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

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.

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

Calculating Displacement via Integration of a Velocity Function

An object moves in one dimension with its velocity described by $$v(t)=4*t-2$$ m/s. Determine the di

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

Car Acceleration on a Highway: Calculus Approach

A car's position along a straight highway is given by $$x(t)= 2*t^3 - 6*t^2 + 4*t$$, where $$x$$ is

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°

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

Designing a Trajectory for a Manufacturing Robot

A robot in a manufacturing plant moves along a straight track with a piecewise position function: Fo

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

Determining Motion from a Sine Position Function

An object's position is given by $$x(t)= 3*\sin(t) + t$$ (meters). Analyze its motion using calculus

Drone Video Analysis of Free Fall

A drone records a free-falling ball dropped from a 100 m height. Video analysis approximates the bal

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 from a Cliff with Calculus Insights

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

Free Fall Kinematics

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

Free Fall under Gravity

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 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: Graphical Analysis of Velocity-Time Data

A researcher collects velocity vs. time data from an object undergoing several phases of motion: acc

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 9: Piecewise Acceleration Motion (HARD)

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

FRQ 13: Comparative Analysis of Two Free Fall Experiments

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

FRQ 15: Circular Motion with Varying Speed

A particle moves along a circular track of radius 5 m with its speed given by $$v(t)= 2 + t$$ (in m/

FRQ 15: Investigating Uniformly Accelerated Motion Using Integrals

A researcher records the acceleration of an object with a sensor, finding that the acceleration vari

FRQ 16: Exploring the Relationships in the Big Five Kinematic Equations

A researcher conducts a series of experiments to test the Big Five kinematic equations, in which eac

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

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

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: Analysis of Motion on an Inclined Plane (MEDIUM)

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

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: Experimental Design – Determining g with a Free Fall Apparatus

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

Graphical Analysis of Distance and Displacement

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

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 with Non-Constant Acceleration

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

Kinematics with Resistive Forces

Design an experiment to study the motion of an object falling under gravity while experiencing a dra

Motion with Changing Direction

An object moves along a straight line with its position given by $$x(t)= t^3 - 6*t^2 + 9*t$$ (meters

Parametric Trajectory Analysis

A particle moves in the plane with its position given by: $$x(t)=2t^2$$ and $$y(t)=20t - 4.9t^2$$, w

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 Experimental Investigation

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

Projectile Motion with Calculus Integration

An object is launched from ground level with an initial speed of 50 m/s at an angle of 40° above the

Rotational Motion: Angular Kinematics

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

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-Dependent Acceleration and Displacement

A particle’s acceleration is given by the function $$a(t)=6-2*t$$ (in $$m/s^2$$) for $$0 \le t \le 4

Uniform Acceleration in One Dimension

An object moves along a straight line with constant acceleration. Its motion is described by the pos

Uniformly Accelerated Motion on a Track

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

Vector Addition in Two-Dimensional Projectile Motion

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

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

Circular Motion with Tangential Work

An object is moving along a circular path of radius 3 m. While the centripetal force (directed towar

Determining Speed of a Roller Coaster Considering Friction

An 800-kg roller coaster car is released from rest at the top of a frictionless 50-m-high hill, then

Efficiency Analysis of a Mechanical System

A motor lifts a 100 kg mass by raising it 10 m in 20 seconds, using an electrical energy input of 15

Energy Analysis in Circular Motion

A 2 kg object moves in a horizontal circular path of radius 5 m. It is subject to a tangential force

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 Loss Due to Position-Dependent Friction

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

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 Transfer in a Bouncing Ball

A ball of mass 0.5 kg is dropped from a height of 10 m and, after hitting the ground, rebounds to a

Experiment on Electric Motor Power Output

Design an experiment to measure the power output of an electric motor used in a small robotic car.

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 14: Calculus Analysis of Work Done on an Object with a Time-Varying Force

An object with a mass of 3 kg is subjected to a force that varies with time according to the equatio

FRQ 14: Elastic Potential Energy in a Spring-Mass System

A news article asserts that the elastic potential energy stored in any deformed spring is always giv

FRQ 15: Energy Conservation in an Oscillating Spring–Mass System

A 2-kg mass attached to a spring (with spring constant k = 200 N/m) oscillates horizontally. A displ

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 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: 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

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

Kinetic Energy and Work-Energy Theorem Application

A block of mass 2 kg undergoes net work in two different scenarios. Use the work–energy theorem to f

Kinetic Energy Gain in a Roller Coaster Ride

A roller coaster car of mass 500 kg is released from rest at the top of a frictionless hill at a hei

Model Rocket Power Measurement Experiment

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

Motion on an Inclined Plane with Friction

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

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

Pendulum Energy Conservation Experiment

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

Potential Energy Curve Analysis

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

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 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 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

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

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\

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 and Velocity: Power and Work Analysis

A machine applies a time-dependent force given by $$F(t)=50+10\,t$$ (N) while the displacement of an

Variable Friction and Kinetic Energy Loss

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

Variable Mass Rocket Energy Analysis

A rocket with an initial mass of 1500 kg expels fuel and decreases its mass linearly to 1200 kg over

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 by Non-Conservative Forces in a Loop

A block experiences a variable nonconservative force along a path given by $$ F(x)= 20 * \sin(x) $$

Work Done along a Curved Path Under Variable Force

A particle moves along a curve defined by $$ y = x^2 $$ in the xy-plane. It is subjected to a force

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-Energy Analysis in Collisions

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

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 in Inelastic Collisions

A 3-kg cart moving at 4 m/s collides inelastically with a stationary 2-kg cart on a frictionless tra

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

Center of Gravity vs. Center of Mass in a Tilted Rod

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

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

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

Center of Mass of a 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 Variable-Density Rod

Consider a thin rod of length $$L=2.0$$ m with a linear density given by $$\lambda(x)=2+3*x$$ (kg/m)

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

Circular Motion: Banked Curve Analysis

A car of mass 1200 kg negotiates a banked curve of radius 50 m with no friction required. The curve

Explosive Fragmentation: Momentum Transfer

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

FRQ 5: Physics of a Football Punt

A football with a mass of 0.4 kg is punted so that its launch speed is 30 m/s, with the kicker’s foo

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

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 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 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 from a Collision with a Wall

A ball of mass $$0.2\,kg$$ strikes a rigid wall and rebounds. During the collision, it experiences a

Impulse in a Collision with Force Graph Analysis

A 0.75 kg object undergoes a collision during which the force acting on it is given by $$F(t)=50-10*

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 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

Impulse with Resistive Force

A 2-kg block on a frictionless surface is subjected to two forces simultaneously over a time interva

Inelastic Collision: Bullet-Block Interaction

A 0.02-kg bullet is fired at 400 m/s into a stationary 2-kg wooden block on a frictionless surface.

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

Mobile Robot Center of Gravity Analysis

A mobile robot features extendable arms that modify its mass distribution. With its arms retracted,

Momentum Analysis in an Asteroid Breakup

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

Momentum and Energy in Elastic Collisions

Two gliders on a frictionless air track undergo an elastic head-on collision. Glider A (mass $$0.8\,

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'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/

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 $

Projectile Explosion and Center of Mass Motion

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

Projectile Motion with 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

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

Rolling Cylinder on an Incline

A uniform solid cylinder (mass = 4 kg, radius = 0.5 m) rolls without slipping down a 30° incline. An

Rotational Dynamics Using Center of Mass

A uniform rod of length $$L=2.0\;m$$ and mass M rotates about a frictionless pivot at one end. (a)

Rotational Impulse and Angular Momentum

A rigid disc of mass 4 kg and radius 0.5 m rotates about its central axis with an initial angular sp

Variable Force Collision Analysis from Graph Data

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

Analysis of Rolling Motion on an Incline

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

Angular Displacement and Kinematics Analysis

A researcher is investigating the kinematics of a rotating disk. The disk rotates about its center,

Angular Momentum Conservation in a Merry-Go-Round Experiment

A child standing on the edge of a rotating merry-go-round (modeled as a disk) provides an opportunit

Angular Momentum Conservation in a Spinning System

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

Angular Momentum Conservation in Figure Skating

A figure skater spins with an initial angular velocity $$\omega_0$$ and moment of inertia $$I_0$$. W

Angular Momentum Transfer in Colliding Rotational Bodies

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

Applying the Parallel Axis Theorem to a Composite Object

A composite object has been tested to determine its moment of inertia about different axes. The foll

Assessment of Rotational Kinematics Equations

Experimental data for a rotating disk include measurements of angular displacement, angular velocity

Calculus-Based Analysis of Angular Impulse

In rotational dynamics, angular impulse is defined as the integral of torque over a time interval, w

Comparative Analysis of Rotational and Translational Dynamics

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

Conservation of Angular Momentum in a Figure Skater's Spin

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

Coupled Rotational and Translational Dynamics in a Rolling Sphere

A sphere is allowed to roll down a curved track without slipping. The experiment examines the coupli

Determining Moment of Inertia of Irregular Objects

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

Determining the Effect of Friction on Rotational Motion

A flywheel with moment of inertia $$I = 1.0\,kg\,m^2$$ is initially spinning at an angular velocity

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 $$

Dynamic Stability of a Rotating Space Station

A rotating space station initially spins with angular velocity $$\omega_i$$ and has a moment of iner

Dynamics of a Damped Flywheel System

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

Dynamics of a Rotating System with Friction

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

Dynamics of Coupled Rotational Systems

Two disks are coupled by a belt: Disk A has a moment of inertia $$I_A = 0.8 \text{ kg m}^2$$ and ini

Equilibrium Analysis in a Rotating Beam System

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

FRQ 2: Rotational Inertia of a Composite System

A thin uniform rod of length L = 2.00 m and mass M = 5.00 kg has two small beads, each of mass m = 1

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

FRQ 7: Equilibrium and Torque on a Seesaw

Consider a seesaw in static equilibrium with a pivot not located at its geometric center. The seesaw

FRQ 10: Comparison of Rotational and Translational Kinetic Energy

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

FRQ 15: Angular Momentum in an Inelastic Collision on a Rotating Platform

A figure skater stands on a frictionless rotating platform with a moment of inertia of \(10.00\,kg\c

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

Parallel Axis Theorem Experiment with a Suspended Bar

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

Rolling Motion on an Inclined Plane

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

Rolling Motion: Energy Partition Analysis on an Inclined Plane

A solid cylinder is released from rest at the top of an inclined plane and allowed to roll without s

Rotational Kinematics from Angular Velocity Graph

A rotating object's angular velocity increases linearly with time. The graph provided shows that $$\

Rotational Kinematics on a Spinning Disk

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

Seesaw Rotational Equilibrium

Two children are sitting on opposite ends of a seesaw (a uniform beam pivoting about its center). Ch

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 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 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

Variable Torque Function Integration

Consider a rotating body with constant moment of inertia I = 5 kg·m². The applied torque is time dep

Verification of the Parallel Axis Theorem

Students test the parallel axis theorem by measuring the moment of inertia of a uniform rod about se

Amplitude and Maximum Speed Relationship in SHM

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

Analyzing Damped Oscillations in a Spring-Mass System

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

Calculating Damped SHM Energy Loss

A student records the amplitude decay of a damped oscillator and calculates the energy using $$U=\fr

Calculus Derivative Analysis in SHM

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

Calculus-Based Derivation of Oscillator Velocity and Acceleration

For an oscillator described by $$y = A\sin(\omega t + \phi_0)$$, derive its velocity and acceleratio

Combined Oscillator: Pendulum with a Spring

A hybrid oscillator is constructed by suspending a 0.5-kg mass from a spring with a force constant o

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

Coupled Oscillations in a Two-Mass Spring System

Consider two masses, $$m_1$$ and $$m_2$$, connected by a spring with spring constant $$k$$. Addition

Damped Oscillation: Logarithmic Decrement Analysis

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

Damped Oscillations: Determining the Damping Coefficient

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

Data Analysis from a Virtual SHM Experiment

A virtual experiment on simple harmonic motion produces the following data for the displacement of a

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 Spring Constant from Oscillation Data

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

Determination of Spring Constant Using SHM Data

An experiment on a mass-spring oscillator provides the following data for different masses and their

Determining the Spring Constant from Oscillation Data

A student conducts an experiment to determine the spring constant $$k$$ of a spring by measuring the

Differentiating SHM: Velocity and Acceleration

A block attached to a spring oscillates on a frictionless track and its position is recorded by a se

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 in Simple Harmonic Motion

A mass-spring system oscillates with a displacement described by $$y(t)=A*\cos(\omega*t)$$. This pro

Energy Analysis of a Simple Pendulum

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

Energy Conservation in a Simple Pendulum

A simple pendulum of length $$L$$ and mass $$m$$ is displaced by a small angle $$\theta$$ from the v

Energy Conservation in Pendulum Motion

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

Energy Exchange in Coupled Oscillators

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

Energy Transformation in SHM

A block of mass $$m = 0.2 \; kg$$ oscillates on a horizontal spring with a force constant of $$k = 1

Error Analysis in SHM Measurements

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

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 Exchanges in SHM

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

FRQ 12: Comparative Analysis of Horizontal and Vertical Oscillators

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

FRQ 18: Pendulum Motion Beyond the Small Angle Approximation

A simple pendulum is tested at various amplitudes, including larger angles where the small angle app

FRQ8: Comparing Spring-Mass and Pendulum Oscillators

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

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

FRQ19: Coupled Oscillators – Normal Modes of a Two-Mass, Three-Spring System

Consider a system in which two identical masses \(m\) are connected in series with three identical s

Graphical Analysis of SHM Experimental Data

A block attached to a horizontal spring oscillates, and its displacement $$x(t)$$ (in meters) is rec

Investigating the Effect of an External Driving Force

An experiment is conducted where a spring-mass system is subjected to an external periodic driving f

Oscillation Frequency's Dependence on Mass and Spring Constant

A research claim suggests that 'doubling the mass of an oscillating system will always decrease the

Pendulum Oscillations for Large Angles

For a simple pendulum with length \(L\) oscillating with a maximum angle \(\theta_{\text{max}}\) tha

Phase Constant and Sinusoidal Motion

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

Phase Difference Between Displacement and Velocity

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

Phase Space Analysis of SHM

For a mass-spring oscillator exhibiting simple harmonic motion with a solution $$x(t) = A\sin(\omega

Resonance and Energy Amplification in Oscillatory Systems

In a driven, damped oscillator, the amplitude as a function of the driving frequency is given by $$

Small-Angle Pendulum Analysis

A simple pendulum consists of a mass attached to a massless string of length $$L = 0.5\,m$$. (a) De

Spring Force and Energy Analysis

A researcher is studying the behavior of a horizontal spring. The spring has a natural length of 12

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

Superposition and Beats in Oscillatory Motion

Two simple harmonic motions are given by $$y_1(t)=A\,\sin(2\pi f_1 t)$$ and $$y_2(t)=A\,\sin(2\pi f_

Uncertainty Analysis in SHM Period Measurements

In an experiment designed to measure the period of a spring-mass oscillator, several sources of unce

Vertical Oscillations of a Mass-Spring System

A vertical spring with a spring constant of $$k = 150\,\text{N/m}$$ supports a block of mass $$m = 2

Vertical Oscillations: Lab Data Analysis

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

Vertical Spring-Block Oscillator: Equilibrium and Oscillations

A block of mass $$m = 1.80\,kg$$ is attached to a vertical spring with force constant $$k = 400\,N/m

Vertical Spring-Mass Oscillator Dynamics

A block of mass $$m = 1.5 \; kg$$ is attached to the end of a vertical spring with force constant $$

Vertical Spring–Block Oscillator Dynamics

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

Analysis of a Gravitational Potential Energy Graph

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

Analyzing Three-Body Gravitational Interactions

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

Angular Momentum Conservation during Gravitational Collapse

An interstellar cloud of gas with initial radius R and angular velocity ω undergoes gravitational co

Angular Momentum Conservation in Orbits

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

Areal Velocity and Angular Momentum in Planetary Motion

A planet of mass $$m$$ orbits a star while conserving its angular momentum $$L$$. Answer the followi

Barycenter of the Sun-Planet System

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

Center of Mass Determination in the Sun-Earth System

A researcher is calculating the barycenter (center of mass) for the Sun-Earth system using a one-dim

Center of Mass of the Sun-Earth System

Consider the Sun-Earth system where the mass of the Earth is $$m = 5.98 \times 10^{24} \text{ kg}$$,

Centripetal Force and Circular Orbits

For an object in a circular orbit around a central mass, gravitational force provides the necessary

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 Gravitational Field due to a Spherical Shell

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

Deriving the Gravitational Potential Energy Function

Starting with Newton's law of gravitation expressed as $$F = - G * \frac{m_1 * m_2}{r^2}$$, derive t

Designing a Modern Cavendish Experiment

A researcher designs an experiment modeled after the Cavendish torsion balance to determine the grav

Determination of Gravitational Constant Using a Compound Pendulum

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

Dynamics of a Binary Star System

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

Eccentricity and Elliptical Orbits

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

Energy Conversion in a Gravitational Slingshot Maneuver

A spacecraft performing a gravitational slingshot maneuver around a planet converts gravitational po

Escape Velocity and Energy Requirements

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

Experimental Analysis of Orbital Decay from a Satellite

A satellite in low Earth orbit is gradually losing altitude due to atmospheric drag. Experimental da

FRQ 5: Energy Conservation in Orbital Transfer

A spacecraft in a lower circular orbit of radius $$r_1$$ performs a burn to initiate a transfer to a

FRQ 9: Kepler’s Second Law – Area Sweep Rate

Kepler’s Second Law states that a line connecting a planet to its star sweeps out equal areas in equ

FRQ 13: Orbital Transfer Maneuver – Hohmann Transfer

A spacecraft in a circular orbit of radius $$r_1$$ wishes to transfer to a higher circular orbit of

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 Field Modeling for Extended Bodies

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

Gravitational Interaction between Two Bodies

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

Gravitational Parameter in Exoplanetary Systems

Using the provided exoplanetary data, analyze the consistency of the gravitational parameter (derive

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 Measurement on a Ramp

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

Gravitational Slingshot Maneuver

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

Kepler's Third Law and Satellite Orbits

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

Newton's Law in Binary Star Systems

Using the provided data for a binary star system, analyze Newton's Law of Gravitation to determine t

Newtonian Approximation of Gravitational Lensing

Although gravitational lensing is accurately described by General Relativity, a simplified Newtonian

Orbital Perturbations and Precession

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

Orbital Speed Variation in Elliptical Orbits

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

Orbital Transfer and the Hohmann Maneuver

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

Planetary Orbits and Energy Considerations

Consider a planet in a highly eccentric orbit. The total orbital energy (kinetic plus potential) is

Satellite Orbit Simulation: Finite Burn and Hohmann Transfer Error

A research team develops a computer simulation to model a satellite's orbital transfer using a Hohma