AP Physics C: Mechanics FRQ Room

Analysis of a Velocity-Time Graph

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

Analysis of Air Resistance on a Falling Object

An object falling under gravity experiences a net acceleration modeled by $$a(t)= 9.8 - 0.5*t$$ (m/s

Block on an Inclined Plane: Kinematic Analysis

A block of mass m is released from rest at the top of a frictionless inclined plane of length L = 10

Calculus-Based Kinematics Derivation

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

Comparative Analysis of Average Speed and Velocity

An object travels at a constant speed of 10 m/s along a circular track of radius 20 m for one comple

Data Analysis from a Velocity-Time Table

An object’s velocity was recorded over time with the following data:

Deriving Velocity and Acceleration from a Position Function

Consider an object moving along a straight line with its position given by $$x(t)=\sin(t)$$, where x

Determining Instantaneous Rates from Discrete Data

A sensor records the position of a moving particle at various times. The recorded data is shown in t

Displacement-Time Graph Analysis for Non-Uniform Motion

A displacement vs. time graph for an object moving in one dimension is given by the function $$s(t)=

Drone Video Analysis of Free Fall

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

Experimental Data and Constant Acceleration

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

Experimental Evaluation of Vector Addition in Two Dimensions

In a laboratory, two displacement vectors are measured. The following table provides their magnitude

Free Fall 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.

FRQ 1: Constant Acceleration Experiment

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

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: Projectile Motion – Maximum Height and Range

A projectile is launched from the ground with an initial speed of 60 m/s at an angle of 30° above th

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 12: Investigating Terminal Velocity in Free Fall with Air Resistance

A researcher examines free fall motion by considering the influence of air resistance. The net force

FRQ 12: Parametric Representation of Projectile Motion

A projectile’s motion is given by the parametric equations: $$x(t) = 5*t$$ and $$y(t) = 4*t - 2*t^2$

FRQ 15: Differentiation of a Cubic Displacement Function (EASY)

An object's displacement is given by $$s(t)=3*t^3-5*t^2+2*t$$ (in m). (a) Find the velocity function

FRQ 15: Investigating Uniformly Accelerated Motion Using Integrals

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

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

Inferring Acceleration from Velocity Data Using Calculus

The following table shows the time and corresponding velocity for an object moving in one dimension,

Investigating Lab Data: Graph Interpretation and Improvements

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

Kinematic Analysis of Circular Motion

A particle moves along a circular path of constant radius R. Its speed increases according to the fu

Kinematics of a Decelerating Vehicle

A car traveling at 30 m/s starts braking and comes to a stop after covering a distance of 120 m unde

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 of a Bus on a Curved Track

A bus moves along a curved track with its acceleration given by $$a(t)= 0.2*t + 0.5*\sin(t)$$ (m/s²)

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}

Optimization of Projectile Range

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

Optimizing the Launch Angle for Maximum Horizontal Displacement on Uneven Terrain

A projectile is launched with an initial speed of 40 m/s from a platform that is 10 m above the grou

Pendulum Energy Conservation Experiment

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

Projectile Motion Analysis

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

Projectile Motion Experimental Investigation

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

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

Relative Motion in Two Dimensions

A boat is moving eastward relative to the water at 5 m/s. The river current flows southward at 3 m/s

Relative Motion: Two Trains on Parallel Tracks

Two trains move on parallel tracks. Train A has its position given by $$x_A(t)=25t$$ and Train B by

Two-Dimensional Motion with Air Resistance

A small spherical object is dropped from a height of 100 m. Its vertical motion is influenced by air

Two-Dimensional Motion with Vector Decomposition

An object moves in the plane and its position is given by the vector function $$\vec{r}(t)= \langle

Uniformly Accelerated Motion: Incorrect Baseline Velocity

A physics lab experiment tracked the displacement of a moving object using a laser sensor, measuring

Vector Addition in Two-Dimensional Projectile Motion

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

Verification of Uniformly Accelerated Motion

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

Assessing Energy Conversion in a Pendulum Experiment

A researcher conducts an experiment with a simple pendulum of length L = 2 m and a bob of mass 0.5 k

Ballistic Kinetic Energy with Air Resistance

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

Calculus-based Integration of Work over a Variable Force

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 5*x$$ N. The particle

Compound Machine Energy Analysis Experiment

A compound machine consisting of a lever and a pulley is used to lift a load, and the work input is

Derivation of the Work-Energy Theorem

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

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

Dissipative Work under Variable Friction

A 5 kg block is sliding on a horizontal surface with an initial speed of 10 m/s. The coefficient of

Energy Analysis of a Pendulum

A simple pendulum consists of a 0.5 kg bob suspended by a 2 m string. It is released from rest at an

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

Free‐Fall Impact Energy Experiment

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

Friction‐Influenced Kinetic Energy Loss Experiment

A 1 kg block is pushed along a horizontal rough surface with a coefficient of kinetic friction μ = 0

FRQ 5: Analysis of Work and Potential Energy in a Spring–Mass System

A mass–spring system is tested in a laboratory. A spring attached to a mass is stretched from its eq

FRQ 12: Analysis of Cyclist Power Output During a Ride

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

FRQ 16: Work and Energy Transformation in a Compound Machine

A 10-kg block is pushed up a frictionless ramp using a compound pulley system. A force sensor record

FRQ 17: Energy Loss Analysis in a Frictional Pendulum

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

FRQ 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

FRQ 20: Evaluating Efficiency in a Conveyor Belt System

In a conveyor belt system used for transporting goods, the work input and the corresponding measured

Horizontal Pulling Work Experiment

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

Measuring Work Done against Air Resistance in Free Fall

A researcher studies the motion of a 1 kg object falling under gravity with air resistance. The drag

Pendulum Oscillation and Air Resistance Experiment

A simple pendulum with a 0.5 kg bob and a 2 m long string swings in air. Over successive oscillation

Potential Energy Curve Analysis

An object has a potential energy given by $$ U(x) = (x-2)^2 - (2*x-3)^3 $$, where U is in joules and

Power Output from a Variable Force: Time-Dependent Problem

A particle is subjected to a time-dependent force given by $$F(t)= 5 \;\cos(0.5*t) \; (\text{N})$$.

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 Work-Energy Analysis in a Flywheel

A flywheel with a moment of inertia $$I = 20\,kg\cdot m^2$$ is accelerated from rest to an angular s

Tidal Energy Extraction Analysis

A tidal turbine is installed in a coastal area where water flow speed varies cyclically. The force e

Variable Force Robotic Arm Power Experiment

In this experiment, a robotic arm exerts a time-varying force on an object as it moves along a horiz

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 by Time-Dependent Force on a Car

A car of mass 1200 kg is subjected to a net force along a straight road given by $$F(t)=2000-50*t$$

Work Done by Friction in Stopping a Car

A 1200 kg car moving at 25 m/s skids to a complete stop on a level road due to a constant frictional

Work Done on an Object by a Central Force

An object moves under a central attractive force given by $$F(r)= -\frac{A}{r^2}$$ with $$A = 50 \;\

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 Application

A 0.5 kg cart increases its speed from 2 m/s to 8 m/s on a frictionless track. Using the work-energy

Work, Energy, and Power in Circular Motion

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

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

Angular Impulse and Rotation

A uniform disk (mass = 4 kg, radius = 0.5 m) rotates initially at 20 rad/s. A constant tangential fo

Automobile Collision and Impulse Analysis

Two cars are involved in a head-on collision. Car 1 (mass = 1200 kg) is traveling east at 20 m/s, an

Block on an Incline: Collision and Momentum

A 2-kg block slides down a frictionless incline of length 4 m, which makes an angle of 30° with the

Calculating Center of Mass Acceleration

A system of mass 10 kg experiences a net external force described by $$F(t)=20+5*t$$ N, where t is i

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 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 Triangular Plate

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

Composite Body Center of Mass Calculation

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

Determination of an Unknown Mass via Collision Data

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

Elastic and Inelastic Collision Analysis

Two balls collide head-on. The red ball (mass $$0.5\,kg$$, velocity $$+4\,m/s$$) and the green ball

Elastic Collision: Two Gliders on an Air Track

Two gliders on an air track experience a head-on elastic collision. Glider X (mass = 1 kg) initially

Experimental Design: COM Independence in Collisions

Design an experiment to test the hypothesis that "the motion of the center of mass (COM) of a system

Experimental Design: Determining the Center of Gravity of a Complex Structure

Design an experiment to determine the center of gravity of a complex structure composed of multiple

Experimental Design: Measuring Impulse with Force Sensors

Propose an experiment to measure the impulse delivered by a non-uniform force applied to a ball. You

Explosive Separation and Momentum Conservation

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

FRQ 6: Elastic Collision Analysis

Two spheres collide elastically along a straight line. Sphere A (mass = 0.5 kg) initially moves at +

FRQ 13: Critical Analysis: Momentum Experiment

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

FRQ 20: Two-Dimensional Collision Analysis

In the xy-plane, Object 1 (mass = 1.5 kg) moves with velocity $$\vec{v}_1 = (3\hat{i} + 2\hat{j})\ m

Impulse Analysis in a Variable Mass Rocket

Consider a simplified rocket system where its mass decreases with time as it expels fuel. The mass i

Impulse and Average Force on a Punted Football

A football (mass = 0.4 kg) is kicked such that its speed increases from 0 to 30 m/s in 8 ms. (a) Use

Impulse and Center of Mass in a Soccer Kick

A soccer ball (mass $$0.45\,kg$$) initially at rest is kicked, resulting in a launch speed of $$25\,

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 from a Variable Force Graph

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

Impulse on Coupled Freight Cars

Two freight cars are coupled and moving on a frictionless track at an initial speed of $$10\,m/s$$.

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 with a Movable Platform

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

Inelastic Collision with Time-Dependent Force

Two gliders on a frictionless air track collide inelastically. The red glider of mass $$0.5\,kg$$ is

Momentum Change under a Non-Constant Force

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

Multiple Collisions in a Figure Skating Routine

In a choreographed figure skating routine, two skaters push off from each other. Skater A has a mass

Non-conservative Forces: Block on an Incline with Friction

A 2 kg block slides down a 30° incline that is 5 m long. The coefficient of kinetic friction between

Projectile and Cart Collision: Trajectory Prediction

A 0.2 kg projectile is launched horizontally at 10 m/s from a 20 m high platform. At the same moment

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 the Tsiolkovsky Rocket Equation

A rocket expels mass continuously and can be modeled as a variable mass system. Starting with a mass

Rocket Propulsion and Variable Mass System

A rocket has an initial mass of $$500$$ kg (including fuel) and expels gas with a constant relative

Rocket Propulsion: Variable Mass System

A rocket with an initial mass of 500 kg (including fuel) expels gas at a constant exhaust velocity o

Stability Analysis: Center of Mass vs. Center of Gravity

A uniform rectangular block of mass 10 kg with dimensions 0.5 m × 0.3 m × 0.2 m rests on a flat surf

Analysis of Gyroscopic Precession

A spinning gyroscope of moment of inertia $$I$$ has an angular momentum $$L$$ and is subject to a gr

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 Kinematics: Modeling a Rotating Spring System

A disk is attached to a torsional spring that exerts a restoring torque given by $$T = -k \times \th

Angular Momentum Conservation in Rotational Collisions

In this experiment, two disks with different moments of inertia and angular velocities are coupled t

Angular Momentum Conservation on a Merry-Go-Round

A child of mass $$m = 30\,kg$$ stands at the edge of a merry-go-round that is modeled as a uniform d

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

Assessment of Rotational Kinematics Equations

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

Calculus Based Determination of Moment of Inertia for a Non-uniform Rod

A rod of length $$L = 2\,m$$ has a linearly varying density given by $$\lambda(x) = \lambda_0 \,(1 +

Calculus-Based Torque Distribution in a Non-uniform Rod

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

Centripetal Force and Angular Velocity Measurement

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

Critical Analysis of Torque in Mechanical Systems

A media report on engine performance claims that a 10% increase in the applied force always results

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 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

Dynamics of a Damped Flywheel System

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

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

Energy Considerations in a Rotating Pendulum

A physical pendulum consisting of a rigid body with moment of inertia $$I$$ rotates about a pivot. T

Energy Dissipation Due to Friction in a Spinning Disk

A disk is spun up to a high angular velocity and then allowed to slow down due to friction. The expe

Equilibrium Analysis in Rotational Systems

A uniform beam is balanced on a pivot, with forces applied at various distances from the pivot to ma

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 17: Moment of Inertia of a Non-Uniform Rod

A rod of length L = 2.00 m has a density that varies with position according to $$\rho(x) = \rho_0 *

FRQ 19: Oscillatory Rotational Motion (Torsional Pendulum)

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

Integration for Moment of Inertia of a Non-Uniform Rod

A rod of length L has a linear mass density given by $$\lambda(x)= \lambda_0 * x$$, where x is measu

Investigating Non-uniform Density Effects on Moment of Inertia

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

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,

Mathematical Modeling of Brake Systems

A braking system applies a constant torque of $$\tau = 15 \text{ Nm}$$ on a flywheel with moment of

Moment of Inertia of a Continuous Rod

Consider a uniform thin rod of length $$L$$ and total mass $$M$$. The rod rotates about an axis perp

Rolling Motion and Energy Conservation: Rolling Cylinder on an Incline

A solid cylinder of mass $$M = 5.0 \text{ kg}$$ and radius $$R = 0.4 \text{ m}$$ rolls without slipp

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 Energy Distribution in a Rolling Object

An experiment investigates a rolling object (such as a cylinder) as it descends an incline. The kine

Rotational Equilibrium of a Beam with Distributed Load

A uniform beam of length $$L = 4.0 \text{ m}$$ and mass 10 kg is hinged at one end. A variable distr

Rotational Inertia of a Uniform Rod

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

Torque and Rotational Inertia: Uniform Rod

A uniform rod of length $$L = 2.0 \text{ m}$$ is pivoted about one end. A force of $$F = 10 \text{ N

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

Torsion Pendulum Method for Irregular Objects' Inertia

This experiment uses a torsion pendulum to determine the moment of inertia of irregular objects. The

Variable Mass Distribution and Moment of Inertia

A rod of length $$L = 3.0 \text{ m}$$ has a linear mass density given by $$\lambda(x) = \lambda_0*(1

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 Dependence in a Nonlinear Oscillator

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

Analyzing Phase Shift and Amplitude Modulation in SHM

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

Calculus-Based Prediction of Maximum Speed

A photogate system is set up to record the speed of a mass-spring oscillator. Answer the following p

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: Spring-Mass System vs. Pendulum Oscillations

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

Conservation of Energy: Integral Approach in SHM

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

Damped Oscillations in a Spring System

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

Damped Oscillatory Motion Analysis

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

Data Analysis of Damped Oscillations

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

Derivation of SHM Equations Using Calculus

Starting with Newton’s second law and Hooke’s law for a mass-spring system, derive the differential

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 Using SHM Data

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

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}

Determining Maximum Speed from Energy Considerations

An oscillator of mass $$m = 0.1 \; kg$$ is attached to a spring with a spring constant of $$k = 250

Driven Oscillations and Resonance in a Mass-Spring System

A mass-spring system of mass $$m$$ is subjected to an external periodic driving force given by $$F(t

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 Conservation in Vertical Oscillators

A media claim states that 'in a vertical spring-mass system, mechanical energy is always conserved r

Energy Conservation Verification Using Calculus

A mass-spring system oscillates with a displacement given by $$x(t) = 0.06 \cos(15t)$$. (a) Derive t

Energy Transformations in a Spring Oscillator

A block attached to a horizontal spring oscillates with amplitude $$A$$. Consider the energy transfo

Evaluating Damped Oscillatory Motion Effects

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

Experimental Determination of Spring Constant

Utilize experimental data from a mass–spring oscillator to determine the spring constant.

Experimental Determination of Spring Constant via SHM

A physics lab report claims that the spring constant, k, of a mass-spring oscillator can be precisel

Frequency Response Analysis from Experimental Data

An experiment with a mass-spring oscillator produces displacement data over time as shown in the pro

FRQ 1: Spring Force Calculation Using Hooke's Law

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

FRQ 8: Energy Exchanges in SHM

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

FRQ 8: Energy Transformation in SHM

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

FRQ 14: Impact of Initial Conditions on SHM

An oscillator is released from an initial displacement of 0.05 m with an initial upward velocity of

FRQ 15: Determination of the Phase Constant

An oscillator with amplitude $$A = 0.05\ m$$ and angular frequency $$\omega = 8\ rad/s$$ is observed

FRQ 17: Pendulum Nonlinear Effects

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

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

FRQ5: Sinusoidal Description of Oscillatory Motion and Phase Determination

A mass-spring system oscillates with an amplitude of $$A = 0.06\,m$$ and a frequency of $$f = 2.0\,H

FRQ20: Energy Dissipation in Damped Pendulum Oscillations

A damped pendulum oscillates with small angles such that its motion is approximately described by $

Graphical Analysis of SHM: Determining Phase and Frequency

A researcher records the oscillatory motion of a block on a spring and generates a position-vs.-time

Integral Calculation of Work Done in SHM

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

Integration Approach to SHM: From Acceleration to Displacement

A mass undergoing simple harmonic motion has an acceleration given by $$a(t) = -\omega^2 * A * \sin(

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

Mass Dependence in Oscillatory Motion

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

Mechanical Energy in SHM

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

Modeling Nonlinearities in Pendulum Motion

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

Momentum and Impulse Analysis in Oscillatory Motion

A block of mass $$m = 0.4 \; kg$$ oscillates on a spring, and its displacement is given by $$x(t)=0

Non-linear Effects in Simple Pendulum Motion

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

Oscillations of a Liquid Column in a U-tube

A U-tube containing a liquid with density $$\rho$$ exhibits oscillatory motion when the liquid is di

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 Period Measurement Experiment

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

Simple Pendulum Energy Analysis

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

Sinusoidal SHM with Phase Shift

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

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

Uncertainty Analysis in SHM Period Measurements

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

Work Done in Spring Oscillation via Calculus

A spring with a constant of $$k = 150\,N/m$$ is stretched from its natural length to a displacement

Analysis of a Gravitational Potential Energy Graph

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

Analysis of Low Earth Orbit Satellite Decay

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

Analyzing 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

Application of Kepler's Third Law in the Solar System

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

Barycenter of the Sun-Earth System

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

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 in the Sun-Earth System

Using the provided data for the Sun-Earth system, analyze the location of the barycenter. Use the ex

Center-of-Mass in the Sun-Earth System

Consider the Sun-Earth system, where the Sun has mass $$ M = 1.99 \times 10^{30} \text{ kg} $$, Eart

Comparative Analysis of Planetary Orbits

Using observational data for two planets, analyze how well their orbital periods conform to Kepler's

Derivation of Escape Velocity Using Calculus

Using the concept of gravitational potential energy, derive the escape velocity from a planet of mas

Deriving the Gravitational Field from a Potential Function

Given the gravitational potential function $$V(r)= -\frac{G*m*M}{r}$$, you are to derive the gravita

Determining the L1 Lagrange Point

In a star-planet system, an object is positioned along the line connecting the two bodies at the L1

Dynamics of a Falling Object in a Gravitational Field

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

Eccentricity and Elliptical Orbits

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

Effects of Non-Spherical Mass Distribution on Satellite Orbits

A planet is not perfectly spherical but exhibits a slight oblateness, introducing a perturbative ter

Energy Analysis in Multi-Body Systems

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

Energy Balance at Apoapsis and Periapsis

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

Energy Conservation in Central Force Motion

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

FRQ 1: Gravitational Force between Two Masses

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

FRQ 4: Gravitational Potential Energy in Satellite Orbits

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

FRQ 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 11: Time-Dependent Gravitational Force in Radial Motion

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

FRQ 20: Determining the Mass of a Central Body from Satellite Orbits

A satellite is observed to orbit a celestial body with a period $$T$$ and at a radius $$r$$. Using t

Gravitational Energy in a Binary Star System

Binary star systems are bound by gravity. The total mechanical energy of such a system includes kine

Gravitational Field Modeling for Extended Bodies

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

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 Variations near Earth

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

Gravitational Slingshot Maneuver

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

Gravitational Slingshot Maneuver

A spacecraft uses a gravitational slingshot maneuver to gain speed by passing near a planet. (a) Des

Gravitational Slingshot Maneuver Analysis

Gravitational slingshot (or flyby) maneuvers are used in space missions to change a spacecraft's vel

Impact of Mass Loss on a Comet's Orbit

A comet loses mass due to sublimation as it approaches the Sun. This variable mass affects its orbit

Investigating Orbital Eccentricity Effects

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

Modeling Tidal Forces with Calculus

Tidal forces arise due to the gradient in gravitational force across an object. These forces can cau

Optimization of Orbital Maneuvers in Multi-Stage Rockets

A multi-stage rocket performs orbital maneuvers to transition from a low Earth orbit to a higher geo

Optimizing Orbital Transfer Maneuvers: Hohmann Transfer

A spacecraft is planning an orbital transfer maneuver from a lower circular orbit to a higher one us

Orbit Transfer and Hohmann Transfer Orbits

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

Orbital Energy Analysis in Elliptical Orbits

The total mechanical energy of a satellite in an elliptical orbit is the sum of its kinetic and grav

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 Mechanics: Applying Kepler's Third Law

A satellite orbits Earth in an elliptical orbit, which for the sake of this problem can be approxima

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 Period Determination Using Kepler's Third Law

Kepler’s Third Law relates the period T of an orbiting body to the semimajor axis a of its orbit. In

Orbital Perturbations and Precession

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

Orbital Speed and Radius in Circular Orbits

For an object in a circular orbit, (a) Derive the expression relating orbital speed $$ v $$ to the

Tidal Forces and their Impact on Orbital Dynamics

A moon orbits a planet and experiences tidal forces. Analyze how these forces are derived and their

Variable Gravitational Field and Calculated Potential Energy

Consider an object moving radially in a gravitational field described by $$F(r) = -\frac{G * M * m}{

Work Done by Gravitational Force on a Falling Object

An object is dropped from a tall structure where the gravitational acceleration decreases with altit