AP Physics C: Mechanics FRQ Room

Analyzing a Parabolic Trajectory

A projectile is launched with an initial speed of 50 m/s at an angle of 45°. Analyze its trajectory

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

Comparative Analysis of Average Speed and Velocity

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

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

Data Analysis from a Velocity-Time Table

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

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 Instantaneous Rates from Discrete Data

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

Displacement-Time Graph Analysis for Non-Uniform Motion

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

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

Dynamics on an Inclined Plane with Friction

A 4.0-kg block is released from rest at the top of a 10.0-m long incline that makes an angle of $$25

Experimental Data and Constant Acceleration

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

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 6: Relative Motion in Two Dimensions (HARD)

In the xy-plane, two particles have position functions given by: $$\vec{r}_A(t)=(2*t, t^2)$$ and $$\

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: Experimental Analysis of Free Fall

Below is experimental data from free fall tests for objects dropped from various heights: | Height

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 16: Integration of a Decaying Velocity Function (HARD)

An object has a velocity function given by $$v(t)=4*e^{-t}-2$$ (in m/s) for $$t\ge0$$, and its initi

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

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

FRQ 18: Motion of a Robot – Programming and Modeling Its Movement

A robotics research laboratory programs a robot to move on a flat plane. The robot's position is mod

FRQ 19: Analysis of Motion on an Inclined Plane (MEDIUM)

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

Graphical Analysis of Distance and Displacement

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

Graphical Analysis of Kinematic Data

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

Investigation of Variable Friction in Curvilinear Motion

Design an experiment to study the motion of an object along a curved path where friction varies with

Kinematics in a SmartLab Setup: Integration Error

In a SmartLab experiment, sensors measured both the acceleration and displacement of an object movin

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 Integrals

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

Motion of a Bus on a Curved Track

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

Motion on an Inclined Plane

A student investigates the motion of a block sliding down a 30° inclined plane initially in a fricti

Newton's Second Law and Force Measurement on a Cart

Design an experiment using a cart on a frictionless track with a pulley system to verify Newton's se

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

Rotational Dynamics: Variable Torque

A solid disk with moment of inertia I is subject to a time-dependent torque given by $$\tau(t)= 5*t$

Round Trip Motion Analysis

An object makes a round trip between points A and B. On the outward journey, it travels at a constan

Sinusoidal Motion

A particle’s position along the x-axis is given by $$x(t)=8\sin(t)$$, where $$0\leq t\leq2\pi$$ (x i

Two-Dimensional Motion with Air Resistance

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

Two-Dimensional Motion with Vector Decomposition

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

Uniformly Accelerated Motion 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 Using Calculus

Design an experiment to investigate the motion of an object under a time-varying force that produces

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 Projectile Motion

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

Work and Energy in Linear Motion

A variable force acts on a 3.0-kg object moving along the x-axis, where the force is given by $$F(x)

Analysis of Potential Energy Curves

Consider the provided graph representing the potential energy function $$U(x)$$ for a diatomic molec

Calculus Application of a Variable Force

A force acting on an object is given by $$F(x) = 5*x^2$$ N. Consider the displacement of the object

Calculus in Friction: Variable Friction Coefficient Analysis

A 10-kg block slides across a surface where the coefficient of friction varies with position as $$\m

Composite System: Roller Coaster Energy Analysis

A roller coaster car of mass 600 kg travels along a frictionless track. The following table provides

Conservation of Mechanical Energy in a Pendulum

A simple pendulum consists of a bob attached to a massless string of length 2 m. The bob is pulled a

Elastic Potential Energy and Hooke’s Law

A spring with a spring constant of $$ k = 200 \;N/m $$ is compressed by 0.1 m from its equilibrium p

Energy Analysis of a Bouncing Ball: Energy Loss and Power

A ball of mass $$m = 0.5 \;\text{kg}$$ is dropped from a height of $$10 \;\text{m}$$ and, after its

Energy Analysis of a Pendulum

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

Energy 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

Experiment on Energy Loss in Frictional Systems

Design an experiment to investigate the relationship between surface roughness and energy loss durin

Experimental Determination of the Coefficient of Friction

A 4 kg block is pulled along a horizontal surface. The applied force, which varies with position, is

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 2: Work-Energy Theorem in Lifting

A news article claims that the work done in lifting an object is independent of the velocity at whic

FRQ 3: Kinetic Energy Measurement in Free Fall

A researcher presents data claiming that objects dropped from rest convert all gravitational potenti

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 11: Analysis of a Bungee Jump – Variable Net Force

A bungee jumper with a mass of 70 kg experiences a net force that changes as a function of vertical

FRQ 11: Deriving Force from a Potential Energy Function

A study posits that the force acting on a particle can be obtained via $$F(x) = - \frac{dU}{dx}$$. E

FRQ 12: Analysis of Cyclist Power Output During a Ride

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

FRQ 13: Energy Loss Analysis in a Bouncing Ball

A 0.5-kg ball is dropped and allowed to bounce on a hard surface. The maximum height reached after e

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 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

Hydraulic Press Work Calculation Experiment

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

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

Measuring Work Done against Air Resistance in Free Fall

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

Nonlinear Spring Energy Experiment

In an experiment with a nonlinear spring, a mass is attached and the force is measured as a function

Optimization of Work in a System with Resistive Force

A particle of mass 2 kg moves along the x-axis under a constant driving force of 10 N and a resistiv

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 Curves and Equilibrium Analysis

An object of mass 4 kg has a potential energy function given by $$U(x) = (x - 2)^2 - (2\,x - 3)^3$$.

Power and Energy Efficiency in a Conveyor Belt Experiment

A researcher investigates the energy efficiency of a conveyor belt system in a manufacturing facilit

Power Output Measurement in an Elevator Experiment

A 500 kg model elevator is accelerated from rest to a speed of 2 m/s over a distance of 3 m under th

Projectile Motion Energy Analysis

A 1-kg projectile is launched with an initial speed of 20 m/s at an angle of 60° above the horizonta

Rocket Engine Power Output Under Variable Thrust

A rocket engine produces a time-varying thrust described by $$T(t) = 5000 + 1000*sin(t)$$ (in newton

Roller Coaster Energy Analysis

A roller coaster car of mass 500 kg is released from rest at a height of 50 m and descends along a t

Rotational Motion Work–Energy Experiment

In a rotational experiment, a disc is accelerated by a motor that applies a measured torque over a s

Solar Energy Mechanical Conversion Experiment

In a lab setup, a solar panel powers an electric motor that lifts a 10 kg mass vertically by 3 m at

Spring Elastic Potential Energy

A spring with a force constant of $$k = 800\,N/m$$ is compressed by 0.1 m.

Tidal Energy Extraction Analysis

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

Variable Force and Velocity: Power and Work Analysis

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

Variable Force Work Calculation

An object moves along the x-axis under the influence of a variable force given by $$ F(x) = 3 * x^2

Work and Energy in Circular Motion

A 2-kg mass is attached to a string and constrained to move on a frictionless vertical circular path

Work 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 Gravitational Force

An object of mass 2 kg moves from a height of 50 m to 30 m above the Earth's surface. The gravitatio

Work Done in a Non-uniform Gravitational Field

An object of mass 500 kg is moved radially outward from the Earth. Assume the Earth’s mass is $$M =

Work with Constant and Variable Forces

An object is acted upon by two different types of forces on separate occasions. In Part (a), a const

Work-Energy Theorem Application

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

Work-Energy Theorem in Inelastic Collisions

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

Work-Energy Theorem with Air Resistance

A 0.2-kg ball is thrown vertically upward with an initial speed of $$40 \;\text{m/s}$$. Air resistan

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 $

Angular Momentum Change in a Disc–Rod Collision Experiment

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

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

Calculating Center of Mass Acceleration

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

Center of Mass Determination for a Composite L-Shaped Object

Students perform an experiment to determine the center of mass of an L-shaped, composite board. The

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 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 Rectangular Plate with Variable Density

A thin rectangular plate extends from $$x=0$$ to $$x=2$$ m and from $$y=0$$ to $$y=3$$ m. Its surfac

Center of Mass of a Triangular Plate

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

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: 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

Explosive Separation of Particle System

A 10 kg object at rest explodes into three fragments with masses 3 kg, 4 kg, and 3 kg. After the exp

Force from Potential Energy Graph

A potential energy function for a system is provided in the graph below, where the potential energy

FRQ 1: Center of Mass of a Non-Uniform Rod

Consider a rod of length $$L = 1.2 \ m$$ with a linear mass density given by $$\lambda(x) = 2 + 3*x$

FRQ 5: Physics of a Football Punt

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

FRQ 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 11: Experimental Evaluation: Measurement of Center of Mass

A media report claims that a new laser-based method can determine the center of mass of irregular ob

Impulse and Kinetic Energy from a Time-Dependent Force on a Car

A 1200 kg car, initially at rest, is subjected to a time-dependent force given by $$F(t)=4000 - 500*

Impulse and Work: Discerning Differences

A particle of mass 3 kg is subjected to a force that varies with position according to $$F(x)=12*x^2

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 from a Variable Force Function

A force acting on an object varies with time according to the function $$F(t) = 4*t^2$$ (in Newtons)

Impulse from Force Sensor Data

In a collision experiment, a force sensor attached to a small car records the force applied during i

Impulse-Momentum in Soccer Kick

A soccer player kicks a ball of mass 0.43 kg. A force sensor attached to the player's foot records a

Impulse-Momentum Theorem with a Non-constant Force

A 0.2 kg hockey puck is struck by a mallet. The force applied by the mallet varies with time and is

Impulsive Collision Involving Rotation

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

Inelastic Collision on a Frictionless Surface

Two gliders on a frictionless air track collide and stick together. Glider A (mass = 2 kg) moves rig

Momentum Analysis of a Variable-Density Moving Rod

A rod of length $$L=1.5\,m$$ has a linear density function $$\lambda(x)=4+3*x$$ (in kg/m) and is mov

Motion of Center of Mass for a Two-Block System with External Force

Two blocks with masses $$m_1 = 3\,\text{kg}$$ and $$m_2 = 2\,\text{kg}$$ are rigidly connected and m

Motion of the Center of Mass under External Force

Consider a two-particle system consisting of masses $$m_1 = 4\,kg$$ and $$m_2 = 6\,kg$$. An external

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

Ramp Push Experiment: Variable Force Integration

In a laboratory experiment, a cart on a frictionless ramp is pushed by a force that varies with time

Rotational Dynamics Using Center of Mass

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

Rotational Impulse and Angular Momentum

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

Structural Stability: Crane Center of Mass Analysis

An engineer is analyzing a crane consisting of a 500-kg horizontal beam whose center of mass is at 5

Time-Varying Force on a Block

A 2 kg block on a frictionless surface is subjected to a time-varying force given by $$F(t) = 15*\si

Two-Dimensional Collision Analysis

Two cars collide on a flat surface. Car A, with a mass of 1200 kg, is traveling east at 15 m/s, and

Two-Dimensional Collision and Momentum Conservation

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

Work Done by a Variable Force and Momentum Change

A 2-kg block on a frictionless surface is subjected to an external force described by $$F(t) = 15e^{

Angular Impulse and Change in Angular Momentum

A stationary flywheel is subjected to a constant torque $$\tau$$ for a time interval $$\Delta t$$.

Angular Momentum Conservation in Figure Skating

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

Angular Momentum in Explosive Separation

A spinning disk with an initial angular velocity $$\omega_i$$ undergoes an explosion that breaks it

Calculus Analysis of Angular Velocity in a Variable Moment of Inertia System

A figure skater’s moment of inertia changes as she pulls her arms in. Assume her moment of inertia v

Calculus-Based Analysis of Angular Impulse

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

Centripetal Force and Angular Velocity Measurement

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

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 Body Rotation

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

Conservation of Angular Momentum in a Figure Skater's Spin

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

Coupled Rotational and Translational Dynamics in a Rolling Sphere

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

Cylinder Rolling Down an Incline

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an incline of angle $$\t

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 Experiment Using a Pulley System

A researcher designs an experiment to measure the rotational inertia of a pulley using a falling mas

Determining Moment of Inertia of Irregular Objects

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

Determining the Effect of Friction on Rotational Motion

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

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

Effects of Non-uniform Mass Distribution on Rotational Inertia

A rod of length $$L$$ has a non-uniform mass density given by $$\lambda(x)=\lambda_0 \left(1 + k \fr

Energy Conservation in Rotational Motion

A hollow sphere of mass $$m = 5\,kg$$ and radius $$R = 0.3\,m$$ rolls without slipping down an incli

FRQ 3: Application of the Parallel Axis Theorem

A solid disk has a mass M = 10.00 kg and a radius R = 0.50 m. Its moment of inertia about its centra

FRQ 6: Angular Momentum Conservation on a Rotating Platform

A 50.0 kg person stands on a frictionless rotating platform that initially has a moment of inertia o

Gyroscopic Precession

A spinning gyroscope with an angular momentum $$L$$ experiences an external torque $$\tau$$ causing

Investigating Non-uniform Density Effects on Moment of Inertia

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

Lever Torque Application

A mechanic uses a long lever to lift a heavy object. A force of $$F = 100 \text{ N}$$ is applied at

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’

Measuring Frictional Torque in a Rotating Apparatus

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

Parallel Axis Theorem Experimental Verification

Design an experiment to verify the parallel axis theorem by measuring the moment of inertia of a com

Parallel Axis Theorem in Compound Systems

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

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 Dynamics Down an Incline

A solid cylinder with mass $$M = 3\,kg$$ and radius $$R = 0.2\,m$$ rolls without slipping down an in

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 with Transition from Slipping to Pure Rolling

A solid sphere of mass m and radius R is initially sliding and rolling down an inclined plane with a

Rolling with Slipping Transition

A solid sphere of mass $$M$$ and radius $$R$$ is released from rest on an inclined plane with an ang

Rotational Dynamics in a Non-Inertial Frame

In a rotating frame (such as on a merry-go-round), fictitious forces arise. Consider a situation whe

Rotational Dynamics: Frictional Torque on a Cylinder

A cylinder of mass $$M = 3.0 \text{ kg}$$ and radius $$R = 0.3 \text{ m}$$ rolls without slipping do

Rotational Energy Distribution in a Compound System

A compound system consists of a uniform disk (mass M and radius R) and an attached thin rod (mass m

Rotational Equilibrium of a Beam with Distributed Load

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

Rotational Inertia Measurement via Pulley Apparatus

A student sets up an experiment to measure the moment of inertia of a uniform disk using a pulley sy

Rotational Inertia of a Hollow Cylinder vs. a Solid Cylinder

Compare the rotational inertias of a solid cylinder and a thin-walled hollow cylinder, both of mass

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 Equilibrium in a Beam

A horizontal beam is pivoted at one end. On the beam, a force F1 = 100 N is applied 0.8 m from the p

Torque, Friction, and Rotational Equilibrium in a Pulley

A pulley with radius 0.3 m and moment of inertia 0.15 kg m^2 is subjected to a tangential force of 2

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

Calculating Damped SHM Energy Loss

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

Comparative Analysis of Oscillator Systems

Consider two oscillator systems: a horizontal spring-block oscillator with mass \(m\) and spring con

Coupled Oscillators Investigation

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

Critical Analysis of Frequency Measurement Techniques in SHM

A newspaper article claims that 'simply timing a few cycles of an oscillator provides an exact measu

Damped Oscillations: Determining the Damping Coefficient

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

Data Analysis and Calculus Estimation in SHM

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

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 Spring Constant via Oscillation Period

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

Determination of the Damping Coefficient from Amplitude Decay

Consider an oscillator with damping such that its displacement is given by: $$x(t) = A e^{-\frac{b}

Differentiating SHM: Velocity and Acceleration

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

Differentiation in SHM: Velocity and Acceleration

The position of an oscillator is given by the function $$y(t)=0.05 * \sin(10*t+0.3)$$ (with $$y$$ in

Energy Analysis in Simple Harmonic Motion

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

Energy 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 Exchange in Coupled Oscillators

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

Estimating Spring Constant from Kinetic Energy Measurements

A spring-mass oscillator's maximum kinetic energy is measured during its motion. Using energy conser

Experimental Analysis of SHM Data

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

Experimental Verification of Hooke's Law

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

FRQ 2: Maximum Speed in SHM

A block of mass $$m = 0.05\ kg$$ oscillates on a spring with a force constant of $$k = 500\ N/m$$ an

FRQ 12: Deriving Velocity and Acceleration Functions

Starting with the position function for a simple harmonic oscillator: $$y = A \sin(\omega t + \phi_0

FRQ 16: Frequency Determination from Oscillatory Data

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

FRQ 17: Determination of Damping Coefficient

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

FRQ 18: Pendulum Motion Beyond the Small Angle Approximation

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

Hooke's Law and Spring Force Calculation

Consider a spring that obeys Hooke’s law, $$F = -k * x$$, where k is the spring constant and x is th

Hooke’s Law and Work in Spring Systems

A spring with a spring constant $$k = 250\,N/m$$ is initially at its natural length. (a) Using Hooke

Horizontal Mass-Spring Oscillator Analysis

A laboratory experiment involves a block of mass $$m = 0.50\,kg$$ attached to a horizontal spring of

Influence of Mass Variation on Oscillation Frequency

In an experiment, different masses are attached to the same spring, and the frequency of oscillation

Modeling Amplitude Reduction Due to Non-Conservative Forces

In a real oscillatory system, non-conservative forces (like friction) result in an energy loss per c

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 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 Oscillations: Small Angle Approximation

A simple pendulum of length $$L=1.5\,\text{m}$$ is set into small-amplitude oscillations. (a) Derive

Phase Angle Determination from Initial Conditions

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

Phase Shift Determination in SHM

In an experiment to determine the phase shift of a simple harmonic oscillator, a block attached to a

Phase Space Analysis of SHM

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

Sinusoidal Description and Phase Constant in SHM

A spring-mass oscillator has an amplitude $$A = 0.04 \; m$$ and a frequency $$f = 5.0 \; Hz$$. The d

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-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

Vertical Oscillations: Lab Data Analysis

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

Vertical Spring Oscillator Analysis

A vertical spring with a force constant of $$k = 400\,N/m$$ supports a mass of $$m = 2.0\,kg$$ and r

Vertical Spring-Mass Oscillator Analysis

In this experiment, a block of mass is attached to a vertical spring. After the block reaches equili

Vertical Spring-Mass Oscillator Dynamics

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

Vertical Spring–Block Oscillator Dynamics

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

Analysis of Tidal Forces Acting on an Orbiting Satellite

A researcher studies the tidal forces acting on a satellite orbiting a massive planet. Due to the fi

Angular Momentum Conservation during Gravitational Collapse

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

Angular Momentum Conservation in Orbital Motion

Angular momentum conservation plays a critical role in determining the properties of orbital motion.

Application of Kepler's Third Law

A planet orbits its star in an almost circular orbit with radius a. Use Kepler's Third Law to analyz

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

Barycenter in a Two-Body System

In a two-body system consisting of masses m₁ and m₂ separated by a distance R, the barycenter (cente

Center-of-Mass in the Sun-Earth System

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

Comparative Analysis of Orbital Periods for Different Planets

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

Comparative Gravitational Forces among Planet Pairs

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

Derivation of Escape Velocity from Earth's Surface Using Calculus

Using the principle of energy conservation and calculus, derive the expression for the escape veloci

Derivation of Gravitational Field due to a Spherical Shell

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

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 Parameter (GM) from Satellite Orbits

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

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

Determining the Gravitational Constant using a Torsion Balance

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

Dynamics of a Binary Star System

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

Elliptical Orbits and Eccentricity Calculation

An object is in an elliptical orbit around a star. (a) Define the eccentricity $$e$$ in relation to

Escape Velocity Derivation

The escape velocity is the minimum speed required for an object to escape from the gravitational fie

Free-Fall Measurement on a Curved Incline

An experiment is designed to measure gravitational acceleration by allowing a small ball to roll dow

FRQ 1: Gravitational Force between Two Masses

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

FRQ 4: Gravitational Potential Energy in Satellite Orbits

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

FRQ 5: Energy Conservation in Orbital Transfer

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

FRQ 8: Elliptical Orbit – Perihelion and Aphelion Distances

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

FRQ 17: Tidal Forces and Differential Gravity

An extended object in a gravitational field experiences differential gravitational forces (tidal for

Gravitational Parameter in Exoplanetary Systems

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

Gravitational Potential Energy Measurement on a Roller Coaster

An amusement park ride features a roller coaster that reaches a maximum height of $$50 \;m$$ above t

Gravitational Potential Energy Variations near Earth

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

Gravitational Potential to Rotational Kinetic Energy Conversion

An experiment uses a rolling cylinder descending an inclined plane to explore the conversion of grav

Gravitational Slingshot Maneuver

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

Impact of Relativistic Effects on Orbital Motion

Discuss how relativistic effects modify the orbital motion of a planet when it orbits close to a ver

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

Modeling Orbital Decay with Differential Equations

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

Orbital Period Determination Using Kepler's Third Law

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

Orbital Periods and Kepler's Third Law

Kepler's Third Law states that the ratio $$\frac{T^2}{a^3}$$ is constant for planets orbiting the sa

Orbital Perturbations and Precession

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

Orbital Perturbations from Impulsive Thrust

A satellite in a circular orbit of radius $$r$$ receives an impulsive radial thrust with magnitude $

Pendulum with Variable Amplitude: Nonlinear Oscillation Effects

In this experiment, a simple pendulum is used to study how the oscillation period changes with ampli

Planetary Orbits and Energy Considerations

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

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