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 Air Resistance Effects on Free Fall

In an experiment on free fall, an object is dropped and its velocity is measured over time. The calc

Analysis of Experimental Data Table

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

Calculating Displacement via Integration of a Velocity Function

An object moves in one dimension with its velocity described by $$v(t)=4*t-2$$ m/s. Determine the di

Conservation of Energy in a Pendulum

Design an experiment to test the conservation of mechanical energy in a simple pendulum. Provide bot

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°

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

Determining Velocity from a Position Function with Differentiation Error

An experiment recorded the position of a particle moving along a straight line, modeled by the funct

Differential Equation of Motion Under Gravity and Drag

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

Experimental Determination of g

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

Free Fall from a Cliff with Calculus Insights

A rock is dropped from an 80-meter cliff. Assuming the only acceleration is due to gravity (with $$g

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: 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 8: Circular Motion Kinematics (MEDIUM)

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

FRQ 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 12: Graphical Analysis of Vertical Motion (MEDIUM)

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

FRQ 13: Analyzing a Two-Dimensional Collision with Projectiles

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

FRQ 15: Circular Motion with Varying Speed

A particle moves along a circular track of radius 5 m with its speed given by $$v(t)= 2 + t$$ (in m/

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 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

Graphical Analysis of Distance and Displacement

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

Kinematics with Calculus: Non-Uniform Acceleration

An object moves along the x-axis under a non-uniform acceleration given by $$a(t) = 4*t - 2$$ m/s²,

Motion on an Inclined Plane

A block is released from rest on a frictionless incline with an angle of $$30^\circ$$. The accelerat

Multi-Phase Rocket Motion Analysis

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

Non-Uniform Acceleration Analysis

A particle's position is given by $$x(t)=\sin(t)-0.5*t^2$$. Analyze its motion using calculus.

Optimization of Projectile Range

A projectile is launched from ground level with a fixed initial speed of 50 m/s. Use calculus to det

Optimizing the Launch Angle for Maximum Horizontal Displacement on Uneven Terrain

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

Oscillating Particle under Uniform Acceleration

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

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 $

Rotational Motion: Angular Kinematics

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

Round Trip Motion Analysis

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

Simultaneous Measurement of Velocity and Acceleration

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

Skydiver with Air Resistance: Variable Acceleration

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

Two-Dimensional Projectile with an Elevated Launch Point

A ball is thrown from the edge of a cliff that is 20 m above the ground with an initial speed of 30

Uniformly Accelerated Motion: Incorrect Baseline Velocity

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

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

Verification of Uniformly Accelerated Motion

A student conducts an experiment on a frictionless track using a cart. The student hypothesizes that

Ballistic Kinetic Energy with Air Resistance

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

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

Conservation of Mechanical Energy with Dissipative Forces

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

Determining Maximum Height using Energy Conservation

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

Elastic Potential Energy and Block Dynamics

A 2 kg block is placed on a frictionless horizontal surface and compresses a spring by 0.1 m. The sp

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

Energy Dissipation in a Bouncing Ball

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

Energy Loss in a Ball with Air Resistance

A ball with a mass of 0.5 kg is thrown vertically upward with an initial speed of 20 m/s. During its

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 3: Kinetic Energy Measurement in Free Fall

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

FRQ 4: Conservation of Mechanical Energy in a Roller Coaster

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

FRQ 7: Energy Loss Due to Friction on a Sliding Object

An object is sliding along a horizontal surface and loses kinetic energy due to friction. A sensor r

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 15: Falling Object Speed in a Varying Gravitational Field

A recent study claims that the speed of an object falling in a varying gravitational field can be de

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

FRQ 19: Analysis of Force–Time Data in a Crash Test

During a crash test, the force experienced by a dummy is recorded as a function of time. The force–t

FRQ 20: Evaluating Efficiency in a Conveyor Belt System

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

Gravitational Potential Energy in a Varying Field

A 5-kg mass is moved vertically in a non-uniform gravitational field where the gravitational acceler

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

Investigating Work on an Inclined Plane

A researcher is investigating the work done on a 5 kg block being pushed up a frictionless inclined

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 +

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 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 Energy Analysis with Air Resistance Correction

A 0.5 kg ball is thrown vertically upward with an initial speed of 20 m/s. Air resistance does negat

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

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

Sliding Block Work‐Energy Experiment

In this experiment, a block of mass $$m$$ is released from rest at the top of a frictionless incline

Time-Varying Velocity and Instantaneous Power Measurement

A vehicle follows a displacement function given by $$x(t)= 4*t^3 - 2*t^2 + 3*t$$ (in meters) while a

Variable Gravitational Acceleration Over a Mountain

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

Work and Energy on an Inclined Plane with Variable Friction

A 4 kg block slides down a 10 m long incline set at 30° from the horizontal. The frictional force al

Work Done by a Variable Force

An object is acted upon by a variable force given by $$F(x)=5\,x^2$$ (in newtons) along the x-axis.

Work in a Variable Force Field along a Curved Path

A particle moves in the xy-plane along the curve defined by $$y = x^2$$ from the point (0, 0) to (2,

Work-Energy Analysis in Collisions

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

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 Experiment in Varying Potential Fields

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

Astronaut Recoil in Space

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

Calculus-Based Analysis of a Variable Density Disk

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

Center of Mass Acceleration under Variable Force

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

Center of Mass in a Coupled Mass-Spring System

Two masses (m1 = 1.5 kg and m2 = 1.5 kg) are connected by a light spring on a frictionless track. In

Center of Mass of a Composite Three-Dimensional Object

A uniform cube (mass $$4$$ kg, side length $$0.5$$ m) and a uniform sphere (mass $$2$$ kg, radius $$

Center of Mass of a Non-Uniform Rod

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

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 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 of a Variable Density Disk

A circular disk of radius R = 0.5 m has a surface mass density that varies with the radial distance

Center-of-Mass Motion Under an External Force

Consider a system composed of two masses, $$m_1=2.0\,kg$$ and $$m_2=3.0\,kg$$, connected by a light,

Complex Rotational and Translational Collision Involving Center of Mass

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

Conservation of Linear Momentum in a Glider Collision

On a frictionless air track, two gliders collide. The experimental data below list the masses and ve

Damped Harmonic Oscillator Analysis

A mass-spring system subject to damping has its displacement described by the function $$x(t)=0.2\,e

Displacement from Variable Acceleration

A 2 kg particle experiences a force given by $$F(t)=10*t$$ N over the time interval from 0 s to 4 s.

Elastic and Inelastic Collision Analysis

Two balls collide head-on. The red ball (mass $$0.5\,kg$$, velocity $$+4\,m/s$$) and the green ball

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

Evaluating Energy Dissipation in an Inelastic Collision

Two vehicles collide and stick together in an inelastic collision. The experimental data below provi

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

Experimental Design: Measuring Impulse with Force Sensors

Propose an experiment to measure the impulse delivered by a non-uniform force applied to a ball. You

Explosive Fragmentation: Momentum Transfer

A stationary object with a mass of 12 kg explodes into three fragments in a frictionless environment

Explosive Separation and Momentum Conservation

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

Explosive Separation in a Multi‐Stage Rocket

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

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 10: Collision with Rotational Motion

A uniform disk of mass $$2 \ kg$$ and radius $$0.5 \ m$$ is rolling without slipping at $$4 \ m/s$$

FRQ 17: Impulse from a Functional Force

A particle experiences a force given by $$F(t) = 4*t$$ (N) over the time interval $$0 \le t \le 5\ s

FRQ 19: Calculating COM for a Variable Density 2D Lamina

A two-dimensional lamina occupies the region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$ in t

Impulse and Angular Momentum in a Collision

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

Impulse and Velocity from a Variable Force

A particle of mass $$m=2.0\,kg$$ initially moves with a velocity $$v_i=2.0\,m/s$$. It is subjected t

Impulse Delivered by a Time-Dependent Damping Force

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

Impulse in a Collision with Force Graph Analysis

A 0.75 kg object undergoes a collision during which the force acting on it is given by $$F(t)=50-10*

Impulse on a Pendulum Bob

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

Inelastic Collision with a Movable Platform

A 0.3 kg ball moving horizontally at 8 m/s collides with and sticks to a 1.2 kg movable platform tha

Integrated Analysis of Momentum and Center of Mass in a Multi-Stage Experiment

In a complex experiment, a projectile traveling at 10 m/s with a mass of 5.0 kg breaks apart mid-fli

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

Non-Uniform Rod Analysis

A 1.0 m long rod has a linear density given by $$\lambda(x) = 10 + 6*x$$ (kg/m) where x is measured

Non-uniform Rod's Center of Mass

A rod of length L = 1.5 m has a non-uniform linear density given by $$\lambda(x) = 4 + 3*x$$ (in kg/

Oscillations: Simple Pendulum Analysis

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

Rocket Propulsion and Center of Mass Dynamics

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

Rocket Propulsion Momentum Problem

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

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

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(

Work-Energy Theorem: Roller Coaster Problem

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

Analysis of a Variable Moment of Inertia System

A rotating disk has a moment of inertia that decreases over time as its arms retract, following the

Analysis of Angular Displacement in a Rotating Disk

In this experiment, several dots are marked along the radius of a rotating disk. The students record

Angular Momentum Changes in a Skater's Spin

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

Angular Momentum in Explosive Separation

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

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

Combined Translational and Rotational Dynamics

A rolling disk collides elastically with a spring, causing the spring to compress before the disk re

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

Composite Object Rotational Dynamics Analysis

A researcher studies the rotational dynamics of a composite object composed of a uniform disk of mas

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

Conveyor Belt Dynamics Driven by a Rotating Drum

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

Determining the Moment of Inertia of a Compound Pendulum

A compound pendulum, consisting of an irregular rigid body pivoted at different locations, is used t

Discrete Mass Distribution and Moment of Inertia

A set of three small beads, each of identical mass $$m$$, is arranged along a light rod of length $$

Dynamic Stability of a Spinning Object

A gyroscope (spinning top) has a moment of inertia $$I=0.1\text{ kg\cdot m}^2$$ and spins with an an

Effect of Changing Moment Arm on System Dynamics

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

Effect of Friction on Rotational Motion

Design an experiment to quantify the torque losses due to friction in a rotating apparatus. Your goa

Energy Dissipation in Rolling Motion with Slipping

A solid sphere of mass $$M=5\text{ kg}$$ and radius $$R=0.2\text{ m}$$ rolls down an incline at an a

Evaluating the Impact of Frictional Torque on Rotational Motion

A researcher studies how a constant frictional torque affects the rotational motion of a spinning ob

Experimental Determination of Torsional Oscillations

Design an experiment to measure the torsional oscillation period of a rod suspended by a wire with a

FRQ 1: Torque Analysis on a Wrench

A mechanic uses a wrench of length L = 0.30 m to loosen a bolt. The mechanic applies a force of F =

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 8: Variable Torque and Angular Acceleration

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

Gyroscopic Precession and its Dependence on Spin Rate: An Experiment

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

Impulse and Angular Momentum: Impact on a Rotating Disk

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

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 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

Parallel Axis Theorem Application in Complex Systems

A composite object consists of a uniform rod of mass $$M = 4\,kg$$ and length $$L = 3\,m$$, and an a

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 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 of a Gyroscope

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

Rotational 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 Equilibrium Analysis of a Beam

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

Rotational Inertia of a Uniform Rod

A uniform, thin rod of length $$L$$ and mass $$m$$ rotates about an axis perpendicular to the rod th

Rotational Kinematics: Angular Displacement via Integration

A motor provides an angular acceleration given by $$\alpha(t) = 4 - 0.5*t$$ (in rad/s²) for $$0 \le

Seesaw Rotational Equilibrium

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

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

Torque and Angular Acceleration Relationship

An experiment measures the response of a rotating object to different applied torques. A graph is pl

Torque and Angular Acceleration: A Variable Force Problem

A rigid rod rotates about a fixed axis. A time-dependent force is applied perpendicular to the rod a

Torque from a Distributed Load

A uniform beam of length $$L$$ has a constant linear weight density $$w$$ (in N/m).

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

Analyzing the Role of Initial Conditions in SHM

In an experiment on a mass-spring oscillator, students set the system in motion with various initial

Calculus Application in SHM: Derivatives and Acceleration

Given the position function of an oscillator, apply calculus to derive its velocity and acceleration

Calculus Approach to Energy Dissipation in a Damped Oscillator

Consider a damped oscillator described by the differential equation $$m\frac{d^2y}{dt^2} + b\frac{dy

Calculus-Derived Velocity and Acceleration in SHM

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

Data Analysis and Calculus Estimation in SHM

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

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 Angular Frequency from Displacement Data

Displacement measurements for a spring-mass oscillator are given by the equation $$y = A\sin(\omega

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 Spring Constant from Experimental Force-Displacement Data

In an experiment to determine the spring constant, a series of force and displacement measurements w

Determining Initial Phase in SHM

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

Differentiation in SHM: Velocity and Acceleration

An oscillator is described by the function $$y = A * \sin(\omega * t + \phi_0)$$. Investigate the ve

Double Spring Oscillator Experiment

In a lab experiment, two springs are attached in series to a block on a frictionless surface. The st

Driven Oscillations and Resonance in a Mass-Spring System

A mass-spring system of mass $$m$$ is subjected to an external periodic driving force given by $$F(t

Elastic Energy and Maximum Speed Calculation

Consider a block attached to a horizontal spring with a spring constant of $$k = 100\,N/m$$. The blo

Energy Analysis and Instantaneous Power in SHM

A block of mass $$m = 0.1\,kg$$ is attached to a spring with force constant $$k = 800\,N/m$$ and osc

Energy Conservation Verification Using Calculus

A mass-spring system oscillates with a displacement given by $$x(t) = 0.06 \cos(15t)$$. (a) Derive t

Energy Conversion in a Spring-Mass Oscillator

Consider an experiment to investigate energy conversion in a spring-mass oscillator. In this experim

Energy Exchange in Coupled Oscillators

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

Energy Exchange in SHM

Consider a mass-spring oscillator with displacement given by: $$x(t) = A * \cos(\omega t)$$, with

Energy Transformations in a Mass-Spring System

A researcher investigates energy transformations in a mass-spring oscillator. The system consists of

Error Analysis in SHM Measurements

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

Evaluating the Role of Calculus in Predicting Oscillator Dynamics

A theoretical paper asserts that 'integrating the equations of motion always leads to an accurate pr

Experimental Analysis of SHM Data

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

Forced Oscillations and Beat Frequency

A mass attached to a spring is simultaneously driven by two periodic forces given by $$F_1(t)=F_0*\c

FRQ 1: Hooke’s Law Experiment

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

FRQ 4: Vertical Motion in a Spring–Block System

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

FRQ 10: Calculus Integration for Work Done in a Spring

Force measurements during the stretching of a spring were recorded as a function of displacement. Us

FRQ 13: Determining Angular Frequency from Oscillation Data

An oscillator’s motion is described by $$y = A \sin(\omega t + \phi_0)$$. A set of position measurem

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

FRQ4: Vertical Spring-Block Oscillator – Equilibrium and Oscillations

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

FRQ14: Oscillations on an Inclined Plane

A block is attached to a spring and placed on a frictionless inclined plane that makes an angle of $

Hooke's Law and Work in Springs

Consider a spring with a spring constant $$k = 200\,N/m$$. A student compresses the spring from its

Horizontal Spring Oscillator: Force and Energy Calculations

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

Impact of Spring Constant Variation on Oscillatory Motion

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

Investigating Damping Effects in a Spring-Mass Oscillator

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

Measuring g with a Simple Pendulum

A researcher uses a simple pendulum to measure the acceleration due to gravity. The length of the pe

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

Nonlinear Characteristics of the Simple Pendulum

The standard formula for the period of a simple pendulum, \(T \approx 2\pi\sqrt{\frac{L}{g}}\), reli

Nonlinear Restoring Force: Effects on the Period of Oscillations

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

Oscillations in a Coupled Mass-Spring System

Two masses, $$m_1 = 0.1 \; kg$$ and $$m_2 = 0.2 \; kg$$, are coupled by a single spring with a force

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 Dynamics Beyond the Small-Angle Approximation

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

Pendulum Motion and the Small Angle Approximation

A simple pendulum of length $$L$$ oscillates with small angular displacements. Analyze its motion us

Phase Difference Between Displacement and Velocity

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

Phase Shift Determination in SHM

A researcher studies a mass-spring oscillator and observes that at time $$t = 0$$ the block is at it

Phase Space Analysis of SHM

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

SHM with a Varying Force Constant

In an experiment on a spring-mass system, a student investigates oscillations at various amplitudes.

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

Vertical Oscillations and Energy Analysis in a Spring–Mass System

Investigate the motion and energy conversion of a vertically oscillating mass–spring system.

Vertical Oscillations on a Spring

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

Vertical Spring Oscillator Investigation

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

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

Analyzing Gravitational Slingshot Maneuvers

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

Analyzing Hohmann Transfer Orbits for Satellite Maneuvers

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

Angular Momentum Conservation in Orbits

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

Calculus Derivation of Kepler's Second Law

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

Calculus-based Derivation of Gravitational Force Variation

The gravitational force between two point masses is given by $$ F(r) = -G * \frac{m_1 * m_2}{r^2} $$

Cometary Orbits: Analyzing Highly Eccentric Trajectories

Comets often exhibit highly eccentric orbits. Their dynamics can provide insight into gravitational

Comparative Analysis of Gravitational Forces

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

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 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 Kepler’s Second Law via Calculus

Kepler’s Second Law states that a line joining a planet and its star sweeps out equal areas in equal

Designing a Satellite Orbit Experiment

An engineering team is planning an experiment to study satellite orbits around Earth. (a) List the

Determination of Gravitational Constant Using a Compound Pendulum

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

Determining Orbital Eccentricity from Observational Data

Astronomers collect data of a planet's distance from its star at various times and wish to determine

Determining the L1 Lagrange Point

In a star-planet system, an object is positioned along the line connecting the two bodies at the L1

Dynamics of Comet Orbits

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

Effective Gravitational Field on an Irregular Asteroid

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

Effects of Eccentricity on Planetary Orbits

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

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 Conservation in a Swinging Mass Experiment

An experiment is performed wherein a mass is swung in a vertical circle to investigate conservation

Escape Velocity Derivation

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

Examining Relativistic Corrections to Newtonian Gravity

In strong gravitational fields, relativistic corrections become significant compared to Newtonian pr

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

Gravitational Force Calculation on a Satellite

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

Gravitational Potential Energy in a Non-Uniform Field

A spacecraft of mass m moves radially from a distance R₀ to a distance R from a planet of mass M. Th

Gravitational Potential Energy Measurement on a Ramp

In a laboratory experiment, a block is released down a long ramp to measure the conversion of gravit

Gravitational Slingshot Maneuver

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

Gravitational Slingshot Maneuver

A spacecraft uses a gravitational slingshot maneuver around a planet to gain additional speed. Answe

Integration of Variable Gravitational Force over an Extended Body

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

Kepler's Third Law and Orbital Analysis

A recent media report claims that the orbital period $$T$$ and the semi‐major axis $$a$$ of satellit

Mass Determination using Orbital Motion and Kepler's Laws

A planet orbits a star in a nearly circular orbit with period $$T$$ and orbital radius $$r$$. (a) De

Modeling Orbital Decay with Differential Equations

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

Modeling Tidal Forces with Calculus

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

Optimization of Orbital Maneuvers in Multi-Stage Rockets

A multi-stage rocket performs orbital maneuvers to transition from a low Earth orbit to a higher geo

Orbit Transfer and Hohmann Transfer Orbits

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

Orbital Decay due to Atmospheric Drag

A satellite in low Earth orbit experiences atmospheric drag, which can be modeled as a force $$F_d =

Orbital Perturbations from Impulsive Thrust

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

Orbital Precession Analysis

Analyze the graph showing the change in orbital orientation of a planet over time and discuss the im

Orbital Transfer and the Hohmann Maneuver

A spacecraft performs a Hohmann transfer to move from a circular orbit of radius $$r_1$$ to a higher

Orbital Transfer Trajectories and Hohmann Transfers

A spacecraft in low Earth orbit (LEO) is planning a Hohmann transfer to reach a geosynchronous orbit

Satellite Orbital Decay with Atmospheric Drag Consideration

An experiment is designed to measure the decay of a satellite's orbit by tracking its altitude over

Variation of Gravitational Force with Distance

Newton’s Law of Gravitation indicates that the gravitational force between two masses varies inverse