AP Physics C: Mechanics FRQ Room

Acceleration from a Given Velocity Function

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

Air Resistance and Projectile Motion

In an experiment, two projectiles with different surface textures (one smooth and one rough) are lau

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

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

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

Determining Launch Angle from Experimental Data

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

Dynamic Cart on Air Track: Misinterpretation of Frictionless Environment

In an experiment, a dynamic cart was set in motion on an air track to study uniformly accelerated mo

Effect of Initial Velocity on Displacement

A student investigates how altering the initial velocity of a cart affects its displacement on a lev

Experimental Evaluation of Vector Addition in Two Dimensions

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

Free Fall with Air Resistance

A 2.0-kg object is dropped from rest and falls under gravity while experiencing air resistance propo

FRQ 6: Motion with Non-Uniform Acceleration

An object experiences a time-dependent acceleration given by $$a(t) = 4*t - 2$$ (in m/s²) and an ini

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 11: Air Resistance and Terminal Velocity

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

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 12: Parametric Representation of Projectile Motion

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

FRQ 14: Differentiation of a Position Function

An object’s position is given by the function $$x(t)= t^3 - 6*t^2 + 9*t$$ (with x in meters and t in

FRQ 17: Experimental Analysis of Uniform Acceleration (MEDIUM)

The following table shows measured velocities of an object at different times: | Time (s) | Velocit

FRQ 20: Real-World Application – Car Braking Analysis

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

Impulse and Momentum with a Variable Force

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

Impulse and Variable Force Analysis

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

Kinematic Analysis of Circular Motion

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

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

Lab Investigation: Effects of Launch Angle on Projectile Range

In a controlled laboratory experiment, a student launches a projectile with a fixed initial speed of

Motion on an Inclined Plane with Friction

A block of mass m slides down an inclined plane making an angle $$\theta$$ with the horizontal. The

Motion with Air Resistance: Approximating Terminal Velocity

A small sphere falling through a medium experiences air resistance proportional to its velocity. Its

Motion with Changing Direction

An object moves along a straight line with its position given by $$x(t)= t^3 - 6*t^2 + 9*t$$ (meters

Multi-Phase Rocket Motion Analysis

A rocket is launched vertically. Its engines provide a constant acceleration for 10 seconds. After e

Projectile Motion Experimental Investigation

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

Relative Motion of Two Vehicles

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

Time-Dependent Acceleration and Displacement

A particle’s acceleration is given by the function $$a(t)=6-2*t$$ (in $$m/s^2$$) for $$0 \le t \le 4

Time-Dependent Force and Work-Energy Theorem

A particle of mass m moves along a straight line under a time-dependent force $$F(t)= 100\,e^{-t}$$

Trajectory Optimization of an Accelerating Car

A car accelerates according to $$a(t)= 6 - 0.5*t$$ (m/s²) with an initial velocity of 10 m/s at $$t=

Uniformly Accelerated Free Fall Analysis

In a free fall experiment, a rock is dropped from a height of 80 m. The following table presents mea

Uniformly Accelerated Motion: Incorrect Baseline Velocity

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

Variable Acceleration and Integration

An object moves along a line with a time-dependent acceleration given by $$a(t)=6*t - 2$$ (in m/s²).

Vector Addition in Two-Dimensional Motion

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

Verifying Free Fall Acceleration

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

Calculating Kinetic Energy from a Velocity Function

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

Calculus 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

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

Efficiency in Energy Conversion

A machine is used to convert electrical energy into mechanical work. It receives a constant electric

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 in a Mass-Spring Oscillator

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

Energy Conservation in 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 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 Loss Due to Position-Dependent Friction

A block of mass $$m = 5 \;\text{kg}$$ slides along a horizontal surface. The coefficient of kinetic

Energy Loss in Inelastic Collisions

Two objects collide and stick together. Object 1 has a mass of 2 kg and is traveling at 4 m/s, while

Energy Loss in Inelastic Collisions Experiment

Two carts on a frictionless track undergo a collision. Cart A (1 kg) moves at 4 m/s and collides wit

Evaluating Work Done on an Object in Rotational Motion

A researcher examines the work done on a rotating disc by a variable torque. The applied torque is d

Experiment on Energy Loss in Frictional Systems

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

Free‐Fall Impact Energy Experiment

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

Friction‐Influenced Kinetic Energy Loss Experiment

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

FRQ 2: Work Done by a Variable Force on a Crate Along a Horizontal Floor

A crate is pulled along a horizontal surface with an applied force that varies with displacement. Th

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

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

FRQ 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 14: Elastic Potential Energy in a Spring-Mass System

A news article asserts that the elastic potential energy stored in any deformed spring is always giv

FRQ 17: Energy Distribution in Car Crash Safety Studies

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

Impulse and Energy Transfer via Calculus

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

Impulse and Work in a Collision

A 1 kg particle is subject to a time-varying force during a collision given by $$F(t)=100-20\,t$$ (N

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

Relationship Between Force and Potential Energy

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

Rotational Work and Energy Experiment

A uniform disk with a moment of inertia $$I = 0.5\,kg\cdot m^2$$ is subjected to a variable torque d

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

Sliding Block on an Incline with Friction

A 5-kg block is released from rest at the top of an incline that is 10 m high. The incline is 20 m l

Spectroscopic Potential Energy Curve Analysis

A spectroscopic experiment is conducted to infer the potential energy curve of a diatomic molecule f

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 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 Force Work Calculation

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

Variable Mass Rocket Energy Calculation

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

Work Done Against Friction on an Inclined Plane

A 5-kg box slides down a 10-m long inclined plane that makes an angle of 30° with the horizontal. Th

Work Done along a Curved Path Under Variable Force

A particle moves along a curve defined by $$ y = x^2 $$ in the xy-plane. It is subjected to a force

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: Calculus Approach

A 5 kg block slides on a horizontal surface. The coefficient of kinetic friction varies with positio

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-Energy Analysis in Collisions

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

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

Analysis of an Oblique Collision

Two ice skaters are initially at rest on a frictionless ice surface. They push off each other; skate

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:

Block on an Incline: Collision and Momentum

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

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 Calculation for a Curved, Variable Density Wire

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

Center of Mass for Discrete Particles

Consider a system of three particles in the xy-plane with the following properties: • Particle A: m

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

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

Center-of-Mass Shift in an Internal Explosion

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

Central Force and Center-of-Mass Motion in a Binary Star System

A binary star system consists of Star A (mass $$2\times10^{30}\,kg$$) and Star B (mass $$3\times10^{

Data Analysis: Momentum from Experimental Graphs

In an experiment, a cart of mass $$2\,kg$$ undergoes a collision event. The following data were reco

Determination of an Unknown Mass via Collision Data

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

Dynamics of a Center-of-Mass System under a Variable Force

A system of total mass $$10$$ kg, initially at rest, is subjected to a time-dependent force given by

Elastic Collision on Air Track

Two gliders are on a near-frictionless air track. Glider A (mass = 0.8 kg) is traveling to the right

Experiment Design: Spring-Loaded Impulse Mechanism

A spring-loaded mechanism is used to deliver an impulse to a 0.5 kg cart. The spring has a constant

Experimental Design: COM Independence in Collisions

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

FRQ 15: Center of Mass versus Center of Gravity

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

Impulse and Center of Mass in a Soccer Kick

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

Impulse 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 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 Variable Force on a Soccer Ball

A soccer ball of mass 0.45 kg is struck by a kick that applies a variable force over a brief time in

Impulse during a Controlled Fall onto an Airbag

A stuntman with a mass of 80 kg falls and lands on an airbag, which decelerates him uniformly from 8

Impulsive Collision Involving Rotation

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

Inclined Plane: Center of Mass and Impulse Analysis

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

Inelastic Collision Analysis with Rolling Carts

In a collision experiment, two carts on a frictionless track collide and their velocities are record

Inelastic Collision of a Pendulum Bob with a Block

A pendulum bob of mass 2 kg is released from rest from a 30° angle from the vertical with a pendulum

Momentum Analysis in an Asteroid Breakup

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

Momentum Analysis in Explosive Fragmentation Simulation

In a simulation of explosive fragmentation, a stationary container bursts into several fragments. Hi

Momentum and Energy in Elastic Collisions

Two gliders on a frictionless air track undergo an elastic head-on collision. Glider A (mass $$0.8\,

Motion of the Center of Mass Under External Force

Consider a system consisting of two point masses connected by a light rod. Mass m1 = 2 kg is located

Motion of the Center of Mass Under External Force

A 10 kg system is subjected to a net external force that varies with time. An experiment records the

Motion of the Center of Mass with External Force

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

Multiple Particle Center of Mass in Two Dimensions

Four particles in a plane have the following properties: Particle A (2 kg) at (0, 0), Particle B (3

Projectile and Cart Collision: Trajectory Prediction

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

Projectile Explosion and Center of Mass Motion

A 5 kg projectile is launched vertically upward. At its highest point, it explodes into two fragment

Rocket Propulsion: Variable Mass System

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

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)

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-Dimensional Collision of Ice Skaters

Two ice skaters initially at rest push off from each other on frictionless ice. Skater A (50 kg) mov

Two-Stage Collision in Coupled Carts

Two carts on a frictionless track undergo a two-stage event. Initially, cart A (mass $$2\,kg$$, velo

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 Kinematics Analysis Using Graphical Data

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

Angular Momentum Conservation on a Merry-Go-Round

A child of mass m = 30 kg stands on the edge of a merry-go-round, modeled as a disk with mass M = 10

Application and Critical Review of the Parallel Axis Theorem

A disk of radius $$R$$ and mass $$M$$ has a moment of inertia about its center of mass given by $$I_

Application of the Parallel Axis Theorem

An object with mass 5 kg has a moment of inertia about its center of mass $$I_{cm} = 0.2\,kg\,m^2$$.

Applying the Parallel Axis Theorem to a Composite Object

A composite object has been tested to determine its moment of inertia about different axes. The foll

Assessment of Rotational Kinematics Equations

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

Calculus Derivation of the Moment of Inertia for a Uniform Disk

Derive the moment of inertia for a uniform solid disk of mass M and radius R about its central axis

Comparative Analysis of Rotational and Translational Dynamics

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

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

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 Explosive Separation

A rotating disk of mass $$M = 5.0 \text{ kg}$$ and radius $$R = 0.5 \text{ m}$$, spinning at $$\omeg

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 Moment of Inertia of a Non-Uniform Rod

A non-uniform rod of length $$L = 2\,m$$ is analyzed to determine its moment of inertia about one en

Dynamics of a Damped Flywheel System

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

Effect of Variable Applied Torque on Angular Acceleration

In a controlled experiment, a variable torque is applied to a rotating disk. The resulting change in

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 Combined Rotational and Translational Motion

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

Energy Conversion in Rolling Motion Experiments

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

Equilibrium Analysis in Rotational Systems

A seesaw, modeled as a uniform beam 4 m long pivoted exactly at its center, is in rotational equilib

Experimental Data: Angular Velocity vs Time Analysis

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

FRQ 5: Rolling Motion on an Incline

A solid cylinder of mass M = 3.00 kg and radius R = 0.20 m rolls without slipping down an inclined p

FRQ 19: Oscillatory Rotational Motion (Torsional Pendulum)

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

Graphical Analysis of Angular Motion

A rotating system has been monitored and the angular velocity (in rad/s) recorded over time (in seco

Inelastic Collision of Rotating Disks

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

Integration for Moment of Inertia of a Non-Uniform Rod

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

Investigating Non-uniform Density Effects on Moment of Inertia

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

Lever Torque Application

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

Non-Uniform Angular Acceleration

A disk has an angular acceleration described by the function $$\alpha(t) = 2 * t$$ (in rad/s²), and

Non-Uniform Angular Velocity: Integration and Differentiation

A rotating disk has an angular velocity given by $$\omega(t)=3*t^2 - 2*t + 1$$ (in rad/s), where t i

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

Physical Pendulum with Offset Mass Distribution

A physical pendulum is constructed from a rigid body of mass $$M$$ with its center of mass located a

Quantitative Analysis of Rolling Down an Incline

An object rolls without slipping down an inclined plane. Measurements are taken at different incline

Rolling Motion with Slipping Transition

A cylinder initially rolls without slipping down an incline. At a certain point, due to a change in

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 Energy Distribution in a Rolling Object

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

Rotational Inertia of a Non-Uniform Disk

A disk of radius R has a surface mass density given by $$\sigma(r)= \sigma_0 \left(1+\frac{r}{R}\rig

Rotational Kinematics on a Spinning Disk

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

Seesaw Rotational Equilibrium

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

Stability Analysis of a Block on a Rotating Platform (Tipping Experiment)

A block is placed on a rotating platform, and the conditions under which the block tips are investig

Time-Resolved Analysis of Angular Acceleration

A wheel's angular velocity is given by $$\omega(t)= 3t^2 + 2t\,rad/s$$. Calculate its angular accele

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 and Equilibrium: Balancing a Non-Uniform Beam

A beam of length $$L$$ has a non-uniform mass distribution such that its center of mass is located a

Torque and Rotational Inertia: Uniform Rod

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

Torque 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

Verification of the Parallel Axis Theorem

Students test the parallel axis theorem by measuring the moment of inertia of a uniform rod about se

Analyzing Damped Oscillations in a Spring-Mass System

An experiment is conducted in which a spring-mass oscillator is exposed to air resistance, introduci

Calculus-Derived Velocity and Acceleration in SHM

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

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 of Horizontal and Vertical Oscillators

Students set up two oscillatory systems: one horizontal spring–block oscillator and one vertical spr

Comparative Analysis of Horizontal and Vertical SHM Systems

A researcher compares the oscillations of a mass attached to a spring in horizontal and vertical con

Comparative Analysis of Oscillator Systems

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

Coupled Oscillators: Two Springs in Parallel

A block is attached to two springs arranged in parallel with spring constants $$k_1 = 250\,\text{N/m

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

Data Analysis and Calculus Estimation in SHM

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

Data Analysis of a Spring-Mass Experiment

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

Derivation and Solution of the Differential Equation for SHM

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

Derivation of SHM Equations Using Calculus

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

Derivation of the SHM Differential Equation

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

Deriving the Equation of Motion Using Calculus

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

Designing an Experiment on the Inverse Relationship between Mass and Period

A researcher designs an experiment to study the relationship $$T = 2\pi * \sqrt{\frac{m}{k}}$$ in a

Determination of Spring Constant from Oscillation Data

A researcher collects oscillation data for different masses attached to a spring. The data is summar

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 Spring Constant from Force-Displacement Data

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

Determining the Spring Constant from Oscillation Data

A student conducts an experiment to determine the spring constant $$k$$ of a spring by measuring the

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 Mass Variation on Oscillation Frequency

A student investigates how the oscillation frequency of a spring-block system varies with the mass a

Energy Exchange in Oscillatory Systems

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

Equation of Motion for a Simple Pendulum Beyond Small Angle Approximation

A researcher examines the motion of a simple pendulum without relying on the small-angle approximati

Error Analysis in SHM Measurements

A student conducting an experiment on a mass-spring oscillator records the following period measurem

Evaluating Experimental Uncertainties in SHM Measurements

Accurate measurement of the oscillation period is crucial in SHM experiments. In this context, uncer

Fourier Analysis of Oscillation Data

In an advanced experiment, the oscillation of a spring-mass system is recorded and analyzed using Fo

Fourier Analysis of Oscillatory Motion

In an advanced experiment, students record the motion of a nonlinear oscillator and attempt to decom

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 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 12: Deriving Velocity and Acceleration Functions

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

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

FRQ5: Sinusoidal Description of Oscillatory Motion and Phase Determination

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

FRQ6: Calculus Derivation of Velocity and Acceleration in SHM

For a mass undergoing simple harmonic motion described by the displacement function $$x(t)= A\sin(\o

FRQ17: Calculus-Based Derivation of Velocity and Acceleration in SHM

Consider a mass undergoing simple harmonic motion described by the displacement function: $$x(t)= A

FRQ19: Coupled Oscillators – Normal Modes of a Two-Mass, Three-Spring System

Consider a system in which two identical masses \(m\) are connected in series with three identical s

Graphical Analysis of SHM: Determining Phase and Frequency

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

Integral Calculation of Work Done in SHM

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

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

Measuring the Spring Constant: An Experimental Investigation

A student performs an experiment to determine the spring constant of a coil spring. The following da

Modeling Nonlinearities in Pendulum Motion

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

Momentum and Impulse Analysis in Oscillatory Motion

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

Pendulum Motion Beyond the Small-Angle Approximation

For a pendulum, the restoring force is accurately given by $$mg\sin(\theta)$$ rather than $$mg\theta

Pendulum Motion Experimental Analysis

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

Period and Frequency Determination from Half Cycle Data

A mass-spring oscillator completes half of a full cycle (i.e. moving from maximum displacement on on

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 Difference Between Displacement and Velocity

For a simple harmonic oscillator with displacement \(x(t) = A \sin(\omega t + \phi)\), (a) different

Phase Shift Analysis in Driven Oscillators

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

Phase Space Analysis of SHM

For a mass-spring oscillator exhibiting simple harmonic motion with a solution $$x(t) = A\sin(\omega

Resonance in a Driven Harmonic Oscillator

Analyze a damped, driven harmonic oscillator and explore the conditions for resonance.

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 SHM with Phase Shift

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

Small-Angle Pendulum Experiment

In a physics lab, a small pendulum of length $$L = 0.80\,m$$ is used to study simple harmonic motion

Vertical Oscillations on a Spring

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

Vertical Oscillations: Energy and Force Analysis

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

Vertical Oscillations: Lab Data Analysis

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

Work Done in Spring Oscillation via Calculus

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

Work Done in Stretching a Nonlinear Spring

A spring exhibits a nonlinear restoring force described by $$F = -k x - \beta x^2$$, where $$k$$ and

Analysis of a Gravitational Potential Energy Graph

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

Analysis of Gravitational Anomalies: Local Variations in g

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

Analyzing Hohmann Transfer Orbits for Satellite Maneuvers

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

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

Calculating Gravitational Potential in a Non-Uniform Planet

A researcher investigates the gravitational potential inside a planet with a radially varying densit

Calculus in Determining Work Against Gravity over Altitude Change

A spacecraft gradually moves from an initial orbital radius r₁ to a higher radius r₂. The work done

Comparative Analysis of Gravitational Forces

Using the data provided, compare the gravitational forces between various pairs of celestial bodies.

Derivation of Escape Velocity Using Calculus

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

Derivation of Orbital Period from Gravitational Force

Using calculus, derive the expression for the orbital period of a planet in circular orbit from Newt

Designing a Cavendish Experiment to Measure the Gravitational Constant

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

Determination of Gravitational Constant Using a Compound Pendulum

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

Effects of Stellar Mass Variation in Binary Systems

In a binary star system, one star gradually loses mass. Analyze the impact on the orbital parameters

Energy Analysis in Multi-Body Systems

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

Energy Balance at Apoapsis and Periapsis

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

Escape Velocity Derivation

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

Experimental Analysis of Orbital Decay from a Satellite

A satellite in low Earth orbit is gradually losing altitude due to atmospheric drag. Experimental da

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 Force Calculation Between Celestial Bodies

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

Gravitational Lensing: Deflection of Light

Using a Newtonian approximation, a light ray passes near a massive object with mass $$M$$ at a close

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

Modeling Orbital Decay with Differential Equations

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

Newton vs. Einstein: Conceptual Analysis of Gravity

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

Orbit Transfer and Hohmann Transfer Orbits

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

Orbital Dynamics and Energy Conservation

Examine the dynamics of a satellite in a circular orbit around the Earth by using energy conservatio

Orbital Motion of a Satellite

A satellite of mass m is in a stable circular orbit around a planet of mass M at a distance r from t

Orbital Perturbations from Impulsive Thrust

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

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

Speed Variation in Elliptical Orbits via Angular Momentum Conservation

In an elliptical orbit, a satellite’s speed varies along its path. Using the conservation of angular

Torsion Balance Gravitational Force Measurement

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