AP Physics C: Mechanics FRQ Room

Analysis of a Velocity vs. Time Graph in a Lab Experiment

In an experiment on an air track, a cart's velocity was recorded and is depicted in the following gr

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

Analyzing a Parabolic Trajectory

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

Analyzing Two-Dimensional Motion Using a High-Speed Camera

In an experiment on projectile motion, a high-speed camera was used to record the two-dimensional mo

Application of the Big Five Kinematic Equations

An experiment provides the following data for an object in motion: initial velocity $$u=5$$ m/s, tim

Block on an Inclined Plane: Kinematic Analysis

A block of mass m is released from rest at the top of a frictionless inclined plane of length L = 10

Calculus in One-Dimensional Kinematics

Consider an object whose position as a function of time is given by $$x(t)=\sin(t)+t^2$$, where x is

Calculus-Based Analysis of a Car’s Accelerating Motion

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

Calculus-Based Analysis of Varying Acceleration

An object moves with a velocity function given by $$v(t)=3*t^2 - 12*t + 5$$. A table below shows cal

Calculus-Based Kinematics Derivation

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

Centripetal Acceleration in Circular Motion

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

Designing a Kinematics Lab Experiment

An engineer is tasked with measuring the constant acceleration of a cart on a frictionless track usi

Determining Launch Angle from Experimental Data

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

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'

Dynamics on an Inclined Plane with Friction

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

Experimental Evaluation of Vector Addition in Two Dimensions

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

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

Free Fall under Gravity

A rock is dropped from the top of a 100-meter tall building. Neglect air resistance.

Free-Fall Motion Analysis

A rock is dropped (from rest) from the top of an 80 m cliff. Assume that the acceleration due to gra

FRQ 4: Projectile Motion – Maximum Height and Range

A projectile is launched from the ground with an initial speed of 60 m/s at an angle of 30° above th

FRQ 5: Calculus-Based Displacement Calculation

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

FRQ 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: Non-Uniform Acceleration: Parabolic Motion

A researcher observes a car whose acceleration is not constant but given by the function $$ a(t) = 2

FRQ 17: Vector Field Analysis – Wind Impact on Projectile Motion

A researcher examines how a time-varying wind affects the horizontal motion of a projectile. In this

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

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

Graphical Analysis of Kinematic Data

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

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

Impulse and Momentum with a Time-Dependent Force

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

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

Instantaneous vs. Average Velocity

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

Investigation of Variable Friction in Curvilinear Motion

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

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 in One Dimension: Variable Acceleration

An object moves along the x-axis with an acceleration given by $$a(t) = 3*t$$ (in m/s²). At time $$t

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

Multi-Dimensional Motion Analysis and Vector Decomposition

An object moves in the plane and its position vector is given by $$\mathbf{r}(t)= (3t^2)\,\mathbf{i}

Oscillating Particle under Uniform Acceleration

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

Oscillatory Motion: Mass-Spring System

A 1.0-kg mass is attached to a spring with a spring constant of $$k = 16\, N/m$$. The mass is displa

Parametric Trajectory Analysis

A particle moves in the plane with its position given by: $$x(t)=2t^2$$ and $$y(t)=20t - 4.9t^2$$, w

Projectile Motion Revisited: Maximum Height and Impact Velocity

An object is projected vertically upward from ground level with an initial speed of $$50\,m/s$$. Ass

Projectile Motion: Determining Initial Conditions

In an experiment, a projectile’s horizontal displacement was measured over time. The recorded data a

Projectile Motion: Height and Flight Time Analysis

A projectile is launched with an initial speed of 50 m/s at a 45° angle. The following table shows o

Projectile Motion: Launch from a Moving Platform

A projectile is launched from the top of a train moving at 20 m/s. The projectile is thrown with an

Projectile Motion: Maximum Height and Range

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

Rotational Dynamics: Variable Torque

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

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

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=

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 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).

Variable Acceleration Analysis Using Calculus

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

Verifying Free Fall Acceleration

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

Analysis of a Potential Energy Curve

An object of mass 3 kg moves along the x-axis in a potential energy field given by $$U(x) = (x-1)(x-

Analysis of Fall Dynamics with Air Resistance

An object with a mass of 0.5 kg is dropped from a height of 50 m. Air resistance is modeled as a dra

Analysis of Force and Velocity Data

An object is subjected to a variable force recorded by a sensor, given by $$F(t)=100-5*t$$ (in newto

Determining Speed of a Roller Coaster Considering Friction

An 800-kg roller coaster car is released from rest at the top of a frictionless 50-m-high hill, then

Energy Analysis in Circular Motion

A 2 kg object moves in a horizontal circular path of radius 5 m. It is subject to a tangential force

Energy Analysis of a Bouncing Ball: Energy Loss and Power

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

Energy Conservation in a Pendulum

A simple pendulum of length 2 m and mass 0.5 kg is released from an initial angle of 30° with respec

Energy Conservation in a Spring–Mass System

In this experiment, a 0.5 kg mass is attached to a vertical spring with a spring constant of 200 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 in a Spring–Mass System

A mass is attached to a spring with a spring constant of $$k = 200\,N/m$$. The spring is compressed

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

Energy Transfer in a Bouncing Ball

A ball of mass 0.5 kg is dropped from a height of 10 m and, after hitting the ground, rebounds to a

FRQ 3: Kinetic Energy Change in a Car's Acceleration

A 1200-kg car accelerates along a straight, level road. Measurements of the car's speed at various d

FRQ 9: Calculus-Based Work Determination in a Braking Scenario

A car undergoing braking experiences a variable force that depends on its displacement. The braking

FRQ 10: Work Done on a Variable Mass System

A recent claim suggests that for systems with variable mass (e.g., rockets), the work done by a non-

FRQ 14: Calculus Analysis of Work Done on an Object with a Time-Varying Force

An object with a mass of 3 kg is subjected to a force that varies with time according to the equatio

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 16: Work and Energy Transformation in a Compound Machine

A 10-kg block is pushed up a frictionless ramp using a compound pulley system. A force sensor record

FRQ 17: Energy Loss Analysis in a Frictional Pendulum

A pendulum bob with a mass of 0.8 kg is released from an initial height corresponding to a potential

Gravitational Potential Energy Conversion

A 10 kg object is dropped from a height of 15 m in a vacuum. Using conservation of energy methods, a

Kinetic Energy and Work-Energy Theorem Application

A block of mass 2 kg undergoes net work in two different scenarios. Use the work–energy theorem to f

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

Motion on an Inclined Plane with Friction

A block of mass 4 kg slides down a 5 m long inclined plane that makes an angle of 20° with the horiz

Oscillations in a Mass-Spring System

A 1 kg mass is attached to a spring with a spring constant of 100 N/m. The mass is displaced 0.1 m f

Oscillatory Motion Energy Exchange Experiment

A mass attached to a vertical spring oscillates up and down, and a sensor records its displacement a

Pendulum Energy Conservation Experiment

A pendulum bob is released from a given initial angle and its speed at the bottom of the swing is re

Potential Energy Curve Analysis

An object has a potential energy given by $$ U(x) = (x-2)^2 - (2*x-3)^3 $$, where U is in joules and

Power Output from a Variable Force: Time-Dependent Problem

A particle is subjected to a time-dependent force given by $$F(t)= 5 \;\cos(0.5*t) \; (\text{N})$$.

Power Output Measurement in an Elevator Experiment

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

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

Rocket Engine Power Output Under Variable Thrust

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

Roller Coaster Energy Analysis

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

Rotational Power in Gear Systems

An experiment measures the power output of a gear train by recording the torque and angular velocity

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

Solar Energy Mechanical Conversion Experiment

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

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

Vertical Lift Work Measurement Experiment

In this vertical lift experiment, an object is raised by a motor and its applied force and displacem

Wind Tunnel Analysis of Mechanical Energy Extraction

In a wind tunnel experiment, a miniature wind turbine is tested. The force exerted by the wind on th

Work Done by a Time-Dependent Force

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

Work Done by Friction: Calculus Approach

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

Work Done on an Object by a Constant Force

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

Work-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 Theorem with Air Resistance

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

Work–Energy Experiment in Varying Potential Fields

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

Work, Energy, and Power in Circular Motion

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

Astronaut Momentum Conservation

An astronaut with a total mass of 89 kg is floating in space near her shuttle. To reorient herself,

Center of Mass Analysis in a Two-Mass Pulley System

In a two-mass pulley system, students aim to determine the center-of-mass motion by measuring accele

Center of Mass Determination for a Composite L-Shaped Object

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

Center of Mass Measurement Using a Suspended Rod

In this experiment, students attempt to determine the center of mass of a non-uniform rod by suspend

Center of Mass of a 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 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 a Variable-Density Rod

Consider a thin rod of length $$L=2.0$$ m with a linear density given by $$\lambda(x)=2+3*x$$ (kg/m)

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 $$\lambda(y)=3.0(1+y)$$

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

Data Analysis: Testing the Linear Relationship between Impulse and Momentum Change

An experiment is conducted where a cart is subjected to several impacts. For each trial, the measure

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.

Dynamics of a Falling Object with Air Resistance

An object of mass 0.1 kg is dropped from a height and experiences air resistance modeled as $$F_{air

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 3: Motion of the Center of Mass under External Force

An object of mass $$10 \ kg$$, initially at rest, is subjected to an external force given by $$F(t)

FRQ 12: Graphical Analysis of Force-Time Data

An experiment measured the force on a 2 kg cart as a function of time. The resulting force-time grap

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

Impulse and Momentum under a Variable Force

A 5 kg cart on a frictionless track is initially at rest. A variable force given by $$F(t)=10*t$$ (N

Impulse and Momentum with Variable Force Function

A ball is struck by a kick where the force exerted on it varies with time according to the function

Impulse from a Variable Force Function

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

Impulse from Force Sensor Data

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

Impulse-Momentum in Soccer Kick

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

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

Mass-Spring Collision Momentum Exchange

A 0.5 kg particle moving horizontally collides briefly and impulsively with a vertically hanging spr

Mobile Robot Center of Gravity Analysis

A mobile robot features extendable arms that modify its mass distribution. With its arms retracted,

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 Change under a Non-Constant Force

A 1.2-kg block on a frictionless surface is subjected to a time-varying force given by $$F(t) = 12*s

Motion of the Center of Mass under Applied Force

Two blocks, with masses 3 kg and 5 kg, are connected by a massless rope on a frictionless surface. A

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

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 Motion with Drag Impulse Analysis

A projectile is launched horizontally and experiences a drag force proportional to its velocity, giv

Ramp Push Experiment: Variable Force Integration

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

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

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

Velocity Determination under a Variable Force

A 2 kg block is pulled on a frictionless surface by a variable force given by $$F(x)=3*x$$ (N) where

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^{

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

Angular Impulse and Change in Angular Momentum

Design an experiment to measure the angular impulse delivered to a rotating object and its resulting

Angular Kinematics Analysis Using Graphical Data

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

Angular Kinematics: Modeling a Rotating Spring System

A disk is attached to a torsional spring that exerts a restoring torque given by $$T = -k \times \th

Angular Momentum in a Variable Moment of Inertia System

A researcher examines the dynamics of a rotating system whose moment of inertia changes with time du

Angular Momentum in Explosive Separation

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

Comparative Angular Momentum in Different Systems

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

Composite Object Rotational Dynamics Analysis

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

Conveyor Belt Dynamics Driven by a Rotating Drum

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

Correlation Between Torque and Rotational Energy via Calculus

A student designs an experiment to investigate the relationship between applied torque and rotationa

Cylinder Rolling on an Incline

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

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

Dynamics of a Rotating System with Friction

A rotating disk with moment of inertia $$I$$ is subject to a frictional torque that is proportional

Energy Conservation in Rotational Motion

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

Energy Conversion in a Rolling Cylinder Experiment

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

Energy Dissipation in a Rotating System with Friction

A turntable with moment of inertia $$I = 0.3 \text{ kg m}^2$$ is initially spinning at $$\omega_0 =

Equilibrium Analysis in Rotational Systems

A uniform beam is balanced on a pivot, with forces applied at various distances from the pivot to ma

FRQ 2: Rotational Inertia of a Composite System

A thin uniform rod of length L = 2.00 m and mass M = 5.00 kg has two small beads, each of mass m = 1

FRQ 4: Rotational Kinematics of a Disk

A disk starts from rest and rotates with a constant angular acceleration. After t = 4.00 s, the disk

FRQ 10: Comparison of Rotational and Translational Kinetic Energy

A solid sphere of mass M = 4.00 kg and radius R = 0.10 m rolls without slipping down a ramp of heigh

FRQ 11: Impact of Mass Distribution on Angular Acceleration

Two wheels have identical mass M and radius R. Wheel A has all its mass concentrated at the rim (\(I

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

Moment of Inertia of a Composite System using Calculus

A composite system consists of a uniform rod of mass $$M$$ and length $$L$$ with two small beads, ea

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 on an Inclined Plane

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an inclined plane from a

Rotational Inertia Determination Using a Torsion Pendulum

You are provided with a torsion pendulum apparatus consisting of a rod suspended by a wire with a kn

Rotational Inertia Measurement with a Disk and Pendulum

In this experiment a flat disk is mounted as a pendulum with its pivot offset from its center. The o

Rotational Inertia of a Hollow Cylinder vs. a Solid Cylinder

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

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

Time-varying Angular Acceleration in a Rotational System

A disk experiences an angular acceleration described by $$\alpha(t)=5\sin(2t) \text{ rad/s}^2$$.

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

Variable Torque Function Integration

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

Analysis of Energy Transitions in an Oscillating Pendulum

An experiment is conducted on a simple pendulum to study the energy transitions between kinetic and

Analysis of Maximum Kinetic Energy in a Spring-Mass Oscillator

For a spring-mass oscillator with spring constant $$k$$ and amplitude $$A$$, the elastic potential e

Analysis of SHM Under Driving Force

A researcher studies a damped, driven harmonic oscillator subjected to an external sinusoidal force

Calculus Approach to Maximum Velocity in SHM

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

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

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

Comparing Vertical and Horizontal Oscillations in Mass-Spring Systems

A claim has been made that 'the behavior of a vertical spring-mass oscillator is identical to that o

Coupled Oscillators: Normal Modes Analysis

Consider a system of two identical masses \(m\) placed on a frictionless surface and connected by th

Critical Analysis of Frequency Measurement Techniques in SHM

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

Damped Oscillations in a Spring System

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

Damped Oscillations: Amplitude Decay Analysis

A mass-spring oscillator with $$m = 0.2\,kg$$ and damping constant $$b = 0.1\,kg/s$$ experiences dam

Data Analysis of Oscillatory Motion with Damping Effects

A student report claims that 'damped oscillations follow the exact same simple harmonic law as undam

Derivation of Total Mechanical Energy Conservation in SHM

For a block-spring system undergoing simple harmonic motion, demonstrate that the total mechanical e

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 from Sinusoidal Oscillation Data

A researcher records the displacement of an oscillator at various time intervals. Use the data provi

Determining the Spring Constant from Oscillation Data

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

Effect of Mass Variation on SHM

A block attached to a spring oscillates, and the period of oscillation is given by $$T = 2\pi\sqrt{\

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 Analysis in Simple Harmonic Motion

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

Energy Conservation 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 Transformations in a Mass-Spring System

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

Estimating Spring Constant from Kinetic Energy Measurements

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

Experimental Analysis of SHM Data

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

Forced Oscillations and Resonance

A mass-spring oscillator is subject to an external periodic driving force given by $$F(t)=F_0\cos(\o

FRQ 1: Spring Force Calculation Using Hooke's Law

A spring has a natural length of $$0.12\ m$$ and a force constant of $$k = 400\ N/m$$. When the spri

FRQ9: Energy Exchanges in a Mass-Spring Oscillator

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

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

FRQ20: Energy Dissipation in Damped Pendulum Oscillations

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

Investigation of Energy Conservation in SHM Using Calculus

A researcher analyses a mass-spring oscillator with its displacement given by $$y(t) = 0.1 * \cos(15

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

Mass-Spring Differential Analysis

Consider a block of mass $$m$$ attached to a horizontal spring with spring constant $$k$$. The block

Nonlinear Restoring Force: Beyond Hooke's Law

Some real-world springs do not obey Hooke's law for large displacements. Consider a spring that exer

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

Pendulum Motion Experimental Analysis

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

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

Time-Derivative Analysis of Displacement in SHM

An experiment aims to determine the velocity of a mass undergoing simple harmonic motion by calculat

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

Vertical Spring Oscillator Investigation

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

Analysis of a Gravitational Potential Energy Graph

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

Calculus Analysis of Gravitational Potential Energy

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

Calculus Derivation of Gravitational Potential Energy

Derive the expression for gravitational potential energy using calculus and compare your result to e

Cannonball Trajectory in a Non-Uniform Gravitational Field

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

Comparative Analysis of Orbital Periods for Different Planets

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

Comparative Analysis of Planetary Orbits

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

Comparative Analysis of Planetary Orbits

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

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 from Gravitational Force

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

Deriving Gravitational Force from Gravitational Potential Energy

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

Deriving the Gravitational Potential Energy Function

Starting with Newton's law of gravitation expressed as $$F = - G * \frac{m_1 * m_2}{r^2}$$, derive t

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

Effects of Non-Spherical Mass Distribution on Satellite Orbits

A planet is not perfectly spherical but exhibits a slight oblateness, introducing a perturbative ter

Elliptical Orbits and Angular Motion

A planet follows an elliptical orbit around its star. Investigate how variations in orbital distance

Energy Conservation in Elliptical Orbits

Energy conservation is a key principle in orbital dynamics, particularly in elliptical orbits where

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

Experimental Verification of Conservation of Energy in a Gravitational Field

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

FRQ 1: Gravitational Force between Two Masses

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

FRQ 14: Work Done in Changing Orbital Radius

The work done against gravity in changing the orbital radius of an object is computed by integrating

FRQ 17: Tidal Forces and Differential Gravity

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

Gravitational Energy in a Three-Body System

Consider three point masses m1, m2, and m3 placed at the vertices of a triangle. Analyze the gravita

Gravitational Potential to Rotational Kinetic Energy Conversion

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

Gravitational Potential via Integration in a Varying Density Sphere

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

Gravitational Slingshot Maneuver Analysis

Gravitational slingshot (or flyby) maneuvers are used in space missions to change a spacecraft's vel

Investigating Tidal Forces and Differential Gravity Effects

Consider a moon orbiting a planet, where tidal forces arise due to the variation in gravitational fo

Kepler's Third Law and Planetary Motion

Consider two planets orbiting the same star with orbital periods $$T_1$$ and $$T_2$$ and semimajor a

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

Orbit Stability from Potential Energy Diagrams

Analyze the provided potential energy diagram and determine the regions corresponding to stable and

Orbit Transfer and Hohmann Transfer Orbits

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

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

Perturbation Analysis in Elliptical Orbits

An elliptical orbit is subject to a small, constant perturbing force that causes a slow change in th

Satellite Maneuver Simulation with Finite Burn Dynamics

An experimental simulation aims to predict a satellite's trajectory during a maneuver that involves

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

Simulating Satellite Orbital Decay and Atmospheric Drag

An experimental simulation is set up to study the orbital decay of a satellite due to atmospheric dr

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 by Gravitational Force on a Falling Object

An object is dropped from a tall structure where the gravitational acceleration decreases with altit