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 Velocity vs. Time Graph in a Lab Experiment

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

Analysis of a Velocity-Time Graph

A velocity vs. time graph for an object moving in one dimension displays a linear increase in veloci

Analyzing Motion with a Nonlinear Acceleration Function

A particle moves along the x-axis with an acceleration given by $$a(t)=4\cos(t)$$ (m/s²). It has an

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

Calculus-Based Kinematics Derivation

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

Comparing Theoretical and Experimental Data in Uniform Acceleration

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

Designing a Kinematics Lab Experiment

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

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 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'

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

Free Fall Kinematics

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

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 1: Constant Acceleration Experiment

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

FRQ 1: One‐Dimensional Constant Acceleration

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

FRQ 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 4: Velocity-Time Graph Analysis (EASY)

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

FRQ 6: Motion on an Inclined Plane

A researcher studies the motion of a block sliding down an inclined plane with friction. The block i

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 11: Air Resistance and Terminal Velocity

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

FRQ 14: Work and Energy in Kinematics – Rolling Ball on an Incline

A researcher is studying a ball rolling down an inclined plane with friction. In addition to the kin

FRQ 16: Integration of a Decaying Velocity Function (HARD)

An object has a velocity function given by $$v(t)=4*e^{-t}-2$$ (in m/s) for $$t\ge0$$, and its initi

FRQ 17: Vector Field Analysis – Wind Impact on Projectile Motion

A researcher examines how a time-varying wind affects the horizontal motion of a projectile. In this

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: Variable Acceleration – An Object Under Changing Forces

A researcher studies an object whose acceleration varies with time according to the function $$ a(t)

Impact Analysis: Collision Avoidance

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

Kinematic Analysis of a Cyclist

A cyclist starts from rest and accelerates uniformly at 1.5 m/s² for 10 seconds, then rides at a con

Kinematics in a SmartLab Setup: Integration Error

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

Kinematics with Non-Constant Acceleration

An object moves along a straight line with a time-dependent acceleration given by $$a(t)=3t - 2\,m/s

Motion Lab Data Analysis

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

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}

Pendulum Energy Conservation Experiment

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

Projectile Motion and Calculus Analysis

A ball is thrown from a platform. Its horizontal and vertical positions are given by $$x(t)=20*t$$ a

Projectile Motion using Calculus

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

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 Experiment

Two carts on parallel tracks move in the same direction with positions given by $$x_A(t)=5*t$$ and $

Simultaneous Measurement of Velocity and Acceleration

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

Skydiver with Air Resistance: Variable Acceleration

A skydiver of mass m experiences air resistance proportional to velocity, characterized by the const

Terminal Velocity Experiment

An experiment involves dropping objects of varying shapes from a tall building to study terminal vel

Terminal Velocity in Free Fall

Design an experiment to determine the terminal velocity of an object in free fall within a fluid med

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

Variable Acceleration: Analyzing a Non-Uniformly Accelerated Motion

An object moves along a straight path with an acceleration given by $$a(t)=\frac{12}{(t+2)^2}$$ m/s²

Vector 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 Fall Dynamics with Air Resistance

An object with a mass of 0.5 kg is dropped from a height of 50 m. Air resistance is modeled as a dra

Calculus‐Based Work Calculation with Constant Force

A constant force of 20 N acts along the direction of displacement over a distance of 3 m. Use calcul

Collision and Energy Loss Analysis

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

Comparative Analysis of Constant and Variable Force Work

Two experiments are performed on a 3 kg object. In Experiment 1, the object is moved under a constan

Conservation of Mechanical Energy in a Pendulum

A simple pendulum consists of a bob attached to a massless string of length 2 m. The bob is pulled a

Determining Instantaneous Power from a Velocity-Time Graph

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

Energy Analysis of a Damped Spring-Mass Oscillator

A spring-mass system consists of a mass $$m = 2 \;\text{kg}$$ attached to a spring with force consta

Energy Conservation in a Pendulum

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

Energy Dissipation in a Bouncing Ball

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

Energy Dissipation in an Oscillatory System

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

Energy Loss Due to Position-Dependent Friction

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

FRQ 6: Work Done on a Crate on an Inclined Plane

A 20-kg crate is moved up a 30° inclined plane. Experimental measurements of the force component alo

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 11: Deriving Force from a Potential Energy Function

A study posits that the force acting on a particle can be obtained via $$F(x) = - \frac{dU}{dx}$$. E

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 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

Instantaneous and Average Power in a Variable Force System

A block is subjected to a variable force and its velocity varies with time. The force acting on the

Interpreting a Diagram of Work–Energy Processes

A detailed diagram is provided that illustrates a block sliding down an inclined plane with friction

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

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

Power and Energy Efficiency in a Conveyor Belt Experiment

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

Power Output Fluctuations in a Jogger

A 70 kg jogger runs along a track with an instantaneous velocity given by $$v(t) = 2 + 0.5\,t$$ (in

Power Output Measurement in an Elevator Experiment

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

Projectile Energy Analysis with Air Resistance Correction

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

Projectile Motion and Energy Conservation

A 0.5 kg projectile is launched from ground level with an initial speed of 20 m/s at an angle of 60°

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

Roller Coaster Energy Transformation Experiment

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

Spectroscopic Potential Energy Curve Analysis

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

Spring Energy Experiment: Measuring Nonlinear Work

A spring exhibits a force-displacement relationship given by $$F(x) = k*x + a*x^3$$, where $$k = 50\

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 Non-uniform Gravitational Field

An object of mass 500 kg is moved radially outward from the Earth. Assume the Earth’s mass is $$M =

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 with Air Resistance

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

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

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

Astronaut Recoil upon Throwing an Object

An astronaut of mass 90 kg, floating in space, throws a 1.5 kg tool directly away from her ship at 5

Billiard Ball Collision and Impulse Analysis

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

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 of an Irregular Lamina in Polar Coordinates

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

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

Complex Rotational and Translational Collision Involving Center of Mass

A uniform rod of length $$2$$ m and mass $$4$$ kg is pivoted frictionlessly about its center. A smal

Conservation of Angular Momentum on a Rotating Platform

An ice skater of mass 50 kg spins with arms extended, having a moment of inertia of 3 kg·m² and an a

Elastic Collision on Air Track

Two gliders are on a near-frictionless air track. Glider A (mass = 0.8 kg) is traveling to the right

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 $$

Experimental Design: Investigating Collision Elasticity

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

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 2: Center of Mass of a Composite Lamina

Consider a composite lamina composed of two rectangular regions in the xy-plane. Region A is 0.8 m b

FRQ 13: Critical Analysis: Momentum Experiment

A research study investigating momentum transfer in vehicle collisions reports that the measured mom

Glancing Collision of Billiard Balls

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

Glider Collision on an Air Track

Two gliders on a frictionless air track collide and stick together. Glider A (mass = $$0.50\,\text{k

Impulse Analysis with Error Bars

In an experiment, a variable force acting on a cart is measured and described by the equation $$F(t)

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 Work: Discerning Differences

A particle of mass 3 kg is subjected to a force that varies with position according to $$F(x)=12*x^2

Impulse Delivered by a Decreasing Force from a Water Jet

A water jet impinging on a stationary nozzle exerts a time-varying force given by $$F(t)=50 - 10*t$$

Impulse Delivered by Variable Thrust Rocket

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

Impulse in a Non-Constant Force Field

A particle moves through a region where the force depends on position, described by $$F(x) = 3 * x$$

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 Theorem with a Non-constant Force

A 0.2 kg hockey puck is struck by a mallet. The force applied by the mallet varies with time and is

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 on a Frictionless Surface

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

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 Analysis in an Asteroid Breakup

An asteroid of total mass 1000 kg fragments into two pieces in deep space. Fragment A (mass unknown)

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

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

Off-Center Collision and Angular Momentum

A small ball (mass $$0.5\,kg$$) moving at $$4\,m/s$$ strikes a uniform rod (mass $$3\,kg$$, length $

Oscillations: Simple Pendulum Analysis

For a simple pendulum of length 1.5 m undergoing small oscillations, the angular displacement is giv

Recoil Dynamics in a Firearm Event

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

Rigid Body Dynamics: Torque and Rotation

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

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

Satellite Debris: Center of Mass and Impulse Effects

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

Stability Analysis Using Center of Mass on a Pivoted Beam

A uniform beam of length 2.0 m and mass 10 kg is pivoted at one end. A 20 kg mass is suspended from

Stability and Center of Mass of a Structure

A T-shaped structure is formed by joining a horizontal rod (mass $$6\,kg$$, length $$2\,m$$) at its

Stability of a Suspended Mobile

A suspended mobile consists of three masses hanging from strings attached to a horizontal bar. The m

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

Two-Stage Collision in Coupled Carts

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

Angular Kinematics with Variable Angular Acceleration

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

Angular Momentum Conservation on a Rotating Platform

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

Angular Momentum Transfer in Colliding Rotational Bodies

A student studies collisions between two rotating bodies—a spinning disk and a smaller rotating whee

Application and Critical Review of the Parallel Axis Theorem

A disk of radius $$R$$ and mass $$M$$ has a moment of inertia about its center of mass given by $$I_

Calculation of Rotational Inertia for Composite System

A uniform disk of mass $$M = 2\,kg$$ and radius $$R = 0.5\,m$$ has two small beads, each of mass $$m

Derivation of Angular Kinematics Equations

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

Derivation of the Moment of Inertia for a Thin Rod

A uniform thin rod of length $$L$$ and mass $$M$$ rotates about an axis perpendicular to the rod thr

Differentiating Between Contact and Rolling Without Slip

An experiment is conducted to analyze the motions of objects rolling down an inclined plane. Both th

Discrete Mass Distribution and Moment of Inertia

A set of three small beads, each of identical mass $$m$$, is arranged along a light rod of length $$

Dynamics of a Rotating System with Friction

A rotating disk with moment of inertia $$I$$ is subject to a frictional torque that is proportional

Effect of Force Angle on Measured Torque

An experiment is performed in which a force of constant magnitude $$F = 50\,N$$ is applied at a cons

Energy Analysis in Rolling Motion

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

Experimental Investigation of Rolling Without Slipping

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

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

FRQ 4: Rotational Kinematics of a Disk

A disk starts from rest and rotates with a constant angular acceleration. After t = 4.00 s, the disk

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,

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

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

Rolling Motion Energy Conversion Experiment

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

Rolling Motion on an Inclined Plane

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

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 Kinematics: Non-Uniform Angular Acceleration of a Disk

A disk rotates such that its angular velocity is given by $$\omega(t) = 3*t^2 - 2*t + 1$$ (in rad/s)

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

Testing the Parallel Axis Theorem

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

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

Torque Measurement and Analysis

A recent experimental study claims that the relationship between force and torque is strictly linear

Torque on a Uniform Rod with Distributed Force

A researcher is studying the effect of a distributed force along a uniform rod of length $$L$$ pivot

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

Calculus Application in SHM: Derivatives and Acceleration

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

Calculus-Based Analysis of Velocity and Acceleration

Consider an oscillator whose displacement is defined by: $$x(t) = 0.1 * \sin(6*t)$$ (a) Differenti

Calculus-Derived Velocity and Acceleration in SHM

For a simple harmonic oscillator described by $$x(t)=A\sin(\omega t+\phi)$$, determine its velocity

Comparative Analysis: Spring-Mass System vs. Pendulum Oscillations

A researcher compares the oscillatory behavior of a horizontal spring-mass system with that of a sim

Comparative Energy Analysis: SHM vs. Pendulum

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

Comparing Sinusoidal and Non-Sinusoidal Oscillatory Motion

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

Comparison of Horizontal and Vertical Oscillations

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

Conservation of Energy: Integral Approach in SHM

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

Conservation of Mechanical Energy in SHM

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

Coupled Oscillators Investigation

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

Coupled Oscillators: Two Springs in Parallel

A block is attached to two springs arranged in parallel with spring constants $$k_1 = 250\,\text{N/m

Damped Oscillations and Energy Decay

A mass-spring system with viscous damping is described by the differential equation $$m*\frac{d^2y}{

Damped Oscillations: Determining the Damping Coefficient

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

Data Analysis of a Spring-Mass Experiment

A researcher experiments with a mass-spring system and records the period of oscillation for differe

Designing an SHM Experiment with Error Analysis

A researcher intends to study the simple harmonic motion of a pendulum using an optical sensor to re

Determination of Gravitational Acceleration Using a Vertical Oscillator

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

Determination of Maximum Elastic Potential Energy

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

Determination of Spring Constant from Oscillation Data

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

Determination of the Damping Coefficient from Amplitude Decay

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

Determination of the Spring Constant from Experimental Force-Displacement Data

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

Determining Oscillation Frequency from Acceleration Data

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

Determining the Phase Constant from Experimental Data

An experiment measuring the displacement of a simple harmonic oscillator produced the following data

Dynamic Equilibrium in a Vertical Oscillator

A researcher studies the oscillatory motion of a block attached to a vertical spring. The block is d

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 Distribution and Phase Analysis

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

Energy Exchange in Coupled Oscillators

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

Error Analysis in SHM Measurements

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

Fourier Analysis of Oscillatory Motion

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

FRQ 1: Hooke’s Law Experiment

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

FRQ 7: Calculus Application in SHM

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

FRQ 8: Energy Transformation in SHM

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

FRQ 13: Determining Angular Frequency from Oscillation Data

An oscillator’s motion is described by $$y = A \sin(\omega t + \phi_0)$$. A set of position measurem

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

FRQ8: Comparing Spring-Mass and Pendulum Oscillators

Compare two classic oscillatory systems: a horizontal spring-mass oscillator (with restoring force $

FRQ10: Damped Oscillations – Amplitude Decay and Velocity Derivation

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

FRQ16: Resonance in a Driven, Damped Oscillator

A damped oscillator is subjected to an external periodic driving force of the form $$F_d \cos(\omega

FRQ18: Effect of Mass Variation on the Oscillation Period

The period \(T\) of a mass-spring oscillator is given by the formula: $$T = 2\pi\sqrt{\frac{m}{k}}.

Hooke’s Law and Work in Spring Systems

A spring with a spring constant $$k = 250\,N/m$$ is initially at its natural length. (a) Using Hooke

Influence of Initial Phase on Oscillator Motion

Consider an oscillator described by $$y = A\sin(\omega t + \phi_0)$$. Explore how variations in the

Influence of Mass Variation on Oscillation Frequency

In an experiment, different masses are attached to the same spring, and the frequency of oscillation

Integral Calculation of Work Done in SHM

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

Investigating Nonlinearity in Large-Amplitude Oscillations

A recent experimental paper claims that 'at large amplitudes, the assumption of simple harmonic moti

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

Mechanical Energy in SHM

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

Non-linear Effects in Simple Pendulum Motion

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

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 Period and Data Analysis

Explore the period of a simple pendulum and compare experimental data with theoretical predictions.

Pendulum Period Measurement Experiment

A group of students measure the period of a simple pendulum by timing multiple oscillations using a

Period and Frequency Determination

A mass on a spring is observed to take $$0.20\,s$$ to move from its maximum displacement on one side

Phase Difference Between Displacement and Velocity

For a simple harmonic oscillator with displacement \(x(t) = A \sin(\omega t + \phi)\), (a) different

Resonance in Forced Oscillations

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

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

Simple Pendulum Energy Analysis

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

Sinusoidal Description and Phase Constant in SHM

A spring-mass oscillator has an amplitude $$A = 0.04 \; m$$ and a frequency $$f = 5.0 \; Hz$$. The d

Sinusoidal SHM with Phase Shift

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

Spring Force Investigation

A researcher investigates the force exerted by a spring using Hooke's law. The aim is to verify the

Vertical Oscillations: Energy and Force Analysis

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

Vertical Spring–Block Oscillator Dynamics

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

Analyzing Hohmann Transfer Orbits for Satellite Maneuvers

Hohmann transfer orbits are used for efficient satellite maneuvers between two circular orbits. Answ

Assessment of Newton's Second Law Along a Gravitational Incline

A small cart is placed on a frictionless inclined plane that makes an angle $$\theta$$ with the hori

Barycenter in a Two-Body System

In a two-body system consisting of masses m₁ and m₂ separated by a distance R, the barycenter (cente

Barycenter of the Sun-Earth System

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

Cannonball Trajectory in a Non-Uniform Gravitational Field

An experiment studies the trajectory of a cannonball launched at a high angle to analyze projectile

Center of Mass in a Two-Body System: Sun-Earth Analysis

Using the values $$m_{Earth} = 5.98 * 10^{24}\,kg$$, $$M_{Sun} = 1.99 * 10^{30}\,kg$$, and the avera

Cometary Orbits: Analyzing Highly Eccentric Trajectories

Comets often exhibit highly eccentric orbits. Their dynamics can provide insight into gravitational

Comparative Analysis of Gravitational Forces

Using the data provided, compare the gravitational forces between various pairs of celestial bodies.

Derivation of Kepler’s Second Law via Calculus

Kepler’s Second Law states that a line joining a planet and its star sweeps out equal areas in equal

Derivation of Orbital Period from Gravitational Force

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

Derivation of 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

Derivation of the Virial Theorem for Gravitational Systems

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

Deriving Gravitational Potential from Gravitational Force

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

Designing a Satellite's Stable Orbit

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

Eccentricity and Elliptical Orbits

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

Effective Gravitational Field on an Irregular Asteroid

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

Effects of Stellar Mass Variation in Binary Systems

In a binary star system, one star gradually loses mass. Analyze the impact on the orbital parameters

Energy Analysis in Multi-Body Systems

Consider a system of three bodies interacting gravitationally. Derive the expression for the total g

Energy Conservation in Elliptical Orbits

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

Energy Conversion in an Asteroid's Elliptical Orbit

A computer simulation is conducted to study the energy conversion in an asteroid's elliptical orbit

Escape Velocity Derivation

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

FRQ 16: Graphical Analysis of Gravitational Potential Energy vs. Height

The graph provided shows experimental data for gravitational potential energy (in joules) versus hei

FRQ 19: Relativistic Corrections and Perihelion Precession

General relativity provides corrections to Newtonian gravity that can explain the observed perihelio

Graphical Analysis of Gravitational Force Variation

A set of experimental data shows how gravitational force varies with distance between two masses. An

Gravitational Energy Trade-offs in a Multi-Body System

Examine the experimental data provided for gravitational potential energies between different pairs

Gravitational Interaction between Two Bodies

Consider two masses m1 and m2 separated by a distance r in deep space. Investigate the gravitational

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 in a Non-Uniform Field

A spacecraft of mass m moves radially from a distance R₀ to a distance R from a planet of mass M. 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 to Rotational Kinetic Energy Conversion

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

Gravitational Slingshot Maneuver

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

Kepler's Third Law and Orbital Analysis

A recent media report claims that the orbital period $$T$$ and the semi‐major axis $$a$$ of satellit

Laboratory Test of Newton's Law of Gravitation using a Torsion Balance

Newton's Law of Gravitation can be tested in a laboratory setting using sensitive apparatus such as

Non-uniform Gravitational Fields in Planetary Interiors

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

Orbit Stability from Potential Energy Diagrams

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

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 Precession Analysis

Analyze the graph showing the change in orbital orientation of a planet over time and discuss the im

Orbital Speed Variation in Elliptical Orbits

Analyze the provided data on orbital speeds at different points in an elliptical orbit. Discuss how

Pendulum Orbital Analog and Kepler's Third Law

In a classroom experiment, a simple pendulum is used as an analog to planetary orbits in order to va

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