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 Experimental Data Table

An experiment on an air track records the displacement of a cart at various times. The data is shown

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-Based Analysis of a Car’s Accelerating Motion

A car traveling along a straight road accelerates from rest with an acceleration given by $$a(t)=2*t

Calculus-Based Kinematics Derivation

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

Centripetal Acceleration in Circular Motion

Design an experiment to measure the centripetal acceleration of an object in circular motion and det

Determining Instantaneous Acceleration from a Displacement Graph

An experiment recorded the displacement of an object as a function of time using a high-precision se

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

Distance vs. Displacement Analysis in One-Dimensional Motion

An experiment recorded the motion of a car along a straight road where its distance traveled and dis

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 5: Calculus-Based Displacement Calculation

An object has a velocity given by the function $$v(t) = 3*t^2 - 2*t + 5$$ (with t in seconds and v i

FRQ 5: Derivation of Motion Equations from Calculus

A researcher aims to derive the standard kinematic equations using calculus for an object moving wit

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 9: Application of the Big Five Equations

An object starts with an initial velocity of 8 m/s and reaches a final velocity of 20 m/s after trav

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 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 12: Graphical Analysis of Vertical Motion (MEDIUM)

A graph of vertical displacement for a projectile is modeled by the function $$y(t)=5*t-4.9*t^2$$ (i

FRQ 13: Analyzing a Two-Dimensional Collision with Projectiles

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

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

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

Impulse and Momentum with a Time-Dependent Force

A baseball (mass m = 0.145 kg) is struck by a bat. The force exerted by the bat is given by $$F(t)=

Investigation of Constant Acceleration in a Car

In an experiment, a motion sensor was set up along a straight track to measure the displacement of a

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

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,

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

Non-linear Position Function Analysis

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

Pendulum Energy Conservation Experiment

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

Piecewise Motion Analysis

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

Projectile Motion using Calculus

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

Projectile Motion with Timing Error

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

Projectile Motion: Maximum Height and Range

An object is launched from ground level at an angle of 30° above the horizontal with an initial spee

Relative Motion Analysis of Two Moving Objects

Two objects move along a straight track with positions given by $$x_A(t)= 3*t^2$$ and $$x_B(t)= 6*t

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: Meeting of Two Objects

Two objects move along a straight line. Object A starts at x = 0 m with a constant velocity of 10 m/

Rotational Motion: Angular Kinematics

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

Simple Harmonic Motion in a Spring-Mass System

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

Simultaneous Measurement of Velocity and Acceleration

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

Skydiver with Air Resistance: Variable Acceleration

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

Slope Analysis in a Velocity-Time Graph

A physics lab recorded an object’s velocity over time using an electronic sensor, and the resulting

Time 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 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

Vector Displacement and Total Distance

An object moves along a straight line in two phases. First, it moves 10 m to the east, then it moves

Verifying Free Fall Acceleration

Design an experiment to verify the acceleration due to gravity using free-fall motion. Detail your m

Analysis of a Potential Energy Curve

A particle of mass 4 kg moves along the x-axis under the influence of a potential energy function gi

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

Calculating Work on an Inclined Plane with Variable Force

A 6 kg box is pushed up a frictionless incline that makes an angle of 30° with the horizontal. The a

Calculus-Based Examination of a Spring System

A spring with a spring constant $$k = 200\,N/m$$ is compressed by a distance x and then released. An

Circular Motion with Tangential Work

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

Conservation of Energy in a Roller Coaster

A 500 kg roller coaster car is released from rest at the top of a hill 50 m above the bottom of a di

Conservation of Mechanical Energy with Dissipative Forces

A 1 kg ball is dropped from a height of 20 m. Experimental measurements indicate that air resistance

Determining Maximum Height using Energy Conservation

A researcher launches a ball of mass 0.06 kg vertically upward with an initial speed of 50 m/s in a

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

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

Elastic Potential Energy in a Spring System

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.25 m from its equilibrium pos

Energy Analysis of a Damped Spring-Mass Oscillator

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

Energy Conservation in Orbital Motion

A satellite of mass 2000 kg is in a circular orbit at a distance of 7000 km from the center of Earth

Energy Dissipation in an Oscillatory System

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

Energy Transformation in a Roller Coaster

A roller coaster car of mass $$m = 500 \;\text{kg}$$ starts from rest at a height $$H = 50 \;\text{m

Explosive Separation and Energy Distribution

A stationary object of mass $$M = 10\,kg$$ undergoes an explosion and splits into two fragments with

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

FRQ 1: Vertical Lifting Experiment – Work Calculation

A lab student lifts a 1.5 kg mass vertically at constant velocity and records the force applied alon

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: 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 10: Conservation of Energy in a Pendulum Experiment

A simple pendulum with a length of 2.0 m is released from an angle of 30° with respect to the vertic

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 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 18: Work–Energy Analysis of a Decelerating Elevator

An elevator with a mass of 1200 kg decelerates uniformly as it approaches a floor. A motion analysis

FRQ 19: Equilibrium Points from a Nonlinear Potential Energy Function

A report presents the nonlinear potential energy function $$U(x) = (x - 2)^2 - (2 * x - 3)^3$$ and c

Gravitational Potential Energy and Free Fall

A 60-kg acrobat climbs to the top of a 50-m tall platform and then jumps off. Neglecting air resista

Model Rocket Power Measurement Experiment

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

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

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

Rotational Work-Energy in a Pulley System

A pulley with a radius of 0.2 m and a moment of inertia $$I = 0.5\,kg\cdot m^2$$ rotates without sli

Spring with Nonlinear Force: Elastic Potential Energy via Integration

A nonlinear spring exerts a restoring force given by $$F(x)= k*x + \alpha*x^3$$, where $$k = 200 \;\

Variable Force with Angular Displacement

A 15 kg crate is pulled along a horizontal floor by a rope. The tension in the rope varies with the

Variable Friction and Kinetic Energy Loss

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

Work Done by a Time-Dependent Force

A 4 kg object on a frictionless surface is subjected to a horizontal force given by $$ F(t) = 10 * 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 in a Resistive Medium

A particle of mass 1 kg moves in a straight line under the influence of two forces: a constant propu

Work Done on an Object by a Constant Force

A 10-kg box is pulled along a frictionless horizontal surface by a constant force of 50 N applied at

Work with a Variable Force on a Straight Path

A particle experiences a variable force along the x-axis given by $$F(x)= 10 + 3*x \; (\text{N})$$.

Work, Energy, and Power in Circular Motion

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

Angular Momentum Change in a Disc–Rod Collision Experiment

In a rotational collision experiment, a spinning disc collides with a stationary rod. Motion sensors

Astronaut Recoil in Space

An astronaut with a total mass of 90 kg, initially at rest relative to her shuttle, throws a 2 kg to

Balancing a Composite System's Center of Mass

A thin uniform rod of length $$3$$ m (mass $$1$$ kg) has two point masses attached to it: a $$2$$ kg

Calculus-Based Analysis of a Variable Density Disk

A thin disk of radius $$R = 0.5$$ m has a surface mass density that varies with radius as $$\sigma(r

Center of Mass Acceleration under Variable Force

Two blocks with masses $$m_1 = 3\,\text{kg}$$ and $$m_2 = 2\,\text{kg}$$ are connected by a light ro

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 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 Non-uniform Circular Disk

A thin circular disk of radius $$R$$ has a surface mass density given by $$\sigma(\theta)= k\,(1+\co

Center of Mass of a Non-Uniform Rod

A thin rod of length $$0.8$$ m has a linear mass density given by $$\lambda(x) = 5 + 3*x^2$$ (kg/m),

Complex Rotational and Translational Collision Involving Center of Mass

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

Composite Body Center of Mass Calculation

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

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

FRQ 14: Derivation of the Continuous Center of Mass Formula

Consider a one-dimensional object with a continuous mass distribution described by the density funct

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 and Angular Momentum in a Collision

A 0.2 kg ball traveling at 5 m/s collides with a thin rod (mass = 2 kg, length = 1.5 m) pivoted abou

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 Momentum Change for a Hockey Puck

A 0.1 kg hockey puck initially has a momentum of 0.5 kg·m/s. It then receives an impulse that increa

Impulse and Momentum in Ball Kicking

In an experiment, a soccer player kicks a 0.4 kg ball. A force sensor records the force exerted by t

Impulse and Swing Angle in a Pendulum

A pendulum bob of mass $$1.0\,kg$$ initially at rest is given a horizontal push by a time-dependent

Impulse 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 Delivered by a Variable Force

A particle experiences a time-dependent force along the x-axis given by $$F(t)=4*t^2 - 12*t + 9$$ N

Impulse Delivered by Variable Thrust Rocket

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

Impulse with Resistive Force

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

Momentum Transfer in Off-Center Collisions on a Frictionless Track

In an experiment, a moving cart collides off-center with a stationary cart on a frictionless track,

Multi-Stage Rocket Propulsion using Momentum Conservation

A rocket with an initial total mass of 1000 kg expels propellant at a constant exhaust velocity of $

Off-Center Collision and Angular Momentum

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

Projectile Motion with Air Resistance Approximation

A 0.2 kg projectile is launched with an initial speed of 15 m/s at an angle of 40° above the horizon

Rocket Propulsion and Center of Mass Dynamics

A rocket has an initial total mass of $$M_0 = 5000\;kg$$ and burns fuel such that its mass decreases

Rocket Propulsion: Variable Mass System

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

Spring-Loaded Collision with Impulsive Force

A 0.5 kg ball moving horizontally at $$8$$ m/s collides with a spring-mounted barrier that exerts a

Two-Ball Collision Dynamics

Two balls collide head-on in a controlled experiment. The red ball (mass = 0.5 kg) moves to the righ

Two-Dimensional Collision and Momentum Conservation

Two ice skaters push off each other on a frictionless surface. Skater A (mass $$60\,kg$$) moves with

Angular Displacement and Kinematics Analysis

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

Angular Kinematics Analysis Using Graphical Data

A rotating disk's angular velocity is given by the graph below. Determine key kinematic quantities f

Angular Momentum Changes in a Skater's Spin

A figure skater initially spins with a moment of inertia $$I_i$$ and angular velocity $$\omega_i$$.

Angular Momentum Conservation in Rotational Collisions

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

Calculus-Based Derivation of Torque from Force Distribution

A beam of length $$L$$ is subject to a continuously distributed force. Consider two cases: (i) const

Comparative Analysis of Rotational and Translational Dynamics

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

Comparative Angular Momentum in Different Systems

Compare the application of conservation of angular momentum in two systems: a spinning mechanical wh

Comparative Calculations for a Composite System

Consider a system of three beads, each of mass $$m$$, arranged along a rod of negligible mass and le

Composite Rod and Point Masses Inertia Analysis

A uniform rod of length L and mass M is pivoted about its left end. Two small beads, each of mass m,

Conservation of Angular Momentum in a Figure Skater's Spin

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

Conservation of Angular Momentum in a Merry-Go-Round Experiment

In this experiment, a child stands on the edge of a rotating merry-go-round. The child then walks to

Coupled Rotational and Translational Dynamics in a Rolling Sphere

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

Cylinder Rolling on an Incline

A solid cylinder of mass M rolls without slipping down an inclined plane of height h and length L (w

Designing a Rotational System with Specified Kinetic Energy

A researcher is tasked with designing a rotational system that must store a specified amount of kine

Determining Moment of Inertia of Irregular Objects

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

Energy Conversion in a Rolling Cylinder Experiment

A cylinder rolls without slipping down an inclined plane. The experiment examines how gravitational

Engine Torque Measurement Analysis

A mechanical engineer is analyzing the torque output of a car engine. The engine uses a lever arm at

Experimental Data: Angular Velocity vs Time Analysis

An experiment records the angular velocity of a rotating object over time. The provided graph shows

FRQ 8: Variable Torque and Angular Acceleration

A rotating wheel with constant moment of inertia \(I = 2.00\,kg\cdot m^2\) experiences a time-depend

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

Gyroscopic Precession and its Dependence on Spin Rate: An Experiment

A spinning wheel mounted on a gimbal is subjected to an applied torque, causing it to precess. The e

Influence of Friction on Rolling Without Slipping

An experiment investigates the effect of surface friction on rolling objects. The angular velocity o

Investigating the Big Five Equations for Rotational Motion

A researcher is verifying the 'Big Five' equations of rotational kinematics under constant angular a

Investigation of Torque in a Lever System

In this experiment a rigid lever, pivoted at one end, is used to measure the torque generated by a c

Kinetic Energy Redistribution in Rotating Systems

A rotating disk initially has two weights attached at its rim, resulting in a moment of inertia $$I_

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,

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’

Mathematical Modeling of Brake Systems

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

Measuring Frictional Torque in a Rotating Apparatus

In this experiment, a rotating apparatus is allowed to decelerate freely due only to friction. By re

Non-uniform Rotational Acceleration: Differentiation from Graph

A rotating disk exhibits a non-uniform angular velocity as a function of time. The experimental grap

Parallel Axis Theorem in Compound Systems

A composite system consists of a uniform disk of mass $$M$$ and radius $$R$$ and a point mass $$m$$

Rotational Dynamics of a Gyroscope

A gyroscope with a moment of inertia of $$I = 0.05\,kg\,m^2$$ is spinning with a spin angular veloci

Rotational Equilibrium Analysis of a Beam

A beam is in static equilibrium under the influence of several forces applied at different distances

Rotational Impulse and Change in Angular Momentum

A flywheel initially at rest receives a constant torque impulse over a brief time interval.

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 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 Kinematics from Angular Velocity Graph

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

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

Seesaw Rotational Equilibrium

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

Simulation Analysis of Rotational Motion with Non-uniform Mass Distribution

A simulation of a rotating flexible system shows that the moment of inertia, $$I$$ (in kg m^2), chan

Testing the Parallel Axis Theorem

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

Time-Varying Torque and Angular Acceleration

A researcher is exploring the effects of a time-varying torque on the rotational motion of a rigid b

Torque Measurement and Analysis

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

Variable Torque Function Integration

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

Amplitude Dependence in a Nonlinear Oscillator

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

Calculus Derivation of Velocity and Acceleration in SHM

Educational materials claim that 'the instantaneous velocity and acceleration of a simple harmonic o

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 Work Done in Stretching a Spring

Investigate the work done in stretching a spring from its natural length using calculus.

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

Coupled Oscillators Investigation

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

Derivation of SHM Equations Using Calculus

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

Determination of Spring Constant via Oscillation Period

An experiment is set up to determine the spring constant k by measuring the period of oscillations f

Determining Phase Shift in Sinusoidal SHM

A simple harmonic oscillator follows the equation $$y = A * \sin(\omega * t + \phi_0)$$ with amplitu

Determining Spring Constant from Force-Displacement Data

In a laboratory experiment, the force exerted by a spring is measured for various displacements. The

Determining the Phase Constant from Experimental Data

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

Differential Equation of Coupled Oscillators

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

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 Simple Pendulum

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

Energy Conservation in Pendulum Motion

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

Energy 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

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

Experimental Verification of Hooke's Law

A physics lab setup involves a horizontal spring-mass system to test Hooke’s law. In this experiment

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

FRQ 1: Hooke’s Law Experiment

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

FRQ 7: Differentiation of SHM to Obtain Velocity and Acceleration

Consider an oscillator described by $$y = A \sin(\omega t + \phi_0)$$. A set of experimental velocit

FRQ 10: Differential Equation of a Horizontal Mass-Spring System

Consider a mass attached to a horizontal spring on a frictionless surface. Answer the following:

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 20: Oscillator with Time-Varying Mass

Consider a spring-mass system in which the mass varies with time according to $$m(t) = m_0 + \alpha

FRQ4: Vertical Spring-Block Oscillator – Equilibrium and Oscillations

A block of mass $$m = 2.0\,kg$$ is attached to a vertical spring with a force constant of $$k = 300\

FRQ9: Energy Exchanges in a Mass-Spring Oscillator

In a frictionless mass-spring oscillator the energy continuously oscillates between kinetic and pote

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

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 Variation and Frequency in SHM

Consider a spring oscillator with a constant spring constant of $$k = 200\,N/m$$. The frequency of o

Oscillatory Motion on an Inclined Plane

A block of mass $$m$$ is attached to a spring (constant $$k$$) on a frictionless inclined plane with

Pendulum Dynamics Beyond the Small-Angle Approximation

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

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

A simple pendulum of length $$L = 0.80\,m$$ is released from a small angle. (a) Using the small-angl

Pendulum Period Measurement Experiment

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

Period and Frequency of a Vertical Oscillator

A block of mass $$m = 1.5 \; kg$$ is suspended from a vertical spring with a force constant of $$k =

Period of a Physical Pendulum: A Calculus Approach

A physical pendulum consists of a uniform thin rod of length $$L$$ and mass $$m$$, pivoted at one en

Phase Shift and Time Determination in SHM

Analyze the effects of phase shift in a sinusoidal oscillator and determine specific times correspon

Sinusoidal Description and Phase Shift in SHM

A spring oscillator has an amplitude of $$A = 0.05\,m$$ and oscillates with a frequency of $$f = 5.0

Spring Force and Energy Analysis

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

Spring Force Investigation

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

Spring Oscillator on an Inclined Plane

A block of mass \(m = 2\,kg\) is attached to a spring with spring constant \(k = 150\,N/m\) on an in

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_

Torsional Oscillator as a Rotational Analogy

A disk with a moment of inertia \(I=0.05\,\text{kg}\cdot\text{m}^2\) is suspended by a wire that pro

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: Energy and Force Analysis

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

Analysis of Gravitational Anomalies: Local Variations in g

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

Analyzing Gravitational Slingshot Maneuvers

A spacecraft uses a gravitational slingshot maneuver around a planet to gain additional speed for an

Analyzing Hohmann Transfer Orbits for Satellite Maneuvers

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

Angular Momentum Conservation during Gravitational Collapse

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

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 Determination in a Sun-Planet Analog with Magnetic Models

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

Calculating the Gravitational Field from a Spherical Mass Distribution

Consider a planet with a spherically symmetric density profile given by $$ \rho(r) = \rho_0 \left(1

Cannonball Trajectory in a Non-Uniform Gravitational Field

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

Comparative Analysis of Planetary Orbits

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

Comparative Gravitational Forces among Planet Pairs

Examine the data comparing gravitational forces between different planet pairs. Use the evidence to

Comparison of Orbital Dynamics: Moon vs. Artificial Satellites

A researcher compares the gravitational forces and orbital characteristics of the Moon and an artifi

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

Designing a Modern Cavendish Experiment

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

Determination of Gravitational Parameter (GM) from Satellite Orbits

An observational study is undertaken to determine the gravitational parameter (GM) of a planet by an

Elliptical Orbit Dynamics: Speed Variation Analysis

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

Energy Analysis in Multi-Body Systems

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

Energy Dissipation in Orbital Decay

A satellite experiences a tangential drag force given by $$F_{drag} = -b * v$$, where $$b$$ is a con

Escape Velocity Derivation

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

Experimental Verification of Conservation of Energy in a Gravitational Field

A pendulum experiment is conducted to verify the conservation of mechanical energy in a gravitationa

FRQ 8: Elliptical Orbit – Perihelion and Aphelion Distances

An object travels in an elliptical orbit with a semimajor axis $$a$$ and eccentricity $$e$$. Answer

FRQ 10: Gravitational Interactions in a Three-Body System

Consider a simplified system with three masses, $$m_1$$, $$m_2$$, and $$m_3$$, located at fixed posi

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 15: Gravitational Anomalies and Their Effects on Orbits

A satellite experiences a small perturbation in the gravitational potential due to a local mass anom

FRQ 19: Relativistic Corrections and Perihelion Precession

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

Gravitational Field Modeling for Extended Bodies

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

Gravitational Field of a Spherical Shell

Using calculus, derive the gravitational field produced by a thin spherical shell of uniform mass M

Gravitational Field Strength Variation

Derive the gravitational field strength as a function of distance from a point mass and analyze how

Gravitational Force Calculation on a Satellite

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

Gravity Assist in Three-Body Dynamics

In a gravitational slingshot (gravity assist) maneuver, a spacecraft can change its velocity by inte

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

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 Second Law and Areal Velocity

Analyze the graph of swept area versus time for a planet in orbit. Use the experimental evidence to

Newton vs. Einstein: Conceptual Analysis of Gravity

Compare and contrast Newton's Law of Gravitation with Einstein's theory of General Relativity. Answe

Non-uniform Gravitational Fields in Planetary Interiors

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

Orbital Period and Semimajor Axis Relationship Using Kepler's Third Law

A researcher collects observational data for various moons orbiting a giant planet. The table below

Orbital Simulation Ignoring Relativistic Effects

A simulation models the orbit of a fast-moving object near a massive body using Newton's law of grav

Planetary Orbit Analysis via Kepler's Third Law

A researcher is studying the orbits of several planets around a distant star. Observations suggest t

Planetary Orbits and Kepler's Laws

Consider a planet orbiting a star under the influence of gravity. The orbit is elliptical with the s

Role of Eccentricity in Orbital Dynamics

Orbital eccentricity (e) quantifies how much an orbit deviates from a circle. (a) Define orbital ec

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

Torsion Balance Gravitational Force Measurement

A research group performs an experiment using a torsion balance to measure the gravitational attract

Work Done in a Variable Gravitational Field

An object of mass $$m$$ is moved radially from a distance $$r_1$$ to $$r_2$$ in the gravitational fi