. The mass is displa

Projectile Motion using Calculus

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

Projectile Motion with Air Resistance

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

Projectile Motion with Drag

Design an experiment to analyze the trajectory of a projectile when including the effects of drag fo

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

Simple Harmonic Motion in a Spring-Mass System

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

Simultaneous Measurement of Velocity and Acceleration

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

Sinusoidal Motion

A particle’s position along the x-axis is given by $$x(t)=8\sin(t)$$, where $$0\leq t\leq2\pi$$ (x i

Uniformly Accelerated Motion on an Incline

A block starts from rest and slides down a frictionless incline of angle 30° and length 5.0 m.

Uniformly Accelerated Motion: Incorrect Baseline Velocity

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

Verification of Uniformly Accelerated Motion

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

Analysis of a Potential Energy Curve

A particle of mass 4 kg moves along the x-axis under the influence of a potential energy function gi

Bouncing Ball Energy Loss Experiment

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

Calculating Kinetic Energy from a Velocity Function

A particle of mass $$m = 1 \;\text{kg}$$ moves along the x-axis with a velocity given by $$v(t)= 3*t

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

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

Determining Instantaneous Power from a Velocity-Time Graph

A car of mass 1200 kg has its velocity measured as a function of time. The graph provided represents

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

Energy Analysis in a Mass-Spring Oscillator

A mass-spring system consists of a 1 kg mass attached to a spring with a spring constant of 100 N/m.

Energy Analysis of a Pendulum

A simple pendulum consists of a 0.5 kg bob suspended by a 2 m string. It is released from rest at an

Energy Conservation in a Pendulum

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

Energy Dissipation in 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 Dissipation in an Oscillatory System

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

Energy Loss in a Ball with Air Resistance

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

Energy Transformation in a Roller Coaster

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

Evaluating Work Done on an Object in Rotational Motion

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

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 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 18: Conservation of Energy in a Variable Gravitational Field Experiment

An experimental report investigates the motion of an object subject to a gravitational field that va

FRQ 19: Equilibrium Points from a Nonlinear Potential Energy Function

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

Inclined Plane Friction Variation Experiment

A block is allowed to slide down an inclined plane over which the coefficient of friction is not con

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

Investigating Work on an Inclined Plane

A researcher is investigating the work done on a 5 kg block being pushed up a frictionless inclined

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

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

Numerical Integration of Work in a Variable Force Field

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

Oscillatory Motion Energy Exchange Experiment

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

Pendulum Energy Conservation Experiment

A pendulum bob is released from a given initial angle and its speed at the bottom of the swing is re

Potential Energy Curve Analysis

An object of mass 4 kg moves in a potential energy field described by $$U(x) = (x-1)^2 - 0.5*(x-2)^3

Potential Energy Curve Analysis

An object of mass m is subject to a potential energy function given by $$U(x) = x^3 - 6*x^2 + 9*x +

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 and Energy Efficiency in a Conveyor Belt Experiment

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

Power Output in Elevator Lifting

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

Rocket Engine Power Output Analysis

A rocket of mass 1000 kg is traveling horizontally at a constant speed of 8.0 m/s under an engine th

Rotational Kinetic Energy in a Rolling Object

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

Rotational Work-Energy Analysis in a Flywheel

A flywheel with a moment of inertia $$I = 20\,kg\cdot m^2$$ is accelerated from rest to an angular s

Solar Energy Mechanical Conversion Experiment

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

Variable Friction and Kinetic Energy Loss

A 5 kg block slides across a horizontal surface and comes to rest. The frictional force acting on th

Vertical Lift Work Measurement Experiment

In this vertical lift experiment, an object is raised by a motor and its applied force and displacem

Work and Power in an Engine

A 1500 kg car is accelerated from rest by an engine whose power output varies with time according to

Work Done by a Variable Exponential Force

A piston is subjected to a variable force described by $$F(x)=500*\exp(-0.5*x)$$ N, where x is in me

Work Done by a Variable Force

An object is acted upon by a variable force given by $$F(x)=5\,x^2$$ (in newtons) along the x-axis.

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 with Constant and Variable Forces

An object is acted upon by two different types of forces on separate occasions. In Part (a), a const

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 with Air Resistance

A 0.2-kg ball is thrown vertically upward with an initial speed of $$40 \;\text{m/s}$$. Air resistan

Billiard Ball Collision and Impulse Analysis

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

Car Collision Analysis

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

Center of Mass of a Non-Uniform Rod

A thin rod of length $$0.8$$ m has a linear mass density given by $$\lambda(x) = 5 + 3*x^2$$ (kg/m),

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 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 Triangular Plate

A thin, homogeneous triangular plate has vertices at (0,0), (4,0), and (0,3) in meters. Determine it

Center of Mass of an Irregular Lamina in Polar Coordinates

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

Composite Body Center of Mass Calculation

A composite system consists of a uniform rectangular block (mass $$5\,kg$$, width $$0.4\,m$$) and a

Data Analysis: Testing the Linear Relationship between Impulse and Momentum Change

An experiment is conducted where a cart is subjected to several impacts. For each trial, the measure

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 a Center-of-Mass System under a Variable Force

A system of total mass $$10$$ kg, initially at rest, is subjected to a time-dependent force given by

Elastic Collision with Time-Dependent Pre-Collision Motion

Particle A (mass = 1 kg) has a velocity given by $$v_A(t)=4-t$$ (m/s) for $$0 \leq t \leq 2$$ s. It

Explosive Separation and Momentum Conservation

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

Explosive Separation of Particle System

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

FRQ 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 and Momentum Change for a Hockey Puck

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

Impulse 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 Delivered by Variable Thrust Rocket

A small model rocket experiences a thrust that varies with time as $$F(t)=50 - 10*t$$ (N) for $$0 \l

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 in a Variable Gravitational Field

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

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 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 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 $

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

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

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

Projectile Explosion and Center of Mass Motion

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

Projectile Motion with Drag Impulse Analysis

A projectile is launched horizontally and experiences a drag force proportional to its velocity, giv

Rigid Body Dynamics: Torque and Rotation

A uniform meter stick (mass = 0.3 kg, length = 1 m) is pivoted at one end. A perpendicular force is

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

Two-Dimensional Collision of Ice Skaters

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

Two-Dimensional Elastic Collision Analysis

A 1 kg particle moving horizontally at 4 m/s collides elastically with a 2 kg particle initially at

Variable Force Collision Analysis from Graph Data

A steel ball (mass = 0.2 kg) collides with a wall, and the force versus time data during the collisi

Analysis of a Variable Moment of Inertia System

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

Analyzing Variable Torque and Angular Acceleration

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

Angular Kinematics from Disk Data

A rotating disk’s angular displacement is recorded over time during a period of uniform angular acce

Angular Kinematics from Experimental Data

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

Angular Kinematics on a Rotating Platform

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

Angular Kinematics with Variable Angular Acceleration

A disk rotates with a non-uniform angular acceleration given by $$\alpha(t) = 4*t$$ (in rad/s²). The

Angular Momentum Conservation on a Rotating Platform

A child stands on a circular merry-go-round. Initially, the child is at 2.5 m from the center and th

Angular Momentum Conservation: Ice Skater

An ice skater is spinning with an initial angular velocity $$\omega_i$$ and an initial moment of ine

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 a Dual-Wheel System

Two wheels, A and B, are coupled so that friction between them eventually brings them to a common an

Calculus Derivation of the Moment of Inertia for a Disk

Using the integral formula $$I = \int r^2\,dm$$, derive the moment of inertia for a uniform circular

Calculus Derivation of the Moment of Inertia for a Uniform Disk

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

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

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

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

Cylinder Rolling Down an Incline

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

Determining Moment of Inertia of Irregular Objects

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

Dynamics of a Damped Flywheel System

A flywheel with moment of inertia $$I$$ is subject to a damping torque proportional to its angular v

Dynamics of a Rotating Flexible Beam

A flexible beam of length $$L = 5\,m$$ and total mass $$M = 10\,kg$$ rotates about one end. The mass

Dynamics of a Wheel under Applied and Frictional Torques

A motor applies a constant torque of 12 Nm to a wheel with a moment of inertia of 0.8 kg m^2. A fric

Energy Analysis in Rolling Motion

A spherical ball of mass $$M$$ and radius $$R$$ rolls without slipping down an inclined plane of ver

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

Experimental Investigation of Rolling Without Slipping

An experimental apparatus is used to study rolling without slipping for various cylindrical objects.

FRQ 5: Rolling Motion on an Incline

A solid cylinder of mass M = 3.00 kg and radius R = 0.20 m rolls without slipping down an inclined p

FRQ 10: Comparison of Rotational and Translational Kinetic Energy

A solid sphere of mass M = 4.00 kg and radius R = 0.10 m rolls without slipping down a ramp of heigh

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 14: Energy Loss in a Rotating Flywheel Due to Friction

A flywheel with moment of inertia \(I = 8.00\,kg\cdot m^2\) is initially spinning at \(\omega_0 = 15

FRQ 18: Rotational Equilibrium of a Beam

A horizontal beam is supported at both ends. Three forces act on the beam: • A force \(F_1 = 100.0\

FRQ 20: Time-Dependent Angular Acceleration with External Torque

A flywheel with moment of inertia \(I = 3.00\,kg\cdot m^2\) experiences an exponentially decaying ex

Graphical Analysis of Angular Motion

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

Impact of Changing Radius on Rotational Motion

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

Integration of Rotational Inertia: Thin Shell vs. Solid Sphere

Derive the moments of inertia for two spherical objects about an axis through their centers: (a) A

Mathematical Modeling of Brake Systems

A braking system applies a constant torque of $$\tau = 15 \text{ Nm}$$ on a flywheel with moment of

Moment of Inertia of a Continuous Rod

Consider a uniform thin rod of length $$L$$ and total mass $$M$$. The rod rotates about an axis perp

Rolling Motion of a Sphere on an Incline

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

Rolling with Slipping Transition

A solid sphere of mass $$M$$ and radius $$R$$ is released from rest on an inclined plane with an ang

Rotational Impulse and Change in Angular Momentum

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

Rotational Inertia Measurement via Pulley Apparatus

A student sets up an experiment to measure the moment of inertia of a uniform disk using a pulley sy

Time-varying Angular Acceleration in a Rotational System

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

Torque 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

Acceleration and Position Relationship in SHM

For an oscillator described by the position function $$x(t) = A \cos(\omega t)$$, analyze the kinema

Analysis of Phase Shift in Oscillator Data

An oscillator is described by the equation $$y = 0.03 \sin(8t + \phi_0)$$. It is experimentally meas

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 Energy Analysis: SHM vs. Pendulum

Compare the energy transformations in a spring-mass oscillator and a simple pendulum (operating unde

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

Comparison of Horizontal and Vertical Oscillations

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

Coupled Oscillators Investigation

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

Coupled Oscillators: Normal Modes Analysis

Consider a system of two identical masses \(m\) placed on a frictionless surface and connected by th

Coupled Oscillators: Normal Modes and Energy Transfer

Consider a system of two identical masses coupled by springs and attached to fixed supports. Analyze

Damped Oscillation: Logarithmic Decrement Analysis

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

Data Analysis of Damped Oscillations

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

Determination of Gravitational Acceleration Using a Vertical Oscillator

A vertical spring-mass oscillator reaches equilibrium when the spring stretches by a distance $$d$$

Determination of Maximum Elastic Potential Energy

A researcher is examining the energy stored in a spring when it is displaced from its equilibrium po

Determination of the Damping Coefficient from Amplitude Decay

Consider an oscillator with damping such that its displacement is given by: $$x(t) = A e^{-\frac{b}

Determination of the Spring Constant from Experimental Force-Displacement Data

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

Determining Spring Constant Through Oscillation Energy Analysis

An experimental report claims that the spring constant k can be precisely determined by equating the

Effects of Mass Variation on Oscillation Frequency

A student investigates how the oscillation frequency of a spring-block system varies with the mass a

Energy Conservation in Pendulum Motion

A pendulum bob of mass $$m=0.5\,\text{kg}$$ is released from rest at an angle of $$20^\circ$$ from t

Energy Conservation via Calculus Integration

In a spring oscillator experiment, energy is exchanged between elastic potential energy and kinetic

Estimating Spring Constant from Kinetic Energy Measurements

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

Experimental Determination of Spring Constant via SHM

A physics lab report claims that the spring constant, k, of a mass-spring oscillator can be precisel

Forced Oscillations and Beat Frequency

A mass attached to a spring is simultaneously driven by two periodic forces given by $$F_1(t)=F_0*\c

Forced Oscillations and Resonance

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

FRQ 4: Vertical Motion in a Spring–Block System

A vertical spring–block system is investigated. The equilibrium displacement for different masses at

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 7: Calculus Application in SHM

Consider a simple harmonic oscillator with its position described by $$y = A \sin(\omega t + \phi_0)

FRQ 10: Calculus Integration for Work Done in a Spring

Force measurements during the stretching of a spring were recorded as a function of displacement. Us

FRQ 12: Comparative Analysis of Horizontal and Vertical Oscillators

Experimental data comparing the oscillation periods of a horizontal spring–block system and a vertic

FRQ 13: Experimental Design for Hooke's Law Verification

Design an experiment to verify Hooke's law using a spring and a set of known masses. Answer the foll

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 15: Graphical Analysis of Restoring Force

A graph showing the restoring force versus displacement for a spring is provided. Analyze the graph

FRQ 17: Determination of Damping Coefficient

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

FRQ6: Calculus Derivation of Velocity and Acceleration in SHM

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

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

FRQ13: Determining Damping Coefficient from Amplitude Decay

A damped oscillator with mass \(m = 0.1\,kg\) exhibits an exponential decay in its amplitude. Initia

Mass Variation and Frequency in SHM

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

Measuring g with a Simple Pendulum

A researcher uses a simple pendulum to measure the acceleration due to gravity. The length of the pe

Momentum Transfer in a Spring-Mass Collision

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

Non-linear Effects in Simple Pendulum Motion

Examine the non-linear behavior of a pendulum when the small-angle approximation is not valid.

Pendulum Dynamics Beyond the Small-Angle Approximation

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

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

Phase Constant and Sinusoidal Motion

A mass-spring system oscillates according to $$x(t) = A \sin(\omega t + \phi_0)$$ with an amplitude

SHM with a Varying Force Constant

In an experiment on a spring-mass system, a student investigates oscillations at various amplitudes.

SHM: Spring Force and Energy Derivation

A spring with force constant $$k = 200 \;\text{N/m}$$ is fixed at one end. When the other end is dis

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

Spring Oscillator on an Inclined Plane

A block of mass \(m = 2\,kg\) is attached to a spring with spring constant \(k = 150\,N/m\) on an in

Vertical Oscillations: Lab Data Analysis

A mass-spring system is arranged vertically in a laboratory setup. The oscillatory motion of the blo

Vertical Spring–Block Oscillator Dynamics

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

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

Angular Momentum Conservation in Orbits

Analyze the relationship between orbital radius and angular momentum as depicted in the provided gra

Barycenter of the Sun-Planet System

Consider the Sun-Earth system, where the Sun's mass is M = 1.99×10^30 kg, the Earth's mass is m = 5.

Center of Mass in a Two-Body System

In a two-body system, such as a planet and its moon, both bodies orbit around their common center of

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.

Comparative Analysis of Planetary Orbits

Two planets orbit the same star with different semimajor axes. Use Kepler's Third Law to analyze and

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 Field due to a Spherical Shell

A thin spherical shell of uniform density with total mass M and radius R produces a gravitational fi

Derivation of Gravitational Potential Energy Difference

A space probe's gravitational potential energy is given by $$U(r) = -\frac{G M m}{r}$$. Answer the f

Derivation of Orbital Period from Gravitational Force

Using calculus, derive the expression for the orbital period of a planet in circular orbit from Newt

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

Deriving Gravitational Potential from Gravitational Force

The gravitational potential \(V(r)\) is related to the gravitational force by calculus. (a) Show th

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 Gravitational Constant using a Torsion Balance

An experimental apparatus utilizing a torsion balance is used to measure the gravitational forces be

Effective Gravitational Field on an Irregular Asteroid

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

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 Produced by a Thin Uniform Disk

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

Gravitational Force Calculation on a Satellite

A satellite with a mass of $$m = 500\,kg$$ orbits the Earth at a distance of $$r = 7.0 * 10^6\,m$$ (

Gravitational Potential Energy Differences in Multi-Body Systems

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

Gravitational Potential to Rotational Kinetic Energy Conversion

An experiment uses a rolling cylinder descending an inclined plane to explore the conversion of grav

Gravitational Potential via Integration in a Varying Density Sphere

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

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 Tidal Forces with Calculus

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

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 Transfer Trajectories and Hohmann Transfers

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

Perturbation Analysis of Satellite Orbits

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

Perturbation in Orbital Motion

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

Planetary Orbits and Kepler's Laws

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