AP Physics C: Mechanics FRQ Room

Acceleration from a Given Velocity Function

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

Analysis of a Velocity Signal in a Laboratory Experiment

In a laboratory experiment, the velocity of a moving cart is found to follow $$v(t)= 5 + 2*\cos(0.5*

Analyzing Circular Motion: Speed and Acceleration

A particle moves with a constant speed of 15 m/s along a circular path of radius 10 m.

Calculus-Based Analysis of Relative Motion

Two objects move along the same straight line with their positions given by $$x_1(t)=3t^2$$ and $$x_

Combined Translational and Rotational Motion Experiment

Design an experiment to study an object that exhibits both translational and rotational motion as it

Conservation of Energy in Projectile Motion

A projectile is launched from the ground with an initial speed $$v_0 = 50$$ m/s at an angle of $$45°

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 Zero Acceleration from a Non-linear Position Function

An object's position is given by the function $$x(t)=t^4-8*t^2$$. Determine at what time the object'

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

Drone Video Analysis of Free Fall

A drone records a free-falling ball dropped from a 100 m height. Video analysis approximates the bal

Experimental Determination of g

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

FRQ 6: Motion on an Inclined Plane

A researcher studies the motion of a block sliding down an inclined plane with friction. The block i

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 7: Effects of Air Resistance in Free Fall

A researcher is examining the motion of an object in free fall where air resistance is not negligibl

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 11: Modeling Motion with Differential Equations

A researcher is modeling the motion of a ski jumper. The dynamics of the jumper's flight can be expr

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

FRQ 19: Comparative Kinematics – Two Launch Angles

Two objects are launched from the same point with the same initial speed of 40 m/s, but at different

Graphical Analysis of Motion: Position to Velocity

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

Gravitational Effects in a Non-Uniform Field

Design an experiment to measure the variation of gravitational acceleration with altitude. Provide a

Inferring Acceleration from Velocity Data Using Calculus

The following table shows the time and corresponding velocity for an object moving in one dimension,

Instantaneous vs. Average Velocity

An object’s position is given by the function $$x(t)=t^4 - 4*t^2$$ (x in meters).

Kinematic Analysis of Circular Motion

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

Motion Along a Curved Track

A roller coaster car moves along a curved track. Its displacement along the track is given by $$s(t)

Motion Analysis Using Integrals

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

Motion with Time-Varying Acceleration (Drag Force Approximation)

An object in free fall experiences a time-dependent acceleration due to air resistance approximated

One-Dimensional Uniform Acceleration Analysis

An object moves along a straight line with its position given by $$x(t) = -2*t^3 + 5*t^2 - 3*t + 1$$

Optimizing the Launch Angle for Maximum Horizontal Displacement on Uneven Terrain

A projectile is launched with an initial speed of 40 m/s from a platform that is 10 m above the grou

Oscillating Particle under Uniform Acceleration

An object exhibits combined motion described by $$x(t)= 5*t^2 + 3*\sin(2*t)$$ (meters). Analyze its

Pendulum Motion and Kinematics

A pendulum of length 2 m oscillates with small amplitude. Its angular displacement is given by $$θ(t

Piecewise Defined Acceleration

A particle moves along a frictionless surface with a piecewise defined acceleration given by: For $

Predicting Motion in a Resistive Medium

An object in free fall experiences a time-dependent acceleration given by $$a(t)=g\left(1-\frac{t}{T

Projectile Motion with Calculus Integration

An object is launched from ground level with an initial speed of 50 m/s at an angle of 40° above the

Relative Motion Experiment

Two carts on parallel tracks move in the same direction with positions given by $$x_A(t)=5*t$$ and $

Sinusoidal Position and Velocity Analysis

Consider an object moving in one dimension whose position function is given by $$x(t)=\sin(t)$$, ove

Time vs. Position Data Analysis: Initial Conditions Overlooked

A student conducted an experiment to study an object’s motion by recording its position over time us

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=

Uniform Acceleration in One Dimension

An object moves along a straight line with constant acceleration. Its motion is described by the pos

Uniformly Accelerated Motion With Non-Zero Initial Velocity

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

Variable Acceleration Analysis

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

Vector Addition in Two-Dimensional Projectile Motion

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

Ballistic Kinetic Energy with Air Resistance

A ball is thrown upward within a controlled chamber while sensors record its velocity as a function

Calculus-based Integration of Work over a Variable Force

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 5*x$$ N. The particle

Conservation of Mechanical Energy in a Pendulum

A pendulum bob of mass 1.2 kg and length 2 m is released from a 60° angle relative to the vertical.

Elastic Potential Energy in a Spring

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.1 m. Analyze the energy store

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 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 Analysis of a Damped Pendulum

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

Energy Dissipation in a Bouncing Ball

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

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

Explosive Separation and Energy Distribution

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

FRQ 4: Potential Energy Curve Analysis for a Diatomic Molecule

A scientific paper provides the potential energy function for a diatomic molecule as $$U(x) = U_0 \l

FRQ 5: Assessing the Independence of Power Output from Time Interval

A magazine article claims that two engines delivering the same work are equally powerful, regardless

FRQ 8: Pendulum Energy Transformations with Damping

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

FRQ 18: Conservation of Energy in a Variable Gravitational Field Experiment

An experimental report investigates the motion of an object subject to a gravitational field that va

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

Impulse and Energy Transfer via Calculus

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

Inclined Plane Friction Variation Experiment

A block is allowed to slide down an inclined plane over which the coefficient of friction is not con

Instantaneous and Average Power of a Rocket Engine

A rocket engine produces a time-dependent force given by $$F(t) = 1000 + 200 * t$$ (N) for t in the

Kinetic Energy Measurement in a Projectile Experiment

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

Minimum Velocity for Orbital Escape

A 1500 kg rocket is in a circular orbit just above the surface of a planet with radius R = 6.37 \(\t

Nonlinear Spring Energy Experiment

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

Potential Energy Curve Analysis

An object of mass m is subject to a potential energy function given by $$U(x) = x^3 - 6*x^2 + 9*x +

Potential Energy Curve Analysis

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

Power in a Repeated Jumping Robot

A robot of mass 50 kg repeatedly jumps vertically. In each jump, its engine does work to convert kin

Power Output in a Variable Force Scenario

A force acting on an object causes work to be done such that the work as a function of time is given

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

Projectile Motion and Energy Conservation

A 0.5 kg projectile is launched from ground level with an initial speed of 20 m/s at an angle of 60°

Roller Coaster Energy Transformation Experiment

A 250 kg roller coaster car is released from rest at the top of a hill of 20 m height. The car then

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 Dynamics and Work-Energy in a Disk

A solid disk of mass 10 kg and radius 0.5 m starts from rest. A constant torque of 5 N·m is applied

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

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

Variable Force and Work on a Block

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

Variable Friction and Kinetic Energy Loss

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

Variable Gravitational Acceleration Over a Mountain

A hiker lifts a 10 kg backpack up a mountain where the gravitational acceleration decreases linearly

Work by Non-Conservative Forces in a Loop

A block experiences a variable nonconservative force along a path given by $$ F(x)= 20 * \sin(x) $$

Work Done by a Variable Exponential Force

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

Work Done on a Variable Inclined Plane

An object of mass $$m = 2 \;\text{kg}$$ is moved along an inclined plane whose angle of inclination

Work Done on an Object by a Central Force

An object moves under a central attractive force given by $$F(r)= -\frac{A}{r^2}$$ with $$A = 50 \;\

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 Analysis on an Inclined Plane

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

Work-Energy Theorem Applied in a Varying Force Field

A particle of mass 1.5 kg moves along the x-axis under a force that varies with position as $$ F(x)=

Work-Energy Theorem in a Variable Force Field

A particle of mass 2 kg is acted on by a position-dependent force given by $$F(x) = 10 - 2*x$$ N, wh

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

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:

Angular Momentum Transfer in a Collision

A 0.5 kg ball moving horizontally at 10 m/s strikes a stationary 0.3 kg rod (length 2 m) pivoted at

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

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 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 of a 2D Plate with Variable Density

A thin plate occupies the region bounded by $$y=0$$, $$x=2$$, and $$y=x$$ in the xy-plane and has a

Center of Mass of a Non-uniform Rod

Consider a rod of length L = 1 m with a linear density given by $$\lambda(x)=10+6*x$$, where x is me

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 Nonuniform Circular Disk

A circular disk of radius $$R$$ has a surface mass density that varies with radial distance $$r$$ ac

Center of Mass of a Variable Density Rod

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

Center of Mass of an L-Shaped Object

An L-shaped object is formed by joining two uniform rods at one end. One rod is horizontal with a le

Center-of-Mass of a Ladder with Varying Linear Density

A ladder of length $$L=2.0\,m$$ has a vertical linear mass density given by $$\lambda(y)=3.0(1+y)$$

Composite Body Center of Mass Calculation

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

Composite Object: Rod with Attached Sphere

A uniform rod of length 1 m and mass 2 kg has a small sphere of mass 1 kg attached to its right end.

Elastic Collision: Conservation of Momentum and Kinetic Energy

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

Explosive Separation and Momentum Conservation

A 2 kg projectile traveling at 15 m/s explodes into two equal fragments (1 kg each). One fragment mo

Explosive Separation in a Multi‐Stage Rocket

A multi‐stage rocket undergoes an explosive separation. Experimentally, Stage 1 (mass = 2000 kg) is

Football Kick: Impulse and Average Force

A 0.4 kg football is punted and achieves a launch speed of 30 m/s as a result of a kick delivered ov

Force from Potential Energy Graph

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

Glancing Collision of Billiard Balls

Two billiard balls, each of mass 0.17 kg, undergo an elastic glancing collision. Initially, Ball 1 m

Impulse Analysis in a Variable Mass Rocket

Consider a simplified rocket system where its mass decreases with time as it expels fuel. The mass i

Impulse and Kinetic Energy 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 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 Delivered by a Decreasing Force from a Water Jet

A water jet impinging on a stationary nozzle exerts a time-varying force given by $$F(t)=50 - 10*t$$

Impulse from Force Sensor Data

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

Impulse in a Variable Gravitational Field

An object of mass 2 kg is thrown vertically upward from the ground with an initial velocity of 50 m/

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: Bullet-Block Interaction

A 0.02-kg bullet is fired at 400 m/s into a stationary 2-kg wooden block on a frictionless surface.

Meteor Impact: Conservation of Momentum and Energy Dissipation

A meteor with a mass of 5000 kg is traveling at 20 km/s (20000 m/s) and impacts the Earth, breaking

Momentum Analysis in Explosive Fragmentation Simulation

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

Motion of the Center of Mass Under an External Force

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

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

Multi-Dimensional Inelastic Collision Analysis

Two particles undergo a perfectly inelastic collision. Particle A (mass = 2 kg) moves with velocity

Non-conservative Forces: Block on an Incline with Friction

A 2 kg block slides down a 30° incline that is 5 m long. The coefficient of kinetic friction between

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 $

Oscillations: Simple Pendulum Analysis

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

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

Rebound Velocity from a Time-Dependent Impact Force

A ball with mass $$m=0.5\,kg$$ is dropped from a height of $$10\,m$$ and approaches the ground with

Rocket Propulsion with Variable Mass

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

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)

Stability Analysis Using Center of Mass on a Pivoted Beam

A uniform beam of length 2.0 m and mass 10 kg is pivoted at one end. A 20 kg mass is suspended from

Stability and Center of Mass of a Structure

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

Torque and Angular Motion of a Rigid Beam

A non-uniform beam of length 2 m is pivoted at one end. Its mass distribution is given by $$\lambda(

Two-Dimensional Collision and Momentum Conservation

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

Variable Density Rod: Mass and Center of Mass Calculation

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

Variable Force Collision Analysis from Graph Data

A steel ball (mass = 0.2 kg) collides with a wall, and the force versus time data during the collisi

Analysis of Rotational Equilibrium in a Beam

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

Angular Acceleration from Variable Torque

Design an experiment where you apply a time-dependent (variable) torque to a rotating object and mea

Angular Displacement and Kinematics Analysis

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

Angular Kinematics from Disk Data

A rotating disk’s angular displacement is recorded over time during a period of uniform angular acce

Angular Kinematics of a Rotating Disk

Consider a rotating disk for which you want to measure angular displacement, angular velocity, and a

Angular Momentum Conservation: Ice Skater

An ice skater is spinning with an initial angular velocity $$\omega_i$$ and an initial moment of ine

Angular Momentum Transfer in Colliding Rotational Bodies

A student studies collisions between two rotating bodies—a spinning disk and a smaller rotating whee

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_

Calculation of Rotational Inertia for Composite System

A uniform disk of mass $$M = 2\,kg$$ and radius $$R = 0.5\,m$$ has two small beads, each of mass $$m

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 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

Calculus-Based Analysis of Angular Impulse

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

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

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 Explosive Separation

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

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 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

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

Differentiating Between Contact and Rolling Without Slip

An experiment is conducted to analyze the motions of objects rolling down an inclined plane. Both th

Dynamic Equilibrium in Rotational Motion

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

Dynamic Stability of a Rotating Space Station

A rotating space station initially spins with angular velocity $$\omega_i$$ and has a moment of iner

Dynamics of a Rotating Rod with Sliding Masses

In this advanced experiment, masses are allowed to slide along a horizontal rod attached to a pivot.

Energy Transfer in Rolling Objects

Design an experiment to study the energy conversion in a rolling object down an incline, by measurin

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 17: Moment of Inertia of a Non-Uniform Rod

A rod of length L = 2.00 m has a density that varies with position according to $$\rho(x) = \rho_0 *

Inelastic Collision of Rotating Disks

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

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

Moment of Inertia of a Continuous Rod

Consider a uniform thin rod of length $$L$$ and total mass $$M$$. The rod rotates about an axis perp

Parallel Axis Theorem in Compound Systems

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

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 Cylinder Down an Incline

A solid cylinder rolls without slipping down an incline. A set of measurements were made at differen

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

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-dependent Torque and Angular Momentum Change

A disk is subjected to a time-dependent torque described by $$\tau(t) = 10 * \cos(t)$$ N*m. The disk

Amplitude Decay in Damped Oscillations

A damped oscillator has its displacement described by the function $$y(t)=A_0*e^{-\frac{b}{2*m}t}*\c

Analyzing Phase Shift and Amplitude Modulation in SHM

An oscillator’s displacement is given by the equation $$y(t)= (0.05 + 0.01 * t) * \sin(8*t+0.2)$$

Calculus Approach to Maximum Velocity in SHM

Consider an oscillator whose displacement is given by the sinusoidal function $$y(t) = A \sin(\omega

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 Analysis of Velocity and Acceleration

Consider an oscillator whose displacement is defined by: $$x(t) = 0.1 * \sin(6*t)$$ (a) Differenti

Calculus-Based Derivation of Oscillator Velocity and Acceleration

For an oscillator described by $$y = A\sin(\omega t + \phi_0)$$, derive its velocity and acceleratio

Calculus-Based Derivation of Work Done in Stretching a Spring

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

Comparative Analysis of Horizontal and Vertical Oscillators

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

Comparative Analysis: Spring-Mass vs. Pendulum Oscillators

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

Conservation of Energy: Integral Approach in SHM

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

Coupled Oscillations in a Two-Mass Spring System

Consider two masses, $$m_1$$ and $$m_2$$, connected by a spring with spring constant $$k$$. Addition

Data Analysis from a Virtual SHM Experiment

A virtual experiment on simple harmonic motion produces the following data for the displacement of a

Determination of Maximum Elastic Potential Energy

A researcher is examining the energy stored in a spring when it is displaced from its equilibrium po

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

Driven Oscillations and Resonant Response

Consider a mass-spring system that is subjected to a periodic driving force given by $$F(t)=F_0*\cos

Dynamic Equilibrium in a Vertical Oscillator

A researcher studies the oscillatory motion of a block attached to a vertical spring. The block is d

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 Transformations in a Spring Oscillator

A block attached to a horizontal spring oscillates with amplitude $$A$$. Consider the energy transfo

Error Analysis in SHM Measurements

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

Evaluating Hooke's Law in Spring Oscillators

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

FRQ 13: Experimental Design for Hooke's Law Verification

Design an experiment to verify Hooke's law using a spring and a set of known masses. Answer the foll

FRQ 16: Maximum Speed in SHM via Energy Methods

Experimental data for a mass–spring oscillator, including amplitude, mass, spring constant, and meas

FRQ 17: Pendulum Nonlinear Effects

Consider a simple pendulum with length $$L = 1\ m$$. While for small angular displacements the perio

FRQ 19: Vertical Oscillator Dynamics

A mass is attached to a vertical spring. When displaced by a distance $$y$$ from its equilibrium pos

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

FRQ10: Damped Oscillations – Amplitude Decay and Velocity Derivation

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

FRQ13: Determining Damping Coefficient from Amplitude Decay

A damped oscillator with mass \(m = 0.1\,kg\) exhibits an exponential decay in its amplitude. Initia

FRQ20: Energy Dissipation in Damped Pendulum Oscillations

A damped pendulum oscillates with small angles such that its motion is approximately described by $

Impact of Spring Constant Variation on Oscillatory Motion

A researcher studies how varying the spring constant affects the oscillatory motion of a block attac

Influence of Initial Phase on Oscillator Motion

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

Kinematics of SHM and Calculus Differentiation

A block-spring oscillator is observed to undergo simple harmonic motion. In an experiment, students

Mechanical Energy in SHM

A researcher attaches a block of mass $$m = 0.05 \; kg$$ to a horizontal spring with force constant

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

Non-conservative Forces in Oscillating Systems

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

Pendulum Energy Dynamics

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

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 Period Measurement Experiment

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

Sinusoidal Analysis of SHM with Phase Shift

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

Spring Force and Elastic Potential Energy

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

Spring Force and Energy Analysis

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

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

Transit Time of a Simple Pendulum in Different Gravitational Fields

A simple pendulum with a length of $$L=2.0\,\text{m}$$ oscillates on Earth (with \(g=9.81\,\text{m/s

Analysis of Low Earth Orbit Satellite Decay

A low Earth orbit (LEO) satellite experiences gradual orbital decay due to atmospheric drag. Analyze

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

Analyzing a Two-Body Gravitational Interaction Using Calculus

Consider two objects of masses $$m_1$$ and $$m_2$$ that are initially at rest and start moving towar

Angular Momentum Conservation in Orbits

Analyze the relationship between orbital radius and angular momentum as depicted in the provided gra

Average Orbital Energy and Angular Momentum

For an orbiting satellite, the specific orbital energy and angular momentum are key conserved quanti

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

Barycenter of the Sun-Earth System

A researcher investigates the center of mass (barycenter) of the Sun-Earth system. Given that the ma

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

Calculus Analysis of Gravitational Potential Energy

Gravitational potential energy (GPE) plays a crucial role in systems influenced by gravity. Answer t

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

Center of Mass in a Two-Body System: Sun-Earth Analysis

Using the values $$m_{Earth} = 5.98 * 10^{24}\,kg$$, $$M_{Sun} = 1.99 * 10^{30}\,kg$$, and the avera

Comparative Analysis of Planetary Orbits

Using observational data for two planets, analyze how well their orbital periods conform to Kepler's

Derivation of Gravitational Potential Energy Difference

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

Derivation of 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

Derivation of the Virial Theorem for Gravitational Systems

Derive the virial theorem for a gravitationally bound system and apply it to a hypothetical star clu

Deriving Gravitational Force from Gravitational Potential Energy

In a region where the gravitational potential energy between two masses is given by $$U(r) = -\frac{

Determination of Gravitational Constant Using a Compound Pendulum

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

Effective Gravitational Field on an Irregular Asteroid

An irregularly shaped asteroid has a non-uniform density distribution. To determine the effective gr

Energy Balance at Apoapsis and Periapsis

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

Energy Conservation in Gravitational Fields: Application to Roller Coaster Dynamics

Although gravitational potential energy is most famously applied in celestial mechanics, the concept

Escape Velocity and Energy Requirements

A spacecraft must reach escape velocity to leave a planet's gravitational field. The escape velocity

FRQ 3: Center of Mass in the Sun-Earth System

In the Sun-Earth system, although both bodies orbit their common center of mass (barycenter), the di

FRQ 13: Orbital Transfer Maneuver – Hohmann Transfer

A spacecraft in a circular orbit of radius $$r_1$$ wishes to transfer to a higher circular orbit of

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 Parameter in Exoplanetary Systems

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

Gravitational Potential via Integration in a Varying Density Sphere

A computational experiment is conducted to calculate the gravitational potential inside a spherical

Kepler's Third Law and Satellite Orbits

Kepler’s Third Law, when applied to satellites in nearly circular orbits, can be used to relate the

Modeling Orbital Decay with Differential Equations

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

Optimizing Orbital Transfer Maneuvers: Hohmann Transfer

A spacecraft is planning an orbital transfer maneuver from a lower circular orbit to a higher one us

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

Orbital Speed and Radius in Circular Orbits

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

Perturbation in Orbital Motion

A small asteroid deviates from a perfect elliptical orbit due to a time-dependent perturbative force

Variation of Gravitational Force with Distance

Consider the gravitational force given by $$F(r) = \frac{G m_1 m_2}{r^2}$$. Answer the following par

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