AP Physics C: Mechanics FRQ Room

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

Analysis of a Velocity vs. Time Graph in a Lab Experiment

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

Analysis of a Velocity-Vs-Time Graph

An object’s velocity is recorded along a straight path. The velocity-time graph indicates that the o

Calculus Analysis of a Parabolic Trajectory

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

Calculus in One-Dimensional Kinematics

Consider an object whose position as a function of time is given by $$x(t)=\sin(t)+t^2$$, where x is

Calculus-Based Kinematics Derivation

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

Critical Evaluation of Experimental Kinematics Data

A researcher claims that the motion of a falling object is characterized by uniform acceleration. Th

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

Differential Equation of Motion Under Gravity and Drag

A particle of mass $$m$$ is falling under gravity and experiences a drag force proportional to its v

Experimental Determination of g

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

Free Fall Kinematics

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

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: 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 7: Projectile Trajectory Analysis

A projectile is fired from ground level with an initial speed of 50 m/s at an angle of 40° above the

FRQ 8: Circular Motion Kinematics (MEDIUM)

An object moves in uniform circular motion with its position given by $$\vec{r}(t)=(R\cos(\omega*t),

FRQ 8: Vector Addition in Two-Dimensional Motion

An object moves in a plane following these displacements in sequence: 4 m east, 3 m north, 5 m west,

FRQ 10: Threshold Velocity in Vertical Projectile Motion (MEDIUM)

An object is launched vertically upward with an initial speed of $$40\,m/s$$. Its velocity as a func

FRQ 13: Analyzing a Two-Dimensional Collision with Projectiles

A researcher conducts an experiment with two projectiles launched simultaneously from different posi

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

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

Impulse and Variable Force Analysis

Design an experiment to measure the impulse delivered to a ball by a bat, given that the force varie

Investigating Lab Data: Graph Interpretation and Improvements

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

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,

Kinematics with Non-Constant Acceleration

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

Motion Analysis Using Integrated Acceleration Data

Researchers used an accelerometer attached to a moving cart to record its acceleration over a period

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

Non-linear Position Function Analysis

A particle moves along the x-axis with a non-linear, decaying oscillatory position given by $$x(t)=e

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

Projectile Motion Revisited: Maximum Height and Impact Velocity

An object is projected vertically upward from ground level with an initial speed of $$50\,m/s$$. Ass

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: Maximum Height and Range

A projectile is launched from ground level with an initial speed of $$v_0 = 60$$ m/s at an angle of

Relative Displacement in Different Frames

A particle moves along the x-axis and its position is described by $$x(t)=7*t-2*t^2$$ (in meters). T

Rotational Motion: Angular Kinematics

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

Skydiver with Air Resistance: Variable Acceleration

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

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

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

Variable Acceleration Analysis

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

Circular Motion with Tangential Work

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

Comparative Analysis of Constant vs. Variable Gravitational Work

An object of mass $$m= 10 \;\text{kg}$$ is lifted from the ground (assume $$x=0$$) to a height of $$

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

Elastic Collision and Energy Transfer

Two blocks, A (2 kg) and B (3 kg), slide without friction on a horizontal surface. Initially, block

Energy Analysis of a Damped Pendulum

A pendulum of length 2 m and mass 1 kg oscillates in air. The damping force due to air resistance is

Energy Conservation on a Frictional Ramp with Calculus Approach

A 2-kg block slides down a straight inclined ramp from a height of $$h = 2 \;\text{m}$$. The ramp is

Energy Dissipation in a Bouncing Ball

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

Energy in a Spring–Mass System

A mass is attached to a spring with a spring constant of $$k = 200\,N/m$$. The spring is compressed

Experiment on Electric Motor Power Output

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

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 4: Conservation of Mechanical Energy in a Roller Coaster

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

FRQ 8: Investigation of Variable Power Output in a Pulley System

A pulley system is used to tow a load with a constant force of 100 N. A sensor records the instantan

FRQ 9: Calculus-Based Work Determination in a Braking Scenario

A car undergoing braking experiences a variable force that depends on its displacement. The braking

FRQ 10: Work Done on a Variable Mass System

A recent claim suggests that for systems with variable mass (e.g., rockets), the work done by a non-

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 20: Evaluating Efficiency in a Conveyor Belt System

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

High-Power Engine Performance Test

An engine is tested on a dynamometer. Its instantaneous force is given by $$F(t)=1000 + 200*t$$ N fo

Hydraulic Press Work Calculation Experiment

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

Impulse and Energy Transfer via Calculus

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

Inclined Plane Energy Transfer Experiment

In an experiment, a 2 kg block is released from rest and slides down a frictionless inclined plane o

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

Potential Energy Curve Analysis

An object of mass 4 kg moves in a potential energy field described by $$U(x) = (x-1)^2 - 0.5*(x-2)^3

Power Output Fluctuations in a Jogger

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

Power Output Variation in a Machine

An object is being moved along a straight path by a constant force of 10 N. The displacement as a fu

Pulley System Work–Energy Verification

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

Relationship Between Force and Potential Energy

For a conservative force, the relationship between force and potential energy is given by $$F(x) = -

Rocket Engine Power Output Analysis

A rocket of mass 1000 kg is traveling horizontally at a constant speed of 8.0 m/s under an engine th

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 Energy Transfer in a Spinning Disc

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

Rotational Power in Gear Systems

An experiment measures the power output of a gear train by recording the torque and angular velocity

Rotational Work and Energy in a Falling Rod

A uniform thin rod of length $$L = 2.0\,m$$ and mass $$m = 4.0\,kg$$ is initially held horizontally

Tidal Energy Extraction Analysis

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

Vertical Lift Work Measurement Experiment

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

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 Against Friction

An 8 kg block slides on a horizontal surface with a kinetic friction coefficient of 0.25. It comes t

Work-Energy Analysis in Collisions

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

Work-Energy Analysis on an Inclined Plane

A 3 kg block slides down a fixed inclined plane that makes an angle of 30° with the horizontal. The

Work–Energy and Friction: Analyzing a Sliding Block

A researcher investigates the effect of friction on a sliding block. A 4 kg block is launched horizo

Work–Energy Experiment in Varying Potential Fields

A researcher studies the motion of an object of mass 4 kg in a potential energy field described by $

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 Multi-Peak Force-Time Graph

A cart of mass 2 kg is subjected to a force that varies with time according to a piecewise function:

Analyzing Momentum Change in a Two-Cart Collision

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

Center of Mass Calculation of a Non-Homogeneous Beam

A horizontal beam of length $$4$$ m has a linear mass density given by $$\lambda(x) = 5 + 4 * x^2$$

Center of Mass for Discrete Particles in the Plane

Three particles are located in the plane with the coordinates and masses given in the table below:

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 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 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,

Conservation of Angular Momentum on a Rotating Platform

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

Conservation of Linear Momentum in Colliding Carts

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

Data Analysis: Testing the Linear Relationship between Impulse and Momentum Change

An experiment is conducted where a cart is subjected to several impacts. For each trial, the measure

Elastic Collision in Two Dimensions

Two particles collide elastically on a frictionless plane. Particle 1 (mass $$m_1=1.0\,kg$$) initial

Elastic Collision of Air Track Gliders

On a frictionless air track, Glider A (mass = 0.8 kg) moves to the right at 2.0 m/s while Glider B (

Elastic Collision: Conservation of Momentum and Kinetic Energy

Two spheres A and B collide elastically. Sphere A has a mass of 0.6 kg and an initial velocity of $$

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

Explosive Separation and Momentum Conservation

An object of total mass $$M$$ at rest explodes into two fragments. One fragment has mass $$m$$ and i

FRQ 13: Critical Analysis: Momentum Experiment

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

FRQ 18: Critical Evaluation: Inelastic Collision Study

A published study on vehicle collisions claims that experimental momentum measurements in inelastic

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 Momentum under a Variable Force

A 5 kg cart on a frictionless track is initially at rest. A variable force given by $$F(t)=10*t$$ (N

Impulse Calculation from a Force-Time Graph

A force acting on a cart is recorded by a sensor and is represented by the following graph: the forc

Impulse Calculation from Force-Time Graph

A force sensor records a time-dependent force acting on an object. The force is modeled by the equat

Impulse Delivered by a Time-Dependent Damping Force

A 3 kg block on a frictionless surface experiences a damping force that varies with time, given by $

Impulse from a Piecewise Force-Time Profile

A 2 kg particle experiences a force whose magnitude is described as follows: from t = 0 s to 2 s, th

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 in a Non-Constant Force Field

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

Impulse 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 Pendulum Bob

A pendulum bob of mass $$1.0$$ kg is initially at rest hanging from a string. An impulsive force is

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: 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.

Momentum Analysis in an Asteroid Breakup

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

Motion of the Center of Mass Under an External Force

A system consists of two particles with masses 2 kg and 3 kg located at x = 0 m and x = 4 m, respect

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

Nonuniform Circular Disk Center of Mass

A circular disk of radius $$R$$ has a nonuniform surface mass density given by $$\sigma(r,\theta)=\s

Oblique Collision of Ice Pucks

Two ice pucks are sliding on a frictionless ice surface. Puck A (mass = 0.2 kg) is moving with a vel

Oblique Collision of Ice-Hockey Pucks

Two ice-hockey pucks collide on frictionless ice. Puck A (mass = 0.2 kg) moves east at 5 m/s, and Pu

Projectile Center-of-Mass Trajectory

A projectile is launched and its trajectory is recorded with emphasis on the motion of its center of

Projectile Motion with Drag Impulse Analysis

A projectile is launched horizontally and experiences a drag force proportional to its velocity, giv

Rocket Propulsion Momentum Problem

A model rocket with an initial mass of $$2\,kg$$ (including fuel) ejects $$0.5\,kg$$ of fuel instant

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

Stability and Center of Mass of a Structure

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

Stability of a Suspended Mobile

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

Vibrational Motion: Coupled Oscillators

Two masses (m1 = 0.5 kg and m2 = 0.5 kg) are connected by a spring with spring constant k = 200 N/m

Work-Energy Theorem: Roller Coaster Problem

A 500 kg roller coaster starts from rest at the top of a 40 m hill and descends to a valley 10 m abo

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.

Analysis of Rotational Equilibrium in a Complex System

A hanging sign is suspended by two cables attached at different points. The sign rotates about a piv

Angular Kinematics on a Rotating Platform

A rotating platform starts from rest and accelerates with a constant angular acceleration $$\alpha$$

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 a Spinning System

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

Angular Momentum Transfer in Coupled Rotating Disks

In an experiment, two disks are coupled so that they eventually rotate together without any external

Comparative Study of Angular Kinematics at Different Radii

In a lab experiment, students measure the angular displacement and corresponding linear displacement

Comparison of Rolling Objects with Different Mass Distributions

Consider two cylinders, one solid and one hollow, each of mass $$M$$ and radius $$R$$, that roll wit

Conveyor Belt Dynamics Driven by a Rotating Drum

A rotating drum of radius R drives a conveyor belt without slipping. Derive the relationship between

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

Determining Angular Acceleration from Time-Resolved Measurements

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

Dynamic Equilibrium in Rotational Motion

Design an experiment to investigate the conditions for rotational equilibrium in a lever system. You

Effect of Changing Moment Arm on System Dynamics

Design an experiment where you systematically vary the moment arm (the distance from the pivot) in a

Energy Conversion in Rolling Motion Experiments

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

Energy Dissipation in a Rotating System with Friction

A turntable with moment of inertia $$I = 0.3 \text{ kg m}^2$$ is initially spinning at $$\omega_0 =

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 13: Dynamics of a Variable Torque System

A rod with moment of inertia \(I = 4.00\,kg\cdot m^2\) is subjected to an angular-dependent torque g

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

Impulse and Angular Momentum: Impact on a Rotating Disk

A solid disk of mass $$M = 2.0 \text{ kg}$$ and radius $$R = 0.25 \text{ m}$$ is spinning with an in

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

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

Nonlinear Angular Acceleration in a Damped Rotational System

A student studies a damped rotating disk where friction causes the angular acceleration to vary non-

Parallel Axis Theorem in Rotational Systems

A uniform disk of mass $$M$$ and radius $$R$$ has a moment of inertia about its central axis given b

Rolling Motion Energy Analysis on an Inclined Plane

A cylinder is allowed to roll down an inclined plane without slipping, converting gravitational pote

Rolling Motion on an Inclined Plane

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

Rotational Inertia Measurement with a Disk and Pendulum

In this experiment a flat disk is mounted as a pendulum with its pivot offset from its center. The o

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

Rotational Kinetic Energy and Work by Friction

A flywheel with a moment of inertia of 2.0 kg m^2 rotates initially at 10 rad/s. It comes to rest du

Torque and Rotational Inertia in Engine Mechanisms

You are exploring the behavior of torque in a small engine. Design an experiment to measure the engi

Using Experimental Data to Evaluate Conservation of Angular Momentum

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

Verification of the Parallel Axis Theorem

A solid disk of mass M = 4 kg and radius R = 0.5 m is considered. The moment of inertia about its ce

Calculus of Oscillatory Motion: Velocity and Acceleration

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

Calculus-Derived Velocity and Acceleration in SHM

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

Comparative Period Analysis: Mass-Spring Oscillator vs. Simple Pendulum

A researcher is comparing the oscillatory behavior of a horizontal mass-spring system and a simple p

Coupled Oscillators: Normal Modes and Energy Transfer

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

Damped Oscillations and Energy Decay

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

Damped Oscillations in a Spring System

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

Derivation and Solution of the Differential Equation for SHM

Starting from Newton's second law, derive the differential equation governing the motion of a spring

Deriving the Equation of Motion Using Calculus

An undamped harmonic oscillator is governed by the differential equation $$\ddot{x} + \omega^2 x = 0

Determination of Gravitational Acceleration Using a Vertical Oscillator

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

Determining Initial Phase in SHM

A simple harmonic oscillator is described by the equation $$x(t)=A\sin(\omega t+\phi_0)$$. An oscill

Differential Equation of Coupled Oscillators

A more advanced experiment involves studying two masses attached by springs (coupled oscillators) to

Differentiation of Sinusoidal Motion

Given the position function of a mass-spring oscillator as $$x(t) = 0.08\,\text{m} \sin(10\,t + 0.2)

Effects of Spring Constant Variation on Oscillatory Motion

A spring-mass system oscillates with motion given by $$y(t)=A*\cos(\omega*t)$$ where $$\omega=\sqrt{

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 Conservation in Vertical Spring Oscillations

A 1.5 kg block is attached to a vertical spring with force constant $$k = 300\,N/m$$. After reaching

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 Transformation in SHM

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

Experimental Determination of Spring Constant

In a lab experiment, students measure the displacement of a spring under various applied forces. The

Forced Oscillations and Resonance

A mass-spring system is driven by an external force of the form \(F(t) = F_0 \cos(\omega_d t)\) and

Frequency Response Analysis from Experimental Data

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

Friction Effects in Horizontal Oscillatory Systems

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

FRQ 2: Energy Conversion in a Spring Oscillator

A block attached to a spring oscillates on a frictionless surface. The following table presents expe

FRQ 4: Vertical Motion in a Spring–Block System

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

FRQ 6: Sinusoidal Description of SHM

A simple harmonic oscillator has an amplitude of $$A = 3.0\ cm$$ and a frequency of $$f = 4.0\ Hz$$.

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 14: Work Done by a Spring via Integration

Using calculus, derive an expression for the work done in stretching a spring from its natural lengt

FRQ 17: Determination of Damping Coefficient

An oscillatory system exhibits damping, with its amplitude decreasing over successive cycles. Analyz

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

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 Experimental Data

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

Horizontal Spring Oscillator: Force and Energy Calculations

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

Influence of Initial Phase on Oscillator Motion

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

Integral Calculation of Work Done in SHM

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

Investigating Damping Effects in a Spring-Mass Oscillator

In real oscillatory systems, damping forces affect the motion of the oscillator. Consider a spring-m

Investigation of Energy Conservation in SHM Using Calculus

A researcher analyses a mass-spring oscillator with its displacement given by $$y(t) = 0.1 * \cos(15

Kinematics of SHM: Period and Frequency Measurements

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

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

Mass-Spring Differential Analysis

Consider a block of mass $$m$$ attached to a horizontal spring with spring constant $$k$$. The block

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

Nonlinear Effects in a Large-Amplitude Pendulum

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

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

Nonlinear Restoring Force: Effects on the Period of Oscillations

A system is characterized by a restoring force that is not perfectly linear: $$F = -k\,x - \alpha\,x

Pendulum Angle Dependence and the Small Angle Approximation

A recent news article claims that 'the period of a pendulum is completely independent of the amplitu

Pendulum Energy Dynamics

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

Pendulum Experiment Analysis

A researcher uses a simple pendulum to measure gravitational acceleration. The pendulum has a length

Pendulum Period and Data Analysis

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

Phase Analysis and Initial Conditions in SHM

A mass attached to a spring oscillates such that its displacement is given by $$y(t) = A \sin(\omega

Phase Shift Analysis in Driven Oscillators

Consider an experiment where a driven oscillator is used to measure phase shifts as the driving freq

SHM: Spring Force and Energy Derivation

A spring with force constant $$k = 200 \;\text{N/m}$$ is fixed at one end. When the other end is dis

Sinusoidal Analysis of SHM with Phase Shift

Examine a sinusoidally described simple harmonic oscillator with a phase shift.

Sinusoidal SHM with Phase Shift

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

Stress Testing of Oscillatory Limits

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

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_

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 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 Oscillator with Offset Equilibrium

A vertical mass-spring system has a mass of $$m = 1.0\,kg$$ attached to a spring with force constant

Vertical Spring Oscillator Investigation

In a vertical spring oscillator experiment, a mass is attached to a spring hanging from a fixed supp

Vertical Spring–Block Oscillator Dynamics

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

Analysis of Gravitational Anomalies: Local Variations in g

Local measurements of gravitational acceleration $$g$$ exhibit small variations due to underlying de

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

Barycenter Determination in a Sun-Planet Analog with Magnetic Models

A lab experiment simulates the Sun-Earth system using scaled models equipped with magnetic component

Center of Mass in a Two-Body System

In a two-body system, such as a planet and its moon, both bodies orbit around their common center of

Comparative Analysis of Orbital Periods for Different Planets

Two planets orbit a star. (a) Derive the relationship $$\frac{T^2}{a^3} = \text{constant}$$ for thes

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

Derivation of Orbital Period in Binary Star Systems

A researcher studies a binary star system in which two stars of masses $$m_1$$ and $$m_2$$ orbit the

Determination of Gravitational Constant Using a Compound Pendulum

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

Determining the Gravitational Constant using a Torsion Balance

An experimental apparatus utilizing a torsion balance is used to measure the gravitational forces be

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 Comet Orbits

A comet follows a highly elliptical orbit around the Sun. Analyze its speed variation along its path

Effects of Eccentricity on Planetary Orbits

A series of simulations have been conducted for planetary orbits with varying eccentricity. The foll

Elliptical Orbit Dynamics: Speed Variation Analysis

For a planet or satellite in an elliptical orbit, the speed varies along the orbit due to conservati

Elliptical Orbit Simulation Error in Barycenter Consideration

A computer simulation is developed to mimic the elliptical orbits of planets using Newton's law of g

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 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

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 Produced by a Thin Uniform Disk

A researcher calculates the gravitational field produced by a thin circular disk of total mass $$M$$

Gravitational Force Calculation on a Satellite

A satellite with a mass of $$m = 500\,kg$$ orbits the Earth at a distance of $$r = 7.0 * 10^6\,m$$ (

Gravitational Parameter in Exoplanetary Systems

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

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

Integration of Variable Gravitational Force over an Extended Body

Consider a uniform rod of length L and total mass m, oriented radially away from the center of a pla

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

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

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

Mathematical Modeling of Tidal Forces

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

Modeling Tidal Forces with Calculus

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

Orbit Stability from Potential Energy Diagrams

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

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 Speed and Radius in Circular Orbits

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

Perturbation Analysis of Satellite Orbits

Study the graph showing small deviations from a nominal circular orbit of a satellite. Analyze the p

Verifying Kepler's Second Law and Angular Momentum Conservation

Kepler’s Second Law states that a line joining a planet and the Sun sweeps out equal areas during eq