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 $$x(t)= 3*t^4 - 2*t^2 + t$$.

Air Resistance and Projectile Motion

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

Analysis of a Complex Position Function

An object has a position defined by $$x(t) = e^{-t} \sin(t)$$ (in meters) for t ≥ 0. Answer the foll

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

Analyzing a Parabolic Trajectory

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

Calculus in One-Dimensional Kinematics

Consider an object whose position as a function of time is given by $$x(t)=\sin(t)+t^2$$, where x is

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

Designing a Kinematics Lab Experiment

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

Determination of Acceleration Due to Gravity

A student drops a small metal ball from a 45 m high platform and records its height over time using

Determination of Maximum Height in Projectile Motion

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

Determining Instantaneous Acceleration from a Displacement Graph

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

Determining Launch Angle from Experimental Data

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

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

Effect of Initial Velocity on Displacement

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

Evaluating an Experimental Claim on Presumed Uniform Acceleration

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

Free Fall with Air Resistance

A 2.0-kg object is dropped from rest and falls under gravity while experiencing air resistance propo

FRQ 1: Constant Acceleration Experiment

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

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 8: Vector Addition in Two-Dimensional Motion

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

FRQ 9: Application of the Big Five Equations

An object starts with an initial velocity of 8 m/s and reaches a final velocity of 20 m/s after trav

FRQ 10: Comparative Analysis of Two Cars with Different Acceleration Profiles

A researcher compares the motion of two cars starting from rest. Car A accelerates at a constant rat

FRQ 11: Air Resistance and Terminal Velocity

An object of mass 2 kg is falling under gravity while experiencing air resistance proportional to it

FRQ 11: Modeling Motion with Differential Equations

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

FRQ 14: Analysis of a Parabolic Projectile Trajectory (MEDIUM)

The vertical motion of a projectile is described by the equation $$h(t)=-4.9*t^2+20*t+5$$ (in m). (a

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 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 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)=

Impulse and Variable Force Analysis

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

Motion with Time-Varying Acceleration (Drag Force Approximation)

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

Non-linear Position Function Analysis

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

Optimization of Projectile Range

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

Piecewise Defined Acceleration

A particle moves along a frictionless surface with a piecewise defined acceleration given by: For $

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 Motion Experimental Investigation

A student investigates projectile motion using a spring-loaded launcher on an elevated platform. The

Projectile Motion with Drag

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

Relative Displacement in Different Frames

A particle moves along the x-axis and its position is described by $$x(t)=7*t-2*t^2$$ (in meters). T

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

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

Two-Dimensional Projectile with an Elevated Launch Point

A ball is thrown from the edge of a cliff that is 20 m above the ground with an initial speed of 30

Uniformly Accelerated Free Fall Analysis

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

Uniformly Accelerated Motion on a Track

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

Uniformly Accelerated Motion With Non-Zero Initial Velocity

An object moves along a straight path with an initial velocity of $$u=5\,m/s$$ and a constant accele

Variable Acceleration and Integration

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

Verification of Uniformly Accelerated Motion

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

Analysis of Elastic and Inelastic Collisions

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

Analysis of Force and Velocity Data

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

Analysis of Mechanical Advantage and Work in a Lever System

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

Conservation of Energy in a Roller Coaster

A 500 kg roller coaster car is released from rest at the top of a hill 50 m above the bottom of a di

Damped Oscillations and Energy Dissipation in a Mass-Spring System

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

Derivation of the Work-Energy Theorem

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

Elastic Potential Energy and Hooke’s Law

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

Energy Analysis of a Bouncing Ball: Energy Loss and Power

A ball of mass $$m = 0.5 \;\text{kg}$$ is dropped from a height of $$10 \;\text{m}$$ and, after its

Energy Loss in Inelastic Collisions Experiment

Two carts on a frictionless track undergo a collision. Cart A (1 kg) moves at 4 m/s and collides wit

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

FRQ 2: Work Done by a Variable Force on a Crate Along a Horizontal Floor

A crate is pulled along a horizontal surface with an applied force that varies with displacement. Th

FRQ 2: Work-Energy Theorem in Lifting

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

FRQ 4: Potential Energy Curve Analysis for a Diatomic Molecule

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

FRQ 11: 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: Analysis of Cyclist Power Output During a Ride

A cyclist’s performance is monitored by a power meter during a ride. The measured power output over

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 17: Energy Loss Analysis in a Frictional Pendulum

A pendulum bob with a mass of 0.8 kg is released from an initial height corresponding to a potential

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 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

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

Gravitational Potential Energy and Free Fall

A 60-kg acrobat climbs to the top of a 50-m tall platform and then jumps off. Neglecting air resista

Gravitational Potential Energy Conversion

A 10 kg object is dropped from a height of 15 m in a vacuum. Using conservation of energy methods, a

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

Integration of Work in a Variable Gravitational Field

A researcher is analyzing the work done on a satellite of mass $$m = 1000\,kg$$ moving away from a p

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

Model Rocket Power Measurement Experiment

In this experiment, a model rocket’s engine power output is determined by measuring its constant spe

Multi‐Phase Cart Energy Experiment

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

Non-Uniform Gravitational Field Work-Energy Calculation

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

Nonlinear Spring Energy Experiment

In an experiment with a nonlinear spring, a mass is attached and the force is measured as a function

Potential Energy Curve of a Diatomic Molecule

The interatomic potential energy of a diatomic molecule can be approximated by the function $$U(r) =

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

Relationship Between Force and Potential Energy

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

Roller Coaster Energy Analysis

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

Rolling Motion on an Incline: Combined Energy Analysis

A solid sphere with mass 3 kg and radius 0.2 m rolls without slipping down an inclined plane of heig

Rotational Motion Work–Energy Experiment

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

Sliding Block on an Incline with Friction

A 5-kg block is released from rest at the top of an incline that is 10 m high. The incline is 20 m l

Work and Energy in Circular Motion

A 2-kg mass is attached to a string and constrained to move on a frictionless vertical circular path

Work Done Against Friction on an Inclined Plane

A 5-kg box slides down a 10-m long inclined plane that makes an angle of 30° with the horizontal. Th

Work Done by a Variable Gravitational Force

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

Work Done in a Variable Gravitational Field

A satellite of mass 500 kg is to be raised from an altitude of 200 km to 400 km above Earth's surfac

Work with a Variable Force on a Straight Path

A particle experiences a variable force along the x-axis given by $$F(x)= 10 + 3*x \; (\text{N})$$.

Work-Energy Analysis 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 Theorem in a Non-Uniform Gravitational Field

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

Work-Energy Theorem with Air Resistance

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

Astronaut Recoil in Space

An astronaut with a total mass of 90 kg, initially at rest relative to her shuttle, throws a 2 kg to

Center of Mass Determination for a Composite L-Shaped Object

Students perform an experiment to determine the center of mass of an L-shaped, composite board. The

Center of Mass in a Coupled Mass-Spring System

Two masses (m1 = 1.5 kg and m2 = 1.5 kg) are connected by a light spring on a frictionless track. In

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 of a Nonuniform Circular Disk

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

Center of Mass of a Triangular Plate

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

Center-of-Mass Motion Under an External Force

Consider a system composed of two masses, $$m_1=2.0\,kg$$ and $$m_2=3.0\,kg$$, connected by a light,

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

Determination of an Unknown Mass via Collision Data

A moving ball of known mass collides with a stationary ball of unknown mass. The experimental data a

Elastic Collision: Conservation of Momentum and Kinetic Energy

Two spheres A and B collide elastically. Sphere A has a mass of 0.6 kg and an initial velocity of $$

Explosive Fragmentation: Momentum Transfer

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

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

Fragmentation and Impulse

A stationary explosive device of total mass $$5\,\text{kg}$$ breaks into two fragments. One fragment

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

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

FRQ 7: Inelastic Collision Analysis

Two carts collide on a frictionless track and stick together. Cart X (mass = 2 kg) moves at +3 m/s a

FRQ 8: Center of Mass and Stability

A uniform beam of length $$5 \ m$$ and mass $$50 \ kg$$ is hinged at one end and held horizontally b

FRQ 10: Collision with Rotational Motion

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

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 on a Rolling Soccer Ball with Piecewise Force

A soccer ball of mass $$0.43\,kg$$ is rolling and experiences friction that acts in two stages: a co

Impulsive Collision Involving Rotation

A 0.5 kg ball traveling at $$8\,m/s$$ horizontally strikes the end of a thin uniform rod of length $

Inelastic Collision with a Movable Platform

A 0.3 kg ball moving horizontally at 8 m/s collides with and sticks to a 1.2 kg movable platform tha

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 Analysis in Explosive Fragmentation Simulation

In a simulation of explosive fragmentation, a stationary container bursts into several fragments. Hi

Momentum Change under a Non-Constant Force

A 1.2-kg block on a frictionless surface is subjected to a time-varying force given by $$F(t) = 12*s

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 Center of Mass Under External Force

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

Multi-Dimensional Inelastic Collision Analysis

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

Multiple Particle Center of Mass in Two Dimensions

Four particles in a plane have the following properties: Particle A (2 kg) at (0, 0), Particle B (3

Non-Uniform Rod Analysis

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

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/

Sequential Collisions in One Dimension

A 1-kg cart moving at $$4\,\text{m/s}$$ collides elastically with a stationary 0.5-kg cart. Immediat

Spring-Loaded Collision with Impulsive Force

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

Structural Stability: Crane Center of Mass Analysis

An engineer is analyzing a crane consisting of a 500-kg horizontal beam whose center of mass is at 5

Time-Varying Force on a Block

A 2 kg block on a frictionless surface is subjected to a time-varying force given by $$F(t) = 15*\si

Two-Dimensional Collision and Momentum Conservation

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

Two-Stage Collision in Coupled Carts

Two carts on a frictionless track undergo a two-stage event. Initially, cart A (mass $$2\,kg$$, velo

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

Angular Kinematics on a Rotating Platform

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

Angular Momentum and Torque in Circular Motion

A particle of mass m moves in a circle of radius r with an angular velocity given by $$\omega(t)= 3t

Angular Momentum Conservation on a Merry-Go-Round

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

Angular Momentum 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

Designing a Rotational Experiment Using a Pulley System

A researcher designs an experiment to measure the rotational inertia of a pulley using a falling mas

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

Effect of Friction on Rotational Motion

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

Energy Analysis in Rolling Motion

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

Engine Torque Measurement Analysis

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

Experimental Determination of Torsional Oscillations

Design an experiment to measure the torsional oscillation period of a rod suspended by a wire with a

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 19: Oscillatory Rotational Motion (Torsional Pendulum)

A torsional pendulum consists of a disk suspended by a wire. The restoring torque is given by $$\tau

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 Mass Distribution on Angular Acceleration

An experiment varies the mass distribution of a rotating rod under a constant applied torque. The ta

Investigation of Torque in a Lever System

In this experiment a rigid lever, pivoted at one end, is used to measure the torque generated by a c

Lever Arm Torque Calculation

A lever arm rotates about a fixed pivot. A force of 50 N is applied at a point 0.8 m from the pivot,

Moments of Inertia for Point Masses on a Rod

Three beads, each of mass 2 kg, are fixed on a massless rod of length 1.2 m. In one configuration, o

Net Torque and Angular Acceleration Calculation

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

Non-Uniform Angular Acceleration

A disk has an angular acceleration described by the function $$\alpha(t) = 2 * t$$ (in rad/s²), and

Non-uniform Rotational Acceleration: Differentiation from Graph

A rotating disk exhibits a non-uniform angular velocity as a function of time. The experimental grap

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 Energy Analysis

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

Rolling Motion on an Inclined Plane

You are tasked with investigating the energy conversion in rolling motion. Design an experiment usin

Rotational Impulse and Change in Angular Momentum

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

Static Equilibrium of a Beam

A uniform beam of length 4 m and weight 200 N is supported at one end by a wall and held horizontal

Time-dependent Torque and Angular Momentum Change

A disk is subjected to a time-dependent torque described by $$\tau(t) = 10 * \cos(t)$$ N*m. The disk

Time-Dependent Torque and Angular Motion

A rotating system is subjected to a time-dependent torque given by $$\tau(t) = \tau_0*e^{-k*t}$$, wh

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

Torque, Friction, and Rotational Equilibrium in a Pulley

A pulley with radius 0.3 m and moment of inertia 0.15 kg m^2 is subjected to a tangential force of 2

Torsional Oscillator Analysis

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

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

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 and Maximum Speed Relationship in SHM

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

Analyzing the Half-Cycle Method in Oscillation Experiments

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

Calculus Approach to Maximum Velocity in SHM

Consider an oscillator whose displacement is given by the sinusoidal function $$y(t) = A \sin(\omega

Comparative Analysis of Horizontal and Vertical SHM Systems

A researcher compares the oscillations of a mass attached to a spring in horizontal and vertical con

Comparative Analysis of Horizontal vs Vertical Oscillations

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

Comparative 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 Oscillatory Systems: Spring vs. Pendulum

A mass-spring system (with mass $$m$$ and spring constant $$k$$) and a simple pendulum (with length

Comparison of SHM in Spring and Pendulum

Compare the simple harmonic motions of a mass-spring oscillator and a simple pendulum (under the sma

Damped Oscillations in a Spring-Mass System

In an experiment exploring damped oscillations, a block attached to a spring is set oscillating on a

Data Analysis of Damped Oscillations

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

Derivation and Solution of the Differential Equation for SHM

Starting from Newton's second law, derive the differential equation governing the motion of a spring

Deriving Equations for a Damped Harmonic Oscillator

An experiment is designed to study the effects of damping in a spring-mass oscillator. This version

Deriving the Equation of Motion Using Calculus

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

Determining Oscillation Frequency from Acceleration Data

An accelerometer attached to a mass-spring system records acceleration data during oscillations. The

Determining Spring Constant Through Oscillation Energy Analysis

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

Driven Oscillator and Resonance

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

Energy Analysis and Instantaneous Power in SHM

A block of mass $$m = 0.1\,kg$$ is attached to a spring with force constant $$k = 800\,N/m$$ and osc

Energy Conservation in a Spring Oscillator

A block of mass $$m = 0.2\,kg$$ oscillates horizontally on a frictionless surface attached to a spri

Energy Distribution and Phase Analysis

An experiment on a spring-mass oscillator is conducted to study the distribution of kinetic and pote

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

Evaluating the Role of Calculus in Predicting Oscillator Dynamics

A theoretical paper asserts that 'integrating the equations of motion always leads to an accurate pr

Experimental Determination of Spring Constant

In a lab experiment, students measure the displacement of a spring under various applied forces. The

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 1: Hooke’s Law Experiment

In a laboratory experiment, the restoring force of a spring was measured for various displacements f

FRQ 1: Spring Force Calculation Using Hooke's Law

A spring has a natural length of $$0.12\ m$$ and a force constant of $$k = 400\ N/m$$. When the spri

FRQ 3: Determining Period and Frequency

An oscillating block moves from its position of maximum stretch to its maximum compression in $$0.25

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

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

FRQ3: Kinematics of SHM – Period and Frequency

A block-spring oscillator is observed to take 0.4 s to travel from its maximum displacement in one d

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.

Integration of Variable Force to Derive Potential Energy

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

Investigating Damping Effects in a Spring-Mass Oscillator

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

Mass Dependence in Oscillatory Motion

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

Mass-Spring Differential Analysis

Consider a block of mass $$m$$ attached to a horizontal spring with spring constant $$k$$. The block

Measuring the Spring Constant: An Experimental Investigation

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

Modeling Amplitude Reduction Due to Non-Conservative Forces

In a real oscillatory system, non-conservative forces (like friction) result in an energy loss per c

Pendulum Angle Dependence and the Small Angle Approximation

A recent news article claims that 'the period of a pendulum is completely independent of the amplitu

Pendulum Energy Dynamics

Analyze the energy dynamics of a simple pendulum using both theoretical derivations and numerical ca

Pendulum Motion Beyond the Small-Angle Approximation

For a pendulum, the restoring force is accurately given by $$mg\sin(\theta)$$ rather than $$mg\theta

Pendulum on a Rotating Platform

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

Phase Shift Analysis in Driven Oscillators

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

Phase Space Trajectories in Simple Harmonic Motion

Phase space diagrams (plots of velocity vs. displacement) offer insight into the dynamics of oscilla

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 Analysis of SHM with Phase Shift

Examine a sinusoidally described simple harmonic oscillator with a phase shift.

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

Stress Testing of Oscillatory Limits

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

Systematic Error Analysis in SHM Experiments

The table below shows measured time intervals and displacements from several trials in an experiment

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

Analyzing Gravitational Slingshot Maneuvers

A spacecraft uses a gravitational slingshot maneuver around a planet to gain additional speed for an

Analyzing Tidal Forces in a Two-Body System

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

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-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.

Calculating Gravitational Potential in a Non-Uniform Planet

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

Calculus Derivation of Gravitational Potential Energy

Derive the expression for gravitational potential energy using calculus and compare your result to e

Comparative Analysis of Planetary Orbits

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

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

Deriving Gravitational Force from Gravitational Potential Energy

In a region where the gravitational potential energy between two masses is given by $$U(r) = -\frac{

Designing a Satellite's Stable Orbit

A space agency plans to deploy a satellite into a stable circular orbit around a planet. The gravita

Determination of Gravitational Parameter (GM) from Satellite Orbits

An observational study is undertaken to determine the gravitational parameter (GM) of a planet by an

Determining Gravitational Potential from Force Field Data

An experiment measures the gravitational force as a function of distance, providing data described b

Effects of Eccentricity on Planetary Orbits

A series of simulations have been conducted for planetary orbits with varying eccentricity. The foll

Energy Balance at Apoapsis and Periapsis

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

Energy Conservation in Gravitational Fields: Application to Roller Coaster Dynamics

Although gravitational potential energy is most famously applied in celestial mechanics, the concept

Escape Velocity and Energy Requirements

A spacecraft must reach escape velocity to leave a planet's gravitational field. The escape velocity

Escape Velocity Derivation

A spacecraft of mass m is located on the surface of a planet with mass M and radius R. Using energy

FRQ 4: Gravitational Potential Energy in Satellite Orbits

A satellite of mass $$m = 1000 \ kg$$ orbits the Earth. The gravitational potential energy of a sate

FRQ 14: Work Done in Changing Orbital Radius

The work done against gravity in changing the orbital radius of an object is computed by integrating

Gravitational Energy in a Three-Body System

Consider three point masses m1, m2, and m3 placed at the vertices of a triangle. Analyze the gravita

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 Change in an Elliptical Orbit

A satellite moves in an elliptical orbit around Earth. Its gravitational potential energy is given b

Gravitational Slingshot and Energy Gain

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

Kepler's Laws and Orbital Dynamics

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

Kepler's Third Law and Satellite Orbits

Kepler’s Third Law, when applied to satellites in nearly circular orbits, can be used to relate the

Orbit Stability from Potential Energy Diagrams

Analyze the provided potential energy diagram and determine the regions corresponding to stable and

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 Perturbations from Impulsive Thrust

A satellite in a circular orbit of radius $$r$$ receives an impulsive radial thrust with magnitude $

Role of Eccentricity in Orbital Dynamics

Orbital eccentricity (e) quantifies how much an orbit deviates from a circle. (a) Define orbital ec

Stability Analysis of a Satellite in Low Earth Orbit

A satellite is in a circular low Earth orbit at an altitude of $$h = 400 \ \text{km}$$. Answer the f

Tidal Forces in Gravitational Fields

An extended object in a gravitational field experiences a differential force across its length, know

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