AP Physics C: Mechanics FRQ Room

Acceleration Calculation by Differentiating a Position Function

In an experiment, the position of an object was modeled by the function $$x(t)= 3*t^4 - 2*t^2 + t$$.

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

Analysis of Experimental Data Table

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

Analyzing a Two-Dimensional Collision

Two objects undergo a collision in a plane. Object A (mass m_A) experiences a force during the colli

Conservation of Energy in a Pendulum

Design an experiment to test the conservation of mechanical energy in a simple pendulum. Provide bot

Coupled Motion: Translation and Rotation

A solid cylinder with mass $$m = 2.0$$ kg and radius $$R = 0.5$$ m is released from rest at the top

Critical Evaluation of Experimental Kinematics Data

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

Determining Velocity from a Position Function with Differentiation Error

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

Determining Zero Acceleration from a Non-linear Position Function

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

Displacement Calculation from a Velocity-Time Graph

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

Experimental Data and Constant Acceleration

A ball rolling down a ramp has its displacement measured at various times as shown in the table belo

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

Free Fall from a Cliff with Calculus Insights

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

FRQ 1: Constant Acceleration Experiment

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

FRQ 3: 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 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 8: Projectile Motion – Targeting a Moving Object

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

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: Kinematics with Acceleration as a Function of Position (HARD)

An object moving along the x-axis has an acceleration that varies with its position: $$a(x)=4*x$$ (i

FRQ 13: Comparative Analysis of Two Free Fall Experiments

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

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 18: Experimental Kinematics Data Analysis

A series of measurements for an object's velocity at various times are recorded as follows: | Time

FRQ 19: Comparative Kinematics – Two Launch Angles

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

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

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

Graphical Analysis of Motion: Position to Velocity

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

Impulse and Momentum with a Time-Dependent Force

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

Instantaneous vs. Average Velocity

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

Investigating Lab Data: Graph Interpretation and Improvements

In a motion sensor experiment, an object’s displacement as a function of time was recorded, resultin

Kinematics with Calculus: Non-Uniform Acceleration

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

Motion Lab Data Analysis

In a laboratory experiment, a car’s position along a straight track was recorded over time. The data

Motion with Time-Varying Acceleration (Drag Force Approximation)

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

Newton's Second Law and Force Measurement on a Cart

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

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

Pendulum Energy Conservation Experiment

Design an experiment to test the conservation of mechanical energy in a simple pendulum system. Your

Piecewise Motion Analysis

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

Projectile Motion Analysis

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

Projectile Motion on Level Ground

An object is launched from ground level at a 45° angle with an initial speed of 30 m/s (neglect air

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: Determining Initial Conditions

In an experiment, a projectile’s horizontal displacement was measured over time. The recorded data a

Projectile Motion: Maximum Height and Range

A projectile is launched from ground level with an initial speed of $$v_0 = 60$$ m/s at an angle of

Projectile Motion: Maximum Height and Range

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

Relative Motion in an Accelerating Frame

Inside an elevator accelerating upward at 2 m/s², an object is dropped. Its motion is recorded relat

Rotational Motion: Angular Kinematics

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

Time-Dependent Acceleration Analysis

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

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

Uniformly Accelerated Motion on an Inclined Plane

A 5.0-kg block is placed on a 30° inclined plane. When released, it slides down with a coefficient o

Uniformly Accelerated Motion: Derivation and Application

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

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 Displacement Measurements

A team conducts an experiment where a cart's displacement in two perpendicular directions is given b

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

Analysis of Force and Velocity Data

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

Calculus Analysis of a Ramp System

A 10 kg block is pushed up a frictionless ramp by an applied force given by $$F(x)=50 - 4\,x$$ (in n

Calculus‐Based Energy Conversion in Elastic Collisions

Two masses, $$m_1$$ and $$m_2$$, undergo an elastic collision. (a) Derive the conservation equatio

Comparative Analysis of Constant vs. Variable Gravitational Work

An object of mass $$m= 10 \;\text{kg}$$ is lifted from the ground (assume $$x=0$$) to a height of $$

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

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

Elastic Potential Energy in a Spring System

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.25 m from its equilibrium pos

Energy Loss in a Damped Pendulum

A pendulum with a length of 1.5 m and a bob of 0.8 kg experiences damping such that its amplitude de

Experiment on Energy Loss in Frictional Systems

Design an experiment to investigate the relationship between surface roughness and energy loss durin

Experimentally Determining the Effect of Angle on Work Done

A crate is pulled over a horizontal surface with a rope, where the angle of the rope with the horizo

Free‐Fall Impact Energy Experiment

In this experiment, a small cart is dropped from a known height and allowed to free-fall until it im

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 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 8: Pendulum Energy Transformations with Damping

An experimental study on a pendulum claims that its mechanical energy is conserved, assuming only gr

FRQ 16: Evaluating Power Output Measurements in a Rocket Launch

A media report asserts that the power output of a rocket engine can be approximated by the formula $

FRQ 18: Work–Energy Analysis of a Decelerating Elevator

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

Inelastic Collision and Energy Dissipation

Two blocks undergo a perfectly inelastic collision. A 3 kg block moving at 5 m/s collides with a 2 k

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

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.

Motion on an Inclined Plane with Friction

A block of mass 4 kg slides down a 5 m long inclined plane that makes an angle of 20° with the horiz

Oscillatory Motion Energy Exchange Experiment

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

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

Rocket Engine Energy Analysis

A rocket of mass 1000 kg is accelerated by a rocket engine providing a constant thrust of 25000 N. N

Spectroscopic Potential Energy Curve Analysis

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

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) =

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 Against Friction

An 8 kg block slides on a horizontal surface with a kinetic friction coefficient of 0.25. It comes 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-Energy Analysis in Collisions

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

Work-Energy Analysis on an Inclined Plane

A 20 kg block slides down an inclined plane of length 8 m, which is inclined at an angle of 30° rela

Work–Energy and Friction: Analyzing a Sliding Block

A researcher investigates the effect of friction on a sliding block. A 4 kg block is launched horizo

Work–Energy Experiment in Varying Potential Fields

A researcher studies the motion of an object of mass 4 kg in a potential energy field described by $

Work, Energy, and Power in Circular Motion

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

Astronaut Momentum Conservation

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

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 Gravity vs. Center of Mass in a Tilted Rod

A uniform rod is supported on one end by a fulcrum. In an experiment, additional weights were placed

Center of Mass of a Composite Object with a Semicircular Cut-out

A thin, uniform rectangular plate has dimensions $$4\,m \times 3\,m$$. A semicircular section with a

Center of Mass of a Non-uniform Rod

A thin rod of length 1.0 m has a linear mass density given by $$\lambda(x) = 5 + 3*x$$ (kg/m), where

Center of Mass of an L-Shaped Object

An L-shaped object is formed by joining two uniform rods at one end. One rod is horizontal with a le

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

Evaluating Energy Dissipation in an Inelastic Collision

Two vehicles collide and stick together in an inelastic collision. The experimental data below provi

Explosive Separation and Momentum Conservation

A 2 kg projectile traveling at 15 m/s explodes into two equal fragments (1 kg each). One fragment mo

Explosive Separation and Momentum Conservation

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

Force from Potential Energy Graph

A potential energy function for a system is provided in the graph below, where the potential energy

Glancing Collision of Billiard Balls

Two billiard balls, each of mass 0.17 kg, undergo an elastic glancing collision. Initially, Ball 1 m

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

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

Impulse 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 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 and Momentum under a Variable Force

A 5 kg cart on a frictionless track is initially at rest. A variable force given by $$F(t)=10*t$$ (N

Impulse Calculation from Force-Time Graph

A force sensor records a time-dependent force acting on an object. The force is modeled by the equat

Impulse from a Time-Varying Force with Graph Stimulus

A force sensor records the force applied to a hockey puck as a function of time while a player strik

Impulse from a Variable Force Graph

A force acting on an object is given by the time-dependent function $$F(t)=200\,\sin\left(\frac{\pi*

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

Meteor Impact: Conservation of Momentum and Energy Dissipation

A meteor with a mass of 5000 kg is traveling at 20 km/s (20000 m/s) and impacts the Earth, breaking

Momentum and Angular Momentum in a Rotational Breakup

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

Momentum Conservation in Glider Collisions on an Air Track

In a laboratory experiment, students investigate conservation of momentum by allowing two gliders to

Momentum Transfer in Off-Center Collisions on a Frictionless Track

In an experiment, a moving cart collides off-center with a stationary cart on a frictionless track,

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 $

Non-uniform Rod's Center of Mass

A rod of length L = 1.5 m has a non-uniform linear density given by $$\lambda(x) = 4 + 3*x$$ (in kg/

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 Motion with Air Resistance Approximation

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

Projectile Motion with Drag Impulse Analysis

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

Two-Dimensional Collision and Momentum Conservation

Two ice skaters push off each other on a frictionless surface. Skater A (mass $$60\,kg$$) moves with

Analysis of Rotational Equilibrium in a Complex System

A hanging sign is suspended by two cables attached at different points. The sign rotates about a piv

Angular Impulse and Change in Angular Momentum

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

Angular Impulse and Change in Angular Momentum

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

Angular Kinematics of a Rotating Disk

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

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

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

Angular Momentum Transfer in Coupled Rotating Disks

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

Application of the Parallel Axis Theorem

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

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-Based Determination of Angular Displacement

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

Calculus-Based Torque Distribution in a Non-uniform Rod

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

Comparative Analysis of Rotational and Translational Dynamics

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

Comparative Study of Rotational Kinetic Energy in Different Shapes

Design an experiment to compare the rotational kinetic energy in different shaped objects (for examp

Comparison of Rolling Objects with Different Mass Distributions

Consider two cylinders, one solid and one hollow, each of mass $$M$$ and radius $$R$$, that roll wit

Conservation of Angular Momentum in a Figure Skater's Spin

A figure skater rotates with an initial angular velocity $$\omega_0$$ with her arms extended. When s

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

Conservation of Angular Momentum in Rotational Collisions

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

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

Cylinder Rolling Down an Incline

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

Determining the Moment of Inertia of a Compound Pendulum

A compound pendulum, consisting of an irregular rigid body pivoted at different locations, is used t

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 Transfer in Rolling Objects

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

Equilibrium Analysis in a Rotating Beam System

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

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

Graphical Analysis of Angular Motion

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

Gyroscopic Precession and its Dependence on Spin Rate: An Experiment

A spinning wheel mounted on a gimbal is subjected to an applied torque, causing it to precess. The e

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

Investigating Non-uniform Density Effects on Moment of Inertia

Design an experiment to determine the moment of inertia for an object with a non-uniform mass distri

Investigating the Big Five Equations for Rotational Motion

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

Moment of Inertia of a Composite System using Calculus

A composite system consists of a uniform rod of mass $$M$$ and length $$L$$ with two small beads, ea

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

Parallel Axis Theorem Experiment with a Suspended Bar

A student conducts an experiment to determine the moment of inertia of a uniform bar by suspending i

Rolling Motion on an Inclined Plane

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an inclined plane from a

Rotational Dynamics of a Gyroscope

A gyroscope with a moment of inertia of $$I = 0.05\,kg\,m^2$$ is spinning with a spin angular veloci

Rotational 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 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 Inertia of a Uniform Rod

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

Rotational Kinematics on a Spinning Disk

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

Testing the Parallel Axis Theorem

An experiment is conducted on a uniform disk with mass $$M$$ and radius $$R$$. The disk's moment of

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

A solid disk of mass M = 4 kg and radius R = 0.5 m is considered. The moment of inertia about its ce

Advanced Pendulum Oscillator: Beyond the Small-Angle Approximation

For a simple pendulum with a large amplitude, the period deviates from the small-angle approximation

Amplitude Dependence in a Nonlinear Oscillator

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

Analysis of SHM Under Driving Force

A researcher studies a damped, driven harmonic oscillator subjected to an external sinusoidal force

Analyzing Damped Oscillations in a Spring-Mass System

An experiment is conducted in which a spring-mass oscillator is exposed to air resistance, introduci

Analyzing Phase Shift and Amplitude Modulation in SHM

An oscillator’s displacement is given by the equation $$y(t)= (0.05 + 0.01 * t) * \sin(8*t+0.2)$$

Anharmonic Effects in a Pendulum

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

Calculus Application in SHM: Derivatives and Acceleration

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

Calculus Derivative Analysis in SHM

Given an oscillator with a position function $$y = 0.05 \sin(12t + \frac{\pi}{6})$$, where $$y$$ is

Comparative Dynamics of Mass-Spring and Pendulum Oscillators

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

Conservation of Energy: Integral Approach in SHM

Utilize calculus to analyze energy conservation in a simple harmonic oscillator.

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

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 Oscillations: Determining the Damping Coefficient

A mass-spring system oscillates vertically but in a medium that exerts a damping force proportional

Damped Oscillatory Motion Analysis

A mass-spring system undergoing damped oscillations has a damping force given by \(F_d = - b * v\),

Data Analysis and Calculus Estimation in SHM

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

Differential Equation of Coupled Oscillators

A more advanced experiment involves studying two masses attached by springs (coupled oscillators) to

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 Analysis of a Simple Pendulum

A simple pendulum with length \(L = 1.0\,m\) and mass \(m = 0.3\,kg\) is released from rest at an in

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 Exchange in Coupled Oscillators

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

Energy Exchange in Oscillatory Systems

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

Equation of Motion for a Simple Pendulum Beyond Small Angle Approximation

A researcher examines the motion of a simple pendulum without relying on the small-angle approximati

Estimating Spring Constant from Kinetic Energy Measurements

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

Experimental Verification of Hooke's Law

A physics lab setup involves a horizontal spring-mass system to test Hooke’s law. In this experiment

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

A mass-spring oscillator is subject to an external periodic driving force given by $$F(t)=F_0\cos(\o

Fourier Analysis of Oscillation Data

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

FRQ 8: Energy Exchanges in SHM

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

FRQ 8: Energy Transformation in SHM

Consider a mass-spring system undergoing simple harmonic motion where mechanical energy is conserved

FRQ 14: Work Done by a Spring via Integration

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

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 16: Frequency Determination from Oscillatory Data

An experiment records the displacement of a mass undergoing simple harmonic motion at various times.

FRQ 17: Pendulum Nonlinear Effects

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

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

Hooke's Law and Spring Force Calculation

Consider a spring that obeys Hooke’s law, $$F = -k * x$$, where k is the spring constant and x is th

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(

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

Kinematics of SHM: Period and Frequency Measurements

Analyze the kinematics of a simple harmonic oscillator using time measurements.

Lagrangian Mechanics of the Simple Harmonic Oscillator

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

Mass Dependence in Oscillatory Motion

A spring with a force constant of $$k = 300 \; N/m$$ is used in two experiments. In the first, a blo

Mass Variation and Frequency in SHM

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

Maximum Speed and Energy Conservation in SHM

A mass-spring oscillator undergoes simple harmonic motion with displacement given by $$x(t)=A \sin(\

Modeling Nonlinearities in Pendulum Motion

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

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

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

Nonlinear Restoring Force: Effects on the Period of Oscillations

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

Oscillatory Motion of a Block on a Horizontal Spring

A block of mass $$m = 0.8 \; kg$$ is attached to a horizontal spring with a spring constant of $$k =

Phase Shift and Time Determination in SHM

Analyze the effects of phase shift in a sinusoidal oscillator and determine specific times correspon

Phase Space Analysis of SHM

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

Resonance in Forced Oscillations

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

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 Description and Phase Shift in SHM

A spring oscillator has an amplitude of $$A = 0.05\,m$$ and oscillates with a frequency of $$f = 5.0

Sinusoidal Description and Phase Shift in SHM

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

Sinusoidal Motion: Phase Constant Determination

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

Small-Angle Pendulum Experiment

In a physics lab, a small pendulum of length $$L = 0.80\,m$$ is used to study simple harmonic motion

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

Vertical Oscillations: Lab Data Analysis

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

Vertical Spring Oscillator Analysis

A vertical spring with a force constant of $$k = 400\,N/m$$ supports a mass of $$m = 2.0\,kg$$ and r

Analysis of a Gravitational Potential Energy Graph

A graph representing gravitational potential energy as a function of distance is provided below. The

Analysis of Low Earth Orbit Satellite Decay

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

Analyzing a Two-Body Gravitational Interaction Using Calculus

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

Angular Momentum Conservation in Orbital Motion

Angular momentum conservation plays a critical role in determining the properties of orbital motion.

Application of Kepler's Third Law in the Solar System

A table below provides the semi-major axis and orbital period for several planets. Use this data to

Comparative Gravitational Forces among Planet Pairs

Examine the data comparing gravitational forces between different planet pairs. Use the evidence to

Comparison of Gravitational and Centripetal Forces

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

Derivation of the Virial Theorem for Gravitational Systems

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

Determination of Gravitational Constant Using a Compound Pendulum

A compound pendulum experiment is conducted to determine the gravitational constant (G) by measuring

Dynamics of a Falling Object in a Gravitational Field

A mass is dropped from a height in a gravitational field and its motion is tracked to study energy c

Eccentricity and Elliptical Orbits

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

Energy Conservation in Elliptical Orbits

Energy conservation is a key principle in orbital dynamics, particularly in elliptical orbits where

Escape Velocity Derivation

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

FRQ 1: Gravitational Force between Two Masses

Consider two masses interacting gravitationally. An object of mass $$m_1 = 5.98 \times 10^{24} \ kg$

FRQ 8: Elliptical Orbit – Perihelion and Aphelion Distances

An object travels in an elliptical orbit with a semimajor axis $$a$$ and eccentricity $$e$$. Answer

FRQ 12: Designing a Geosynchronous Satellite Orbit

A geosynchronous satellite must orbit the Earth with an orbital period equal to one sidereal day (\(

FRQ 17: Tidal Forces and Differential Gravity

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

Gravitational Field of a Spherical Shell

Using calculus, derive the gravitational field produced by a thin spherical shell of uniform mass 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 Energy Measurement on a Ramp

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

Gravitational Potential Energy Variations near Earth

An object of mass $$m$$ is elevated in Earth's gravitational field. (a) Derive the expression $$PE =

Gravitational Potential via Integration in a Varying Density Sphere

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

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 Period and Semimajor Axis Relationship Using Kepler's Third Law

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

Orbital Perturbations and Precession

Investigate how small perturbative forces lead to the precession of a planet's orbit.

Orbital Transfer and the Hohmann Maneuver

A spacecraft performs a Hohmann transfer to move from a circular orbit of radius $$r_1$$ to a higher

Satellite Maneuver Simulation with Finite Burn Dynamics

An experimental simulation aims to predict a satellite's trajectory during a maneuver that involves

Satellite Orbital Decay with Atmospheric Drag Consideration

An experiment is designed to measure the decay of a satellite's orbit by tracking its altitude over