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

Analyzing Circular Motion: Speed and Acceleration

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

Calculus Analysis of a Parabolic Trajectory

A projectile is launched with the equations of motion given by $$x(t)=10*t$$ and $$y(t)=50*t-4.9*t^2

Calculus-Based Kinematics Derivation

Consider an object moving along a straight line with constant acceleration. Use calculus to derive e

Comparative Analysis of Kinematic Equations

A researcher claims that the kinematic equation $$s=ut+\frac{1}{2}at^2$$ is universally valid for al

Conservation of Energy in Projectile Motion

A projectile is launched from the ground with an initial speed $$v_0 = 50$$ m/s at an angle of $$45°

Dynamics on an Inclined Plane with Friction

A 4.0-kg block is released from rest at the top of a 10.0-m long incline that makes an angle of $$25

Experimental Determination of g

In a free-fall experiment, a student fits the data to obtain the position function $$h(t)= 4.9*t^2$$

Free Fall Kinematics

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

Free Fall with Air Resistance

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

Free-Fall Experiment Analysis

A rock is dropped from a 50-meter high cliff. Neglecting air resistance and using $$g= 9.8\; m/s^2$$

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 5: Projectile from an Elevated Platform (HARD)

A ball is launched from the edge of a cliff 50 m high with an initial speed of $$20\,m/s$$ at an ang

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: Experimental Analysis of Free Fall

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

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 19: Analysis of Motion on an Inclined Plane (MEDIUM)

An object slides down a frictionless inclined plane that makes an angle $$\theta$$ with the horizont

Graphical Analysis of Distance and Displacement

A student records the position of a runner during a 10-second race, resulting in the following table

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

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

Kinematics in a SmartLab Setup: Integration Error

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

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,

Lab Investigation: Effects of Launch Angle on Projectile Range

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

Motion Lab Data Analysis

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

Motion on an Inclined Plane

A block is released from rest on a frictionless incline with an angle of $$30^\circ$$. The accelerat

Motion with Time-Varying Acceleration (Drag Force Approximation)

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

Multi-Dimensional Motion Analysis and Vector Decomposition

An object moves in the plane and its position vector is given by $$\mathbf{r}(t)= (3t^2)\,\mathbf{i}

Multi-Phase Rocket Motion Analysis

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

One-Dimensional Uniform Acceleration Analysis

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

Pendulum Motion and Kinematics

A pendulum of length 2 m oscillates with small amplitude. Its angular displacement is given by $$θ(t

Piecewise Defined Acceleration

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

Projectile Motion from a Cliff

An object is launched from the edge of a 20-meter high cliff with an initial speed of 40 m/s at an a

Projectile Motion on an Inclined Plane

A ball is launched with an initial speed of $$30\,m/s$$ at an angle of $$40^\circ$$ above the horizo

Projectile Motion: Launch from a Moving Platform

A projectile is launched from the top of a train moving at 20 m/s. The projectile is thrown with an

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

Projectile Range Analysis with Angular Misinterpretation

An experiment was conducted to analyze the range of a projectile launched at various angles. A fixed

Simultaneous Measurement of Velocity and Acceleration

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

Slope Analysis in a Velocity-Time Graph

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

Time vs. Position Data Analysis: Initial Conditions Overlooked

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

Vector Addition in Two-Dimensional Projectile Motion

Design an experiment to test the principles of vector addition by analyzing the two-dimensional moti

Vector Decomposition in Displacement Measurements

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

Ballistic Kinetic Energy with Air Resistance

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

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

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 and Hooke’s Law

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

Elastic Potential Energy in a Spring System

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

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 Conservation in Orbital Motion

A satellite of mass 2000 kg is in a circular orbit at a distance of 7000 km from the center of Earth

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

Energy Loss in Inelastic Collisions

Two objects collide and stick together. Object 1 has a mass of 2 kg and is traveling at 4 m/s, while

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

FRQ 7: Roller Coaster Energy Conversion and Energy Loss Analysis

A roller coaster car is reported to convert all its gravitational potential energy into kinetic ener

FRQ 15: Falling Object Speed in a Varying Gravitational Field

A recent study claims that the speed of an object falling in a varying gravitational field can be de

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

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

Horizontal Pulling Work Experiment

A crate is pulled along a horizontal floor by a worker using a rope that makes a constant angle with

Hydraulic Press Work Calculation Experiment

A hydraulic press compresses a metal block. The pressure in the hydraulic fluid varies with displace

Impulse and Energy Transfer via Calculus

A 2-kg object experiences a time-dependent force given by $$F(t)= 4*t$$ N for $$0 \leq t \leq 5\,s$$

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

Optimization of Work in a System with Resistive Force

A particle of mass 2 kg moves along the x-axis under a constant driving force of 10 N and a resistiv

Particle Dynamics in a Variable Force Field

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 12 - 2*x$$ (in newton

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 Curve Analysis

A particle moves in a one-dimensional potential given by $$U(x) = (x-2)^2 - (2x-3)^3$$. A graph of t

Power and Efficiency in a Wind Turbine

A wind turbine with a rotor radius of 40 m extracts energy from wind. The wind speed varies with hei

Power in a Repeated Jumping Robot

A robot of mass 50 kg repeatedly jumps vertically. In each jump, its engine does work to convert kin

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 Sphere Energy Experiment

A solid sphere is rolled without slipping down a tilted ramp, and its kinetic energy is measured at

Rotational Dynamics and Work-Energy in a Disk

A solid disk of mass 10 kg and radius 0.5 m starts from rest. A constant torque of 5 N·m is applied

Rotational Energy Transfer in a Spinning Disc

A disc of mass 2 kg and radius 0.5 m has a moment of inertia given by $$I = \frac{1}{2} m R^2$$ and

Sliding Block Work‐Energy Experiment

In this experiment, a block of mass $$m$$ is released from rest at the top of a frictionless incline

Time-dependent Power and Differential Equations

A machine's power output, $$P(t)$$ in watts, is governed by the differential equation $$\frac{dP}{dt

Time-Varying Velocity and Instantaneous Power Measurement

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

Variable Force Work Calculation

An object moves along the x-axis under the influence of a variable force given by $$ F(x) = 3 * x^2

Variable Gravitational Acceleration Over a Mountain

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

Work and Energy in a Pulley System

A researcher investigates a two-mass system connected by a massless, frictionless pulley. Mass m1 =

Work and Energy on an Inclined Plane with Variable Friction

A 4 kg block slides down a 10 m long incline set at 30° from the horizontal. The frictional force al

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 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-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 Rotational System

A solid disk with moment of inertia $$I = 0.5 \;\text{kg·m}^2$$ is subjected to a variable torque gi

Analysis of an Oblique Collision

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

Analyzing a Force-Time Graph: Impulse and Momentum

A hockey puck of mass 0.15 kg is struck by a hockey stick. The force exerted on the puck during the

Analyzing Momentum Change in a Two-Cart Collision

Two carts on a frictionless track collide inelastically and stick together. Cart A (mass = 2 kg) mov

Assessing the Effects of Impact Duration on Impulse

In an experiment, a baseball is struck with varying impact durations. The impulse delivered during e

Balancing a Composite System's Center of Mass

A thin uniform rod of length $$3$$ m (mass $$1$$ kg) has two point masses attached to it: a $$2$$ kg

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 Non-Uniform Rod

A vertical rod of length 1.0 m has a linear density given by $$\lambda(x)=5+2*x$$ (kg/m), where x is

Center of Mass Analysis in a Two-Mass Pulley System

In a two-mass pulley system, students aim to determine the center-of-mass motion by measuring accele

Center of Mass of a Non-uniform Rod

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

Center of Mass of a 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 Variable Density Two-Dimensional Lamina

Consider a right triangular lamina with a base of length $$b$$ and a height of $$h$$. The density of

Center-of-Mass of a Ladder with Varying Linear Density

A ladder of length $$L=2.0\,m$$ has a vertical linear mass density given by $$\lambda(y)=3.0(1+y)$$

Composite Body Center of Mass Calculation

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

Conservation of Linear Momentum in Colliding Carts

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

Derivation of the Rocket Equation Using Momentum Conservation

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

Experiment Design: Spring-Loaded Impulse Mechanism

A spring-loaded mechanism is used to deliver an impulse to a 0.5 kg cart. The spring has a constant

Experimental Design: Investigating Collision Elasticity

Design a laboratory experiment to compare the kinetic energy retention in elastic and inelastic coll

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 in a Multi‐Stage Rocket

A multi‐stage rocket undergoes an explosive separation. Experimentally, Stage 1 (mass = 2000 kg) is

FRQ 3: Motion of the Center of Mass under External Force

An object of mass $$10 \ kg$$, initially at rest, is subjected to an external force given by $$F(t)

FRQ 12: Graphical Analysis of Force-Time Data

An experiment measured the force on a 2 kg cart as a function of time. The resulting force-time grap

FRQ 15: Center of Mass versus Center of Gravity

A popular science article claims that for a complex-shaped satellite orbiting Earth, the center of m

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 Swing Angle in a Pendulum

A pendulum bob of mass $$1.0\,kg$$ initially at rest is given a horizontal push by a time-dependent

Impulse Delivered by a Variable Force

A particle experiences a time-dependent force along the x-axis given by $$F(t)=4*t^2 - 12*t + 9$$ N

Impulse in a Rebounding Ball

A 0.5-kg ball is dropped from a height of 2 m onto a rigid surface. It rebounds with a speed of 1.2

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/

Impulse-Momentum in Soccer Kick

A soccer player kicks a ball of mass 0.43 kg. A force sensor attached to the player's foot records a

Inelastic Collision on a Frictionless Surface

Two gliders on a frictionless air track collide and stick together. Glider A (mass = 2 kg) moves rig

Motion of the Center of Mass Under External Force

Consider a system consisting of two point masses connected by a light rod. Mass m1 = 2 kg is located

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

Nonuniform Rod: Total Mass and Center of Mass

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

Oblique Collision of Two Billiard Balls

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

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

Rebound Velocity from a Time-Dependent Impact Force

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

Rocket Propulsion and Center of Mass Dynamics

A rocket has an initial total mass of $$M_0 = 5000\;kg$$ and burns fuel such that its mass decreases

Rotational Dynamics of a Composite Object

A composite object consists of two connected rods: Rod A is 1 m long and has uniform density, while

Rotational Impulse in a Spinning Disc Experiment

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

Satellite Debris: Center of Mass and Impulse Effects

In Earth orbit, three pieces of debris are observed. Their properties are recorded in the following

Three-Body Collision on a Frictionless Table

Three particles with masses 1 kg, 2 kg, and 3 kg are initially placed along the x-axis at x = 0 m, 4

Analysis of Rolling Motion on an Incline

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

Analyzing Rotational Equilibrium

A researcher is investigating conditions for rotational equilibrium in a beam subject to multiple fo

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 Changes in a Skater's Spin

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

Angular Momentum Conservation in a Spinning System

Design an experiment to verify the conservation of angular momentum using a rotating platform and mo

Calculus Derivation of Moment of Inertia for a Thin Ring

Derive the moment of inertia for a thin ring of radius $$R$$ and mass $$m$$ using calculus.

Calculus-Based Derivation of Torque from Force Distribution

A beam of length $$L$$ is subject to a continuously distributed force. Consider two cases: (i) const

Centripetal Force and Angular Velocity Measurement

Design an experiment to measure the centripetal force acting on an object in circular motion and rel

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

Determining Angular Acceleration from Time-Resolved Measurements

A researcher measures the angular velocity of a rotating wheel at several time intervals. The follow

Determining the Moment of Inertia of a Non-Uniform Rod

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

Dynamic Analysis of a Gyroscope: Precession

A gyroscope consists of a spinning disk mounted on a pivot. The center of mass is offset from the pi

Effect of Variable Applied Torque on Angular Acceleration

In a controlled experiment, a variable torque is applied to a rotating disk. The resulting change in

Effects of Non-uniform Mass Distribution on Rotational Inertia

A rod of length $$L$$ has a non-uniform mass density given by $$\lambda(x)=\lambda_0 \left(1 + k \fr

Energy Conversion in Rolling Motion Experiments

In an experiment, a sphere rolls without slipping down an inclined plane of height $$h$$. Measuremen

Experimental Investigation of Rolling Without Slipping

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

FRQ 6: Angular Momentum Conservation on a Rotating Platform

A 50.0 kg person stands on a frictionless rotating platform that initially has a moment of inertia o

FRQ 16: Composite Rotational Inertia via Integration

A thin rod of length L = 3.00 m has a linear mass density that varies along its length according to

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

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

Measuring Frictional Torque in a Rotating Apparatus

In this experiment, a rotating apparatus is allowed to decelerate freely due only to friction. By re

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

Rolling Cylinder Down an Incline

A solid cylinder rolls without slipping down an incline. A set of measurements were made at differen

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 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 Impulse and Change in Angular Momentum

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

Rotational Inertia of a Hollow Cylinder vs. a Solid Cylinder

Compare the rotational inertias of a solid cylinder and a thin-walled hollow cylinder, both of mass

Rotational Kinematics: Angular Displacement via Integration

A motor provides an angular acceleration given by $$\alpha(t) = 4 - 0.5*t$$ (in rad/s²) for $$0 \le

Stability Analysis of a Block on a Rotating Platform (Tipping Experiment)

A block is placed on a rotating platform, and the conditions under which the block tips are investig

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-Varying Torque and Angular Acceleration

A researcher is exploring the effects of a time-varying torque on the rotational motion of a rigid b

Torque and Equilibrium: Balancing a Non-Uniform Beam

A beam of length $$L$$ has a non-uniform mass distribution such that its center of mass is located a

Torque on a Lever Arm

A lever arm is used to apply force at an angle. Consider a force of $$50*N$$ applied at an angle of

Advanced Pendulum Oscillator: Beyond the Small-Angle Approximation

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

Analysis of Energy Transitions in an Oscillating Pendulum

An experiment is conducted on a simple pendulum to study the energy transitions between kinetic and

Calculus Application in SHM: Derivatives and Acceleration

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

Comparative Analysis of Horizontal and Vertical Oscillators

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

Comparative Dynamics of Mass-Spring and Pendulum Oscillators

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

Comparing Sinusoidal and Non-Sinusoidal Oscillatory Motion

In some experiments, the oscillatory motion observed may deviate from a pure sinusoid. Consider a sy

Conservation of Mechanical Energy in SHM

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

Coupled Oscillators: Normal Modes and Energy Transfer

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

Damped Oscillations in a Spring-Mass System

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

Dependence of Maximum Speed on Amplitude

For a spring-mass oscillator undergoing simple harmonic motion, analyze how the maximum speed $$v_{m

Derivation of the SHM Differential Equation

Starting from basic principles, derive the differential equation that governs the motion of a mass a

Determination of Spring Constant from Oscillation Data

A researcher collects oscillation data for different masses attached to a spring. The data is summar

Determining Spring Constant Through Oscillation Energy Analysis

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

Driven Oscillations and Resonant Response

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

Elastic Energy and Maximum Speed Calculation

Consider a block attached to a horizontal spring with a spring constant of $$k = 100\,N/m$$. The blo

Elastic Potential Energy and Maximum Speed Calculation

A block of mass $$m = 0.10\,\text{kg}$$ is attached to a spring with a spring constant of $$k = 200\

Energy Analysis and Instantaneous Power in SHM

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

Energy Conservation in a Mass–Spring Oscillator

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

Energy Distribution and Phase Analysis

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

Error Analysis in SHM Measurements

A student conducting an experiment on a mass-spring oscillator records the following period measurem

Evaluating Damped Oscillatory Motion Effects

A mass-spring oscillator with mass $$m = 0.5 \; kg$$, spring constant $$k = 100 \; N/m$$, and dampin

Evaluating Experimental Uncertainties in SHM Measurements

Accurate measurement of the oscillation period is crucial in SHM experiments. In this context, uncer

Evaluating Hooke's Law in Spring Oscillators

A recent media report claims that 'any spring, when compressed by 5 cm, always exerts the same resto

Fourier Analysis of Oscillatory Motion

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

Friction Effects in Horizontal Oscillatory Systems

A block attached to a horizontal spring oscillates with amplitude $$A$$, but friction is present. Th

FRQ 4: Vertical Motion in a Spring–Block System

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

FRQ 4: Vertical Oscillations of a Spring-Block System

A block of mass $$m = 1.5\ kg$$ is attached to a vertical spring with force constant $$k = 300\ N/m$

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: Graphical Analysis of Restoring Force

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

FRQ 20: Oscillator with Time-Varying Mass

Consider a spring-mass system in which the mass varies with time according to $$m(t) = m_0 + \alpha

FRQ1: Hooke’s Law in a Horizontal Spring-Mass System

A horizontal spring with force constant $$k = 250\,N/m$$ is fixed at one end. A block attached to th

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

FRQ17: Calculus-Based Derivation of Velocity and Acceleration in SHM

Consider a mass undergoing simple harmonic motion described by the displacement function: $$x(t)= A

Hooke's Law and Work in Springs

Consider a spring with a spring constant $$k = 200\,N/m$$. A student compresses the spring from its

Horizontal Mass-Spring Oscillator Analysis

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

Horizontal Spring Oscillator: Force and Energy Calculations

A mass is attached to a light spring on a frictionless horizontal surface. The spring has a force co

Impact of Spring Constant Variation on Oscillatory Motion

A researcher studies how varying the spring constant affects the oscillatory motion of a block attac

Mass Variation and Frequency in SHM

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

Nonlinear Effects in a Large-Amplitude Pendulum

A researcher studies the behavior of a simple pendulum at large amplitudes where the small-angle app

Pendulum Motion and the Small Angle Approximation

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

Pendulum Motion: Small Angle Approximation and Beyond

A simple pendulum consists of a mass $$m = 0.3 \; kg$$ attached to a massless rod of length $$L = 1.

Pendulum Oscillation under Small-Angle Approximation

A simple pendulum consists of a bob of negligible size suspended from a pivot by a massless string o

Pendulum Oscillations: Small Angle Approximation

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

Phase Space Analysis of SHM

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

SHM with a Varying Force Constant

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

Simple Pendulum Energy Analysis

Consider a simple pendulum of length $$L$$ and mass $$m$$ undergoing small oscillations. Answer the

Spring Force and Energy Analysis

A spring with a force constant $$k = 350 \; N/m$$ and a natural (unstretched) length of 20 cm is str

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

Time-Dependent Length in a Variable-Length Pendulum

In an experiment, a pendulum has a length that varies with time according to the relation $$L(t)=L_0

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

Vertical Spring-Mass Oscillator Analysis

In this experiment, a block of mass is attached to a vertical spring. After the block reaches equili

Vertical Spring–Block Oscillator Dynamics

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

Analysis of Tidal Forces Acting on an Orbiting Satellite

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

Analyzing Multi-body Interactions in a Three-Body Problem

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

Analyzing Three-Body Gravitational Interactions

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

Calculus Analysis of Gravitational Potential Energy

Gravitational potential energy (GPE) plays a crucial role in systems influenced by gravity. Answer t

Center of Mass Analysis in Two-Body System

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

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 $$m = 5.98 \times 10^{24} \text{ kg}$$,

Derivation of Equations of Motion in a Gravitational Field Using Lagrangian Mechanics

A researcher analyzes the motion of a particle of mass $$m$$ moving radially under the influence of

Derivation of Escape Velocity from Earth's Surface Using Calculus

Using the principle of energy conservation and calculus, derive the expression for the escape veloci

Dynamics of a Binary Star System

Binary stars orbit around their common center of mass. Consider two stars with masses $$M_1$$ and $$

Elliptical Orbits and Eccentricity Calculation

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

Energy Conservation in Central Force Motion

A particle of mass $$m$$ moves under the gravitational influence of a large mass $$M$$. Analyze its

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

Experimental Analysis of Gravitational Acceleration

An experiment was conducted to measure the acceleration due to gravity $$g$$ at various altitudes. (

FRQ 11: Time-Dependent Gravitational Force in Radial Motion

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

Gravitational Assist Maneuver Simulation

Gravitational assist maneuvers, which use the gravity of a planet to alter a spacecraft’s trajectory

Gravitational Field Modeling for Extended Bodies

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

Gravitational Potential Energy Change for a Satellite

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

Gravitational Potential Energy 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)

Gravitational Slingshot Maneuver

A spacecraft performs a gravitational slingshot maneuver around a planet of mass M that is moving wi

Impact of Relativistic Effects on Orbital Motion

Discuss how relativistic effects modify the orbital motion of a planet when it orbits close to a ver

Inferring Mass Distribution of a Galaxy through Orbital Dynamics

The rotation curves of galaxies can reveal information about their mass distribution and the possibl

Investigating Orbital Eccentricity Effects

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

Investigating Tidal Forces and Differential Gravity Effects

Consider a moon orbiting a planet, where tidal forces arise due to the variation in gravitational fo

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

Kepler's Third Law and Satellite Orbits

Using the provided data for satellite orbits, analyze Kepler’s Third Law. Determine the relationship

Non-uniform Gravitational Fields in Planetary Interiors

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

Orbit Transfer and Hohmann Transfer Orbits

A spacecraft is transitioning between two circular orbits using a Hohmann transfer orbit. (a) Descri

Orbital Energy and Conservation Laws

For a satellite in a circular orbit of radius $$r$$ around a planet, the kinetic energy is given by

Orbital Motion of a Satellite

A satellite of mass m is in a stable circular orbit around a planet of mass M at a distance r from t

Orbital Transfer Trajectories and Hohmann Transfers

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

Planetary Orbits and Kepler's Laws

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

Role of Eccentricity in Orbital Dynamics

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

Satellite Maneuver Simulation with Finite Burn Dynamics

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

Simulating Satellite Orbital Decay and Atmospheric Drag

An experimental simulation is set up to study the orbital decay of a satellite due to atmospheric dr

Tidal Forces in Gravitational Fields

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

Variation of Gravitational Force with Distance

Newton’s Law of Gravitation indicates that the gravitational force between two masses varies inverse

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