AP Physics C: Mechanics FRQ Room

Acceleration from a Given Velocity Function

An object moves along a straight line with its velocity described by $$v(t)= 5*t^2 - 3*t + 2$$ (m/s)

Analysis of a Ballistic Trajectory with Inaccurate Symmetry Assumption

In a projectile motion experiment, a ball was launched at a known angle and its trajectory was recor

Application of the Big Five Kinematic Equations

An experiment provides the following data for an object in motion: initial velocity $$u=5$$ m/s, tim

Average vs. Instantaneous Quantities

A particle’s displacement is given by the integral function $$x(t)= \int_0^t e^{-\tau} \cos(\tau)\,d

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-Based Kinematics Derivation

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

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

Determination of Acceleration Due to Gravity

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

Determination of Maximum Height in Projectile Motion

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

Determining Launch Angle from Experimental Data

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

Determining Velocity from a Position Function with Differentiation Error

An experiment recorded the position of a particle moving along a straight line, modeled by the funct

Dynamic Motion Analysis: Cubic Position Function

A particle's position along the x-axis is given by $$x(t) = t^3 - 6t^2 + 9t$$ with time t in seconds

FRQ 2: Projectile Motion – Launch Experiment

A researcher uses a projectile launcher to study the flight of a ball launched at an angle. The ball

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 7: Motion with Variable Acceleration (MEDIUM)

An object moving along the x-axis experiences an acceleration given by $$a(t)=6*t-4$$ (in m/s²), and

FRQ 9: Piecewise Acceleration Motion (HARD)

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

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

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

FRQ 11: Air Resistance and Terminal Velocity

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

FRQ 20: Real-World Application – Car Braking Analysis

A car traveling at 30 m/s begins braking uniformly until it comes to a complete stop. Sensors record

Graphical Analysis of Kinematic Data

Consider the following velocity vs. time graph for an object in motion. Use the graph to answer the

Impact Analysis: Collision Avoidance

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

Impulse and Momentum with a Variable Force

A cart of mass $$m = 5.0\,kg$$ is subjected to a time-dependent force described by $$F(t)=5*t²$$ (in

Kinematics with Calculus: Non-Uniform Acceleration

An object moves along the x-axis under a non-uniform acceleration given by $$a(t) = 4*t - 2$$ m/s²,

Motion Analysis Using Integrals

An object moves along a straight line with an acceleration given by $$a(t)=6-2*t$$ (m/s²) for $$0\le

Motion with Variable Acceleration

An object has a time-dependent acceleration given by $$a(t)= 6*t - 4$$ (in m/s^2) and starts from re

Photogate Timer in Free Fall

A student uses a photogate timer to record the free fall of an object dropped from a height of 1.5 m

Projectile Motion with Air Resistance

Design an experiment to study the effect of launch angle on the horizontal range of a projectile in

Projectile Motion with Timing Error

In an outdoor lab experiment, a projectile launcher was used to fire a ball at a 45° angle relative

Relative Motion in an Accelerating Frame

Inside an elevator accelerating upward at 2 m/s², an object is dropped. Its motion is recorded relat

Relative Motion of Two Vehicles

Two vehicles start from the same point and travel along a straight road in opposite directions. Vehi

Simple Harmonic Motion in a Spring-Mass System

Design an experiment to investigate simple harmonic motion (SHM) using a spring-mass system. Describ

Time of Flight Measurement Using Video Analysis: Frame Rate Miscalibration

A student recorded a projectile's motion using a digital video camera to measure its time of flight,

Two-Dimensional Projectile with an Elevated Launch Point

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

Uniformly Accelerated Motion on a Track

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

Uniformly Accelerated Motion With Non-Zero Initial Velocity

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

Variable Acceleration Analysis

An object experiences variable acceleration described by $$a(t) = 2*t$$ (in m/s²).

Vector Addition in Two-Dimensional Motion

An object is launched with a horizontal velocity of $$30\,m/s$$ and a vertical velocity of $$40\,m/s

Vector Addition in Two-Dimensional Projectile Motion

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

Vector Decomposition in Displacement Measurements

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

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

Block Under a Varying Force

A 2 kg block moves along a frictionless horizontal surface under the influence of a variable force g

Calculating Kinetic Energy from a Velocity Function

A particle of mass $$m = 1 \;\text{kg}$$ moves along the x-axis with a velocity given by $$v(t)= 3*t

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

Calculus‐Based Energy Conversion in Elastic Collisions

Two masses, $$m_1$$ and $$m_2$$, undergo an elastic collision. (a) Derive the conservation equatio

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

Comparing Work–Energy Analysis Across Different Reference Levels

A researcher examines the impact of choosing different reference levels for potential energy calcula

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 a Mass-Spring Oscillator

A mass-spring system consists of a 1 kg mass attached to a spring with a spring constant of 100 N/m.

Energy Conservation in a Pendulum

A simple pendulum of length 2 m and mass 0.5 kg is released from an initial angle of 30° with respec

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 1: Analysis of Work at an Angle

A media report claims that when a constant force is applied at an angle to the displacement, the wor

FRQ 3: Kinetic Energy Change in a Car's Acceleration

A 1200-kg car accelerates along a straight, level road. Measurements of the car's speed at various d

FRQ 8: Pendulum Energy Transformations with Damping

An experimental study on a pendulum claims that its mechanical energy is conserved, assuming only gr

FRQ 9: Interpreting Drop Test Kinetic and Potential Energy Data

A study provides experimental data for a 3 kg ball dropped from various heights, with the measured s

FRQ 16: Evaluating Power Output Measurements in a Rocket Launch

A media report asserts that the power output of a rocket engine can be approximated by the formula $

FRQ 17: Energy Distribution in Car Crash Safety Studies

A study on car crash safety claims that the kinetic energy of a moving car is completely dissipated

Instantaneous Power in a Variable Force Scenario

An object is subjected to a time-dependent force given by $$F(t)=5*t$$ N, and its displacement is de

Kinetic Energy Measurement in a Projectile Experiment

A researcher is studying the change in kinetic energy of a small projectile. A ball of mass $$m = 0.

Model Rocket Power Measurement Experiment

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

Non-Uniform Gravitational Field Work-Energy Calculation

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

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 of a Diatomic Molecule

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

Power and Energy in High-Speed Systems: Rocket Launch Analysis

A researcher is analyzing the power requirements of a rocket engine during a launch phase. A rocket

Power Output from a Variable Force: Time-Dependent Problem

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

Pulley System Work–Energy Verification

A two-mass pulley system is used to verify the work–energy theorem. Velocities of both masses are re

Rolling Through a Loop-the-Loop

A roller coaster car of mass 500 kg starts from rest at a height of 50 m above the bottom of a verti

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 Work on a Block

A 4 kg block is pushed along a horizontal, frictionless surface by a variable force given by $$F(x)

Variable Force Work Calculation and Kinetic Energy Analysis

Consider an object moving along the x-axis under the influence of a variable force given by $$F(x) =

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

Variable Mass Rocket Energy Calculation

A rocket burns fuel at a constant rate so that its mass decreases with time according to $$m(t)= M_0

Vertical Lift Work Measurement Experiment

In this vertical lift experiment, an object is raised by a motor and its applied force and displacem

Work and Power in an Engine

A 1500 kg car is accelerated from rest by an engine whose power output varies with time according to

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 by a Variable Exponential Force

A piston is subjected to a variable force described by $$F(x)=500*\exp(-0.5*x)$$ N, where x is in me

Work Done by a Variable Force

An object is acted upon by a variable force given by $$F(x)=5\,x^2$$ (in newtons) along the x-axis.

Work–Energy Analysis in a Rotating Space Station

A rotating space station uses thrusters to adjust its rotation rate. The work required to change the

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

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 Mass Calculation for a Curved, Variable Density Wire

Students attempt to determine the center of mass of a flexible wire whose density varies along its l

Center of Mass Measurement Using a Suspended Rod

In this experiment, students attempt to determine the center of mass of a non-uniform rod by suspend

Center of Mass of a Non‐Uniform Rod

A non‐uniform rod of length $$L = 1$$ m has a linear density that is modeled by $$\lambda(x) = 5 + 2

Center of Mass of a System of Particles

Three particles are located along the x-axis at positions: Particle 1 (mass $$2\,kg$$ at $$x=1\,m$$)

Center of Mass of a System of Particles in 3D Space

Three point masses are located in 3D space with the following properties: - Mass 1: 2 kg at coordin

Center of Mass of a Variable Density Rod

A rod of length $$L = 1\,\text{m}$$ has a linear density given by $$\lambda(x) = 5 + 3*x$$ (in kg/m)

Center-of-Mass Motion Under an External Force

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

Center-of-Mass Shift in an Internal Explosion

Three masses are arranged on a frictionless horizontal surface: m1 = 1 kg at (0,0), m2 = 2 kg at (1,

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

Combined Translational and Rotational Motion Analysis

A uniform rod of length $$2\,m$$ and mass $$4\,kg$$ is pivoted at one end. Initially at rest, an imp

Conservation of Linear Momentum in a Glider Collision

On a frictionless air track, two gliders collide. The experimental data below list the masses and ve

Displacement from Variable Acceleration

A 2 kg particle experiences a force given by $$F(t)=10*t$$ N over the time interval from 0 s to 4 s.

Elastic Collision Analysis

Two balls undergo an elastic head-on collision. Ball A has a mass of $$0.6\,\text{kg}$$ and an initi

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: Investigating Collision Elasticity

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

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 Fragmentation: Momentum Transfer

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

Fragmentation and Impulse

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

FRQ 9: Rocket Propulsion and Momentum Conservation

A rocket in space initially has a mass of $$500 \ kg$$ and is traveling at $$20 \ m/s$$. It ejects $

FRQ 15: Center of Mass versus Center of Gravity

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

Impulse 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 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 Velocity from a Variable Force

A particle of mass $$m=2.0\,kg$$ initially moves with a velocity $$v_i=2.0\,m/s$$. It is subjected t

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 from a Time-Varying Force with Graph Stimulus

A force sensor records the force applied to a hockey puck as a function of time while a player strik

Impulse Measurement via Force-Time Graph Analysis

A student attaches a force sensor to a baseball bat to record the force exerted on a ball during imp

Impulse on a Rolling Soccer Ball with Piecewise Force

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

Impulsive Collision Involving Rotation

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

Inclined Plane: Center of Mass and Impulse Analysis

A block of mass $$3$$ kg rests on a frictionless inclined plane with an angle of $$30^\circ$$. The i

Inelastic Collision Energy Loss Analysis

Two carts on a frictionless track undergo a completely inelastic collision. Cart A has a mass of $$1

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)

Momentum Conservation in a Skaters' Push-Off

Two ice skaters start from rest on frictionless ice. Skater A has a mass of 50 kg and, after pushing

Momentum Conservation in Glider Collisions on an Air Track

In a laboratory experiment, students investigate conservation of momentum by allowing two gliders to

Motion of the Center of Mass with External Force

Consider two blocks (masses 3 kg and 5 kg) connected by a light spring on a frictionless surface. In

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

Nonuniform Rod: Total Mass and Center of Mass

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

Oscillations: Simple Pendulum Analysis

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

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 with Variable Mass

A rocket has an initial mass of $$M_0 = 50$$ kg (including fuel) and ejects fuel such that its mass

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

Three-Body Collision on a Frictionless Table

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

Variable Density Rod: Mass and Center of Mass Calculation

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

Analysis of a Variable Moment of Inertia System

A rotating disk has a moment of inertia that decreases over time as its arms retract, following the

Analysis of Rotational Equilibrium in a Beam

A uniform beam of length $$L = 4\,m$$ is balanced on a frictionless pivot located 1 m from one end.

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 Derivation of Moment of Inertia for a Thin Ring

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

Centripetal Force and Angular Velocity Measurement

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

Complex Rotational Motion: Gyroscopic Precession

A spinning top has a spin angular momentum of $$L = 0.15 \text{ kg m}^2/\text{s}$$ and experiences a

Composite Body Rotation

A composite object is formed by welding a solid disk to a thin rod. The disk has mass $$M$$ and radi

Composite Object Rotational Dynamics Analysis

A researcher studies the rotational dynamics of a composite object composed of a uniform disk of mas

Conservation of Angular Momentum in Rotational Collisions

Two disks (Disk A and Disk B) rotate independently and are then brought into contact, eventually rot

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

Design and Analysis of a Flywheel Energy Storage System

A flywheel, modeled as a solid disk, is used for energy storage. The flywheel has a mass $$M=50 \tex

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 Wheel under Applied and Frictional Torques

A motor applies a constant torque of 12 Nm to a wheel with a moment of inertia of 0.8 kg m^2. A fric

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 Conservation in Combined Rotational and Translational Motion

A sphere is made to roll down an incline without slipping, converting gravitational potential energy

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 14: Energy Loss in a Rotating Flywheel Due to Friction

A flywheel with moment of inertia \(I = 8.00\,kg\cdot m^2\) is initially spinning at \(\omega_0 = 15

FRQ 20: Time-Dependent Angular Acceleration with External Torque

A flywheel with moment of inertia \(I = 3.00\,kg\cdot m^2\) experiences an exponentially decaying ex

Impact of Changing Mass Distribution on Angular Acceleration

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

Inelastic Collision of Rotating Disks

Two disks mounted on a common frictionless axle collide and stick together. Disk A has a moment of i

Integration of Rotational Inertia: Thin Shell vs. Solid Sphere

Derive the moments of inertia for two spherical objects about an axis through their centers: (a) A

Investigating the Parallel Axis Theorem

A researcher examines the effect of changing the axis of rotation on the moment of inertia of a rigi

Investigation of Torque on a Rotating Pulley

In an experiment, a student applied a constant force of $$F = 40\,N$$ at varying distances (moment a

Mass Redistribution and Kinetic Energy in Rotating Systems

In a rotating system, a person on a rotating platform moves closer to the axis, reducing the system’

Non-uniform Mass Distribution Effects on Rotational Inertia

Consider a rod of length $$L$$ whose linear mass density varies as $$\lambda(x) = \lambda_0 * (1 + x

Parallel Axis Theorem Application in Complex Systems

A composite object consists of a uniform rod of mass $$M = 4\,kg$$ and length $$L = 3\,m$$, and an a

Rolling Motion Energy Analysis

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

Rolling Motion: 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 Inertia Determination Using a Torsion Pendulum

You are provided with a torsion pendulum apparatus consisting of a rod suspended by a wire with a kn

Rotational Kinematics on a Spinning Disk

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

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

Time-varying Angular Acceleration in a Rotational System

A disk experiences an angular acceleration described by $$\alpha(t)=5\sin(2t) \text{ rad/s}^2$$.

Torque and the Right-Hand Rule Verification Experiment

Design an experiment to verify the direction of the torque vector as predicted by the right-hand rul

Torque from a Distributed Load

A uniform beam of length $$L$$ has a constant linear weight density $$w$$ (in N/m).

Torque Measurement and Angular Acceleration Experiment

In this experiment, you will investigate the relationship between applied force, moment arm, and the

Torsion Pendulum and Restoring Torque Error

In a torsion pendulum experiment, a disk is suspended from a wire and its angular displacement due t

Using Experimental Data to Evaluate Conservation of Angular Momentum

An experimental setup involves a rotating platform where the moment of inertia and angular velocity

Work Done by Torque and Rotational Kinetic Energy

An engine applies a constant torque to a flywheel, causing it to rotate from rest through an angular

Amplitude and Maximum Speed Relationship in SHM

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

Amplitude Dependence in a Nonlinear Oscillator

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

Analysis of Maximum Kinetic Energy in a Spring-Mass Oscillator

For a spring-mass oscillator with spring constant $$k$$ and amplitude $$A$$, the elastic potential e

Analyzing a Mass-Spring System on an Inclined Plane

A block of mass $$m = 1.0\,\text{kg}$$ is attached to a spring with spring constant $$k = 100\,\text

Calculus of Oscillatory Motion: Velocity and Acceleration

A researcher analyzes the displacement of a mass-spring oscillator given by the function $$y(t) = 0.

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

Comparative Analysis: Spring-Mass vs. Pendulum Oscillators

An experiment compares the oscillatory behavior of a spring-mass system and a simple pendulum. Answe

Comparison of Oscillatory Systems: Spring vs. Pendulum

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

Conservation of Energy: Integral Approach in SHM

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

Damped Oscillations and Energy Decay

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

Data Analysis and Calculus Estimation in SHM

Using discrete experimental data for a harmonic oscillator, estimate derivatives and analyze acceler

Data Analysis of a Spring-Mass Experiment

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

Derivation and Solution of SHM Differential Equation

A mass-spring system exhibits simple harmonic motion. Derive the differential equation governing the

Derivation of the SHM Differential Equation

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

Deriving the General Solution of SHM

Derive and analyze the general solution for simple harmonic motion from the governing differential e

Energy Exchange in Coupled Oscillators

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

Energy Exchange in Oscillatory Systems

A new research article claims that 'the maximum speed of a block on a spring is invariant with respe

Energy Exchange in SHM

Consider a mass-spring oscillator with displacement given by: $$x(t) = A * \cos(\omega t)$$, with

Evaluating Hooke's Law in Spring Oscillators

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

Experimental Analysis of SHM Data

The displacement data for a mass-spring system were recorded during a laboratory experiment. The fol

Experimental Determination of Spring Constant

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

Forced Oscillations and Beat Frequency

A mass attached to a spring is simultaneously driven by two periodic forces given by $$F_1(t)=F_0*\c

Forced Oscillations and Resonance

An oscillator is driven by an external force and is modeled by the equation $$m\ddot{x} + kx = F_0 \

FRQ 5: Period of a Simple Pendulum

An ideal simple pendulum has a length of $$L = 1.0\ m$$ and swings with a maximum angular displaceme

FRQ 10: Calculus Integration for Work Done in a Spring

Force measurements during the stretching of a spring were recorded as a function of displacement. Us

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 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 16: Frequency Determination from Oscillatory Data

An experiment records the displacement of a mass undergoing simple harmonic motion at various times.

FRQ7: Period of a Simple Pendulum under the Small-Angle Approximation

A simple pendulum consists of a mass attached to a massless string of length \(L\). Answer the foll

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 Oscillatory Data

A researcher records and graphs the displacement of a mass-spring oscillator as a function of time.

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

Kinematics and Phase Angle Determination

An oscillator is described by the equation $$y = A\sin(\omega t + \phi_0)$$ with an initial conditio

Kinematics of SHM: Period and Frequency Measurements

Analyze the kinematics of a simple harmonic oscillator using time measurements.

Non-conservative Forces in Oscillating Systems

In an experiment with a spring-mass oscillator, students study the effect of friction on the oscilla

Nonlinear Pendulum Oscillations and Error Analysis

A pendulum with a length of $$L = 2.0\,m$$ is released from an initial angle of 45° (approximately 0

Oscillatory Motion of a Block on a Horizontal Spring

A block of mass $$m = 0.8 \; kg$$ is attached to a horizontal spring with a spring constant of $$k =

Pendulum Dynamics Beyond the Small-Angle Approximation

Investigate the dynamics of a pendulum when the small-angle approximation begins to break down.

Pendulum Energy Dynamics

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

Pendulum Motion and the Small Angle Approximation

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

Pendulum Motion Experimental Analysis

A simple pendulum experiment is used to measure the period of oscillation. A bob is attached to a ma

Pendulum Period and Data Analysis

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

Phase Angle Determination from Initial Conditions

An oscillator exhibits motion described by $$y(t) = A * \sin(\omega*t + \phi_0)$$ with amplitude $$A

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

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

Phase Space Trajectories in Simple Harmonic Motion

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

Sinusoidal Motion: Phase Constant Determination

An oscillator’s motion is described by the equation $$y = A \sin(\omega t + \phi_0)$$ with an amplit

Spring Force and Elastic Potential Energy

A spring with a force constant $$k = 300\,N/m$$ and a natural length of 0.50 m is stretched to a len

Spring-Mass Oscillator on an Inclined Plane

A mass $$m = 0.5\,kg$$ is attached to a spring with force constant $$k = 150\,N/m$$, and the assembl

Stress Testing of Oscillatory Limits

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

Systematic Error Analysis in SHM Experiments

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

Time-Dependent Length in a Variable-Length Pendulum

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

Vertical Oscillations and Energy Analysis in a Spring–Mass System

Investigate the motion and energy conversion of a vertically oscillating mass–spring system.

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 on a Spring

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

Vertical Oscillations: Lab Data Analysis

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

Vertical Oscillator in a Gravitational Field

A block of mass $$m = 2.0 \;\text{kg}$$ is attached to a vertical spring with force constant $$k = 4

Analysis of Orbital Transfer Maneuvers Using Calculus

A spacecraft is initially in a circular orbit of radius $$ r_1 $$ and is to be transferred to a circ

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

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

Comparative Analysis of Planetary Orbits

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

Derivation of 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 Potential Energy

Starting from Newton's law of gravitation given by $$F(r) = -G * \frac{M * m}{r^2}$$, derive the exp

Derivation of Gravitational Potential Energy Difference

A space probe's gravitational potential energy is given by $$U(r) = -\frac{G M m}{r}$$. Answer the f

Derivation of Kepler's Second Law from Angular Momentum Conservation

Using the principle of conservation of angular momentum, derive Kepler's Second Law which states tha

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

Designing a Cavendish Experiment to Measure the Gravitational Constant

A student plans to design a version of the Cavendish experiment to measure the gravitational constan

Designing a Modern Cavendish Experiment

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

Designing a Satellite Orbit Experiment

An engineering team is planning an experiment to study satellite orbits around Earth. (a) List the

Determining Planetary Mass from Satellite Orbital Data

Satellite orbital data can be used to determine the mass of the planet they orbit. Answer the follow

Escape Velocity Derivation

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

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

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

Graphical Analysis of Gravitational Force Variation

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

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 Energy in a Three-Body System

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

Gravitational Force Calculation Between Celestial Bodies

Consider two celestial bodies with masses $$m_1$$ and $$m_2$$ separated by a distance $$r$$. Newton'

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 Slingshot Maneuver Analysis

Analyze the gravitational slingshot maneuver used by spacecraft to gain speed during a flyby of a pl

Investigating Orbital Eccentricity Effects

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

Kepler's Third Law and Planetary Motion

Consider two planets orbiting the same star with orbital periods $$T_1$$ and $$T_2$$ and semimajor a

Mathematical Modeling of Tidal Forces

Using the provided data on tidal forces measured at different distances, analyze how the tidal force

Modeling Orbital Decay with Differential Equations

A satellite in orbit experiences a drag force proportional to its velocity, leading to orbital decay

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