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 Ballistic Trajectory with Inaccurate Symmetry Assumption

In a projectile motion experiment, a ball was launched at a known angle and its trajectory was recor

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*

Comparative Analysis of Kinematic Equations

A researcher claims that the kinematic equation $$s=ut+\frac{1}{2}at^2$$ is universally valid for al

Critical Evaluation of Experimental Kinematics Data

A researcher claims that the motion of a falling object is characterized by uniform acceleration. Th

Designing an Experiment: Motion on an Inclined Air Track

You are asked to design an experiment to determine the coefficient of kinetic friction on an incline

Determining Instantaneous Rates from Discrete Data

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

Determining 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

Evaluating Non-Uniform Acceleration from Experimental Data

A student records the following velocity data for an object undergoing non-uniform acceleration:

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

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 2: Distance vs. Displacement in Variable Motion (MEDIUM)

An object moves along the x-axis with a velocity given by $$v(t)=3*t-6$$ (in m/s). (a) Determine the

FRQ 3: Graphical Analysis of Velocity-Time Data

A researcher collects velocity vs. time data from an object undergoing several phases of motion: acc

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 11: Kinematics with Acceleration as a Function of Position (HARD)

An object moving along the x-axis has an acceleration that varies with its position: $$a(x)=4*x$$ (i

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 17: Experimental Analysis of Uniform Acceleration (MEDIUM)

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

FRQ 18: Experimental Kinematics Data Analysis

A series of measurements for an object's velocity at various times are recorded as follows: | Time

FRQ 20: Experimental Design – Determining g with a Free Fall Apparatus

In a physics laboratory, researchers design an experiment using a free fall apparatus to measure the

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

Investigating Motion on an Inclined Plane

A lab experiment was set up to study the motion of a cart on an inclined air track. The cart’s displ

Lab Investigation: Effects of Launch Angle on Projectile Range

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

Motion 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 of a Bus on a Curved Track

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

Non-Uniform Acceleration Analysis

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

Photogate Timer in Free Fall

A student uses a photogate timer to record the free fall of an object dropped from a height of 1.5 m

Polynomial Position Function Analysis

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

Projectile Launch from an Elevated Platform

A ball is launched from a platform 10 meters above the ground with an initial speed of 30 m/s at an

Projectile Motion Analysis

An object is launched with an initial speed of $$60\,m/s$$ at an angle of $$30^\circ$$ above the hor

Projectile Motion on an Inclined Plane

A ball is launched with an initial speed of $$30\,m/s$$ at an angle of $$40^\circ$$ above the horizo

Projectile Motion on Level Ground

An object is launched from ground level at a 45° angle with an initial speed of 30 m/s (neglect air

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

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

Relative Displacement in Different Frames

A particle moves along the x-axis and its position is described by $$x(t)=7*t-2*t^2$$ (in meters). T

Sinusoidal Position and Velocity Analysis

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

Skydiver with Air Resistance: Variable Acceleration

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

Uniformly Accelerated Motion on a Track

Design an experiment to test the hypothesis that in uniformly accelerated motion, the displacement i

Uniformly Accelerated Motion on an Inclined Plane

A 5.0-kg block is placed on a 30° inclined plane. When released, it slides down with a coefficient o

Variable Acceleration and Integration

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

Verifying Free Fall Acceleration

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

Work and Energy in Linear Motion

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

Analysis of Mechanical Advantage and Work in a Lever System

A lever is used to lift a 500 N weight. The operator applies a force that varies with the angle, giv

Analysis of Potential Energy Curves

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

Calculating Work on an Inclined Plane with Variable Force

A 6 kg box is pushed up a frictionless incline that makes an angle of 30° with the horizontal. The a

Calculus Analysis of a Ramp System

A 10 kg block is pushed up a frictionless ramp by an applied force given by $$F(x)=50 - 4\,x$$ (in n

Calculus in Friction: Variable Friction Coefficient Analysis

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

Collision and Energy Loss Analysis

Two objects collide inelastically and stick together. Object A has mass 3 kg moving at 4 m/s, and Ob

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

Damped Pendulum Energy Dissipation

A pendulum with a bob of mass 0.5 kg and length 2 m is subject to a damping force proportional to it

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 on a Frictional Ramp with Calculus Approach

A 2-kg block slides down a straight inclined ramp from a height of $$h = 2 \;\text{m}$$. The ramp is

Energy Loss in Inelastic Collisions Experiment

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

FRQ 7: Roller Coaster Energy Conversion and Energy Loss Analysis

A roller coaster car is reported to convert all its gravitational potential energy into kinetic ener

FRQ 11: Analysis of a Bungee Jump – Variable Net Force

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

FRQ 15: Energy Conservation in an Oscillating Spring–Mass System

A 2-kg mass attached to a spring (with spring constant k = 200 N/m) oscillates horizontally. A displ

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 20: Non-Constant Force Work Calculation via Integration

An experiment claims that for a non-constant force, the work done on an object can be accurately com

Hydraulic Press Work Calculation Experiment

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

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

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 in a Variable Force System

A block is subjected to a variable force and its velocity varies with time. The force acting on the

Integrating Power over Time for Energy Consumption

A machine operates with a time-dependent power output given by $$ P(t)= 500 + 100*t $$ (in watts) ov

Integration of Work in a Variable Gravitational Field

A researcher is analyzing the work done on a satellite of mass $$m = 1000\,kg$$ moving away from a p

Investigating Power Output in a Mechanical System

A researcher measures the power output of a machine that exerts a constant force while moving an obj

Investigation of Non-Conservative Forces in a Roller Coaster Model

A researcher develops a model of a roller coaster of mass 500 kg moving along a track with both grav

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

Pulley System Work–Energy Verification

A two-mass pulley system is used to verify the work–energy theorem. Velocities of both masses are re

Relationship Between Force and Potential Energy

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

Rolling Motion on an Incline: Combined Energy Analysis

A solid sphere with mass 3 kg and radius 0.2 m rolls without slipping down an inclined plane of heig

Rotational Work-Energy Analysis in a Flywheel

A flywheel with a moment of inertia $$I = 20\,kg\cdot m^2$$ is accelerated from rest to an angular s

Rotational Work-Energy in a Pulley System

A pulley with a radius of 0.2 m and a moment of inertia $$I = 0.5\,kg\cdot m^2$$ rotates without sli

Sliding Block Work‐Energy Experiment

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

Spring with Nonlinear Force: Elastic Potential Energy via Integration

A nonlinear spring exerts a restoring force given by $$F(x)= k*x + \alpha*x^3$$, where $$k = 200 \;\

Variable Gravitational Acceleration Over a Mountain

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

Work Done Against Friction on an Inclined Plane

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

Work Done in a Non-uniform Gravitational Field

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

Work-Energy Principle in a Frictional System

A 5 kg block is moving upward along a 10-degree incline of length 5 m with an initial speed of 7 m/s

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 Non-Uniform Gravitational Field

A particle of mass 1 kg is raised from the surface of a planet where the gravitational acceleration

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 Momentum Conservation

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

Center of Mass Calculation of a Non-Homogeneous Beam

A horizontal beam of length $$4$$ m has a linear mass density given by $$\lambda(x) = 5 + 4 * x^2$$

Center of Mass for Discrete Particles in the Plane

Three particles are located in the plane with the coordinates and masses given in the table below:

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 Nonuniform Circular Disk

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

Derivation of the Rocket Equation Using Momentum Conservation

A rocket moving in space expels mass continuously. Assume that during an infinitesimal time interval

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: Determining the Center of Gravity of a Complex Structure

Design an experiment to determine the center of gravity of a complex structure composed of multiple

Experimental Design: Investigating Collision Elasticity

Design a laboratory experiment to compare the kinetic energy retention in elastic and inelastic coll

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

FRQ 6: Elastic Collision Analysis

Two spheres collide elastically along a straight line. Sphere A (mass = 0.5 kg) initially moves at +

FRQ 7: Inelastic Collision Analysis

Two carts collide on a frictionless track and stick together. Cart X (mass = 2 kg) moves at +3 m/s a

FRQ 10: Collision with Rotational Motion

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

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 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 Calculation from Force-Time Graph

A particle is subjected to a time-varying force represented by the graph provided. Using calculus, d

Impulse from a Time-Varying Force with Graph Stimulus

A force sensor records the force applied to a hockey puck as a function of time while a player strik

Inelastic Collision of a Pendulum Bob with a Block

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

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

Momentum Analysis in Explosive Fragmentation Simulation

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

Momentum Analysis of a Variable-Density Moving Rod

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

Momentum and Angular Momentum in a Rotational Breakup

A rotating disk in space breaks apart into two fragments. Experimental measurements record both the

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

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

Multiple Collisions in a Figure Skating Routine

In a choreographed figure skating routine, two skaters push off from each other. Skater A has a mass

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

Recoil of an Astronaut after Throwing a Tool

An astronaut with a total mass of 90 kg, initially stationary in space, throws a 2 kg tool at a spee

Rocket Propulsion Momentum Problem

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

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

Spring-Loaded Collision with Impulsive Force

A 0.5 kg ball moving horizontally at $$8$$ m/s collides with a spring-mounted barrier that exerts a

Two-Dimensional Collision Analysis

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

Two-Dimensional Collision and Momentum Conservation

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

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 Kinematics Analysis Using Graphical Data

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

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 Conservation on a Merry-Go-Round

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

Angular Momentum Conservation: Merry-Go-Round with a Moving Child

A child of mass $$m = 30 \text{ kg}$$ stands on a merry-go-round modeled as a solid disk of mass $$M

Calculus Based Determination of Moment of Inertia for a Non-uniform Rod

A rod of length $$L = 2\,m$$ has a linearly varying density given by $$\lambda(x) = \lambda_0 \,(1 +

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

Combined Translational and Rotational Dynamics

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

Comparative Analysis of Rotational and Translational Dynamics

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

Comparative Study of Rotational Kinetic Energy in Different Shapes

Design an experiment to compare the rotational kinetic energy in different shaped objects (for examp

Conservation of Angular Momentum in Rotational Collisions

Two disks (Disk A and Disk B) rotate independently and are then brought into contact, eventually rot

Effect of Friction on Rotational Motion

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

Engine Torque Measurement Analysis

A mechanical engineer is analyzing the torque output of a car engine. The engine uses a lever arm at

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 9: Experimental Determination of Moment of Inertia

A student performs an experiment to determine the moment of inertia of a uniform disk by measuring i

Impact of Changing Mass Distribution on Angular Acceleration

An experiment varies the mass distribution of a rotating rod under a constant applied torque. The ta

Investigation of Torque in a Lever System

In this experiment a rigid lever, pivoted at one end, is used to measure the torque generated by a c

Investigation of Torque on a Rotating Pulley

In an experiment, a student applied a constant force of $$F = 40\,N$$ at varying distances (moment a

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

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 Experimental Verification

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

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

Relation Between Linear and Angular Velocity on a Rotating Disk

In an experiment, a rotating disk is used to measure the linear speed of points located at different

Rolling Cylinder Down an Incline

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

Rolling Motion Down an Inclined Plane

A solid cylinder of mass m and radius R rolls without slipping down an inclined plane of height h, s

Rolling Motion Energy Analysis on an Inclined Plane

A cylinder is allowed to roll down an inclined plane without slipping, converting gravitational pote

Rolling Motion Energy Conversion Experiment

A researcher investigates the energy conversion in rolling motion without slipping. A solid cylinder

Rolling Motion with Transition from Slipping to Pure Rolling

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

Rolling Motion: Energy Partition Analysis on an Inclined Plane

A solid cylinder is released from rest at the top of an inclined plane and allowed to roll without s

Rotational Equilibrium of a Beam with Distributed Load

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

Rotational Inertia 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 Kinetic Energy and Work by Friction

A flywheel with a moment of inertia of 2.0 kg m^2 rotates initially at 10 rad/s. It comes to rest du

Time-Varying Torque and Angular Acceleration

A researcher is exploring the effects of a time-varying torque on the rotational motion of a rigid b

Torque and its Direction: Vector Analysis

A force of magnitude $$F = 25 \text{ N}$$ is applied at an angle of $$30^\circ$$ above the horizonta

Torque from a Distributed Load

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

Torsional Oscillator Analysis

A torsional pendulum consists of a disk suspended by a wire with torsion constant $$k$$. The system

Acceleration and Position Relationship in SHM

For an oscillator described by the position function $$x(t) = A \cos(\omega t)$$, analyze the kinema

Analyzing Damped Oscillations in a Spring-Mass System

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

Analyzing the Half-Cycle Method in Oscillation Experiments

A media report asserts that 'timing just half a cycle of a pendulum is sufficient to determine its f

Anharmonic Effects in a Pendulum

A simple pendulum of length $$L = 0.8 \; m$$ is released from an initial angle of $$15^\circ$$. For

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 Approach to Maximum Velocity in SHM

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

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 Work Done in Stretching a Spring

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

Comparative Analysis of Horizontal vs Vertical Oscillations

Two identical mass-spring systems have a mass of $$m = 0.5\,\text{kg}$$ and a spring constant of $$k

Comparative Period Analysis: Mass-Spring Oscillator vs. Simple Pendulum

A researcher is comparing the oscillatory behavior of a horizontal mass-spring system and a simple p

Damped Oscillations: Determining the Damping Coefficient

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

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 the SHM Differential Equation

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

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 Maximum Elastic Potential Energy

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

Determining Initial Phase from Sinusoidal Oscillation Data

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

Determining Spring Constant Through Oscillation Energy Analysis

An experimental report claims that the spring constant k can be precisely determined by equating the

Determining the Phase Constant from Experimental Data

An experiment measuring the displacement of a simple harmonic oscillator produced the following data

Determining the Spring Constant from SHM Measurements

A block of mass $$m$$ is attached to a horizontal spring on a frictionless surface. When displaced f

Effect of Amplitude on Acceleration in SHM

Consider a simple harmonic oscillator described by \(y(t) = A\sin(\omega t)\). (a) Differentiate to

Effect of Amplitude on the Period of an Oscillator

An experiment is conducted to investigate if the period of a spring-mass oscillator depends on the a

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 of a Simple Pendulum

A simple pendulum with length \(L = 1.0\,m\) and mass \(m = 0.3\,kg\) is released from rest at an in

Energy Conservation in a Spring Oscillator

A block of mass $$m = 0.2\,kg$$ oscillates horizontally on a frictionless surface attached to a spri

Energy Conservation in Vertical Oscillators

A media claim states that 'in a vertical spring-mass system, mechanical energy is always conserved r

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 SHM

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

Error Analysis in SHM Measurements

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

Evaluating the Impact of Initial Conditions on SHM Motion

An educational resource asserts that 'the initial displacement and velocity of a mass-spring system

Experimental Determination of Spring Constant

In a lab experiment, students measure the displacement of a spring under various applied forces. The

FRQ 2: Energy Conversion in a Spring Oscillator

A block attached to a spring oscillates on a frictionless surface. The following table presents expe

FRQ 15: Graphical Analysis of Restoring Force

A graph showing the restoring force versus displacement for a spring is provided. Analyze the graph

FRQ 16: Frequency Determination from Oscillatory Data

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

FRQ 19: Vertical Oscillator Dynamics

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

FRQ6: Calculus Derivation of Velocity and Acceleration in SHM

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

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}

FRQ20: Energy Dissipation in Damped Pendulum Oscillations

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

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

Impact of Varying Spring Constants on Oscillatory Behavior

Two identical blocks of mass $$m = 0.2 \; kg$$ are attached to two different springs with spring con

Influence of Initial Phase on Oscillator Motion

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

Integration of Variable Force to Derive Potential Energy

A non-linear spring exerts a force given by $$F(x)= - k * x - \alpha * x^3$$, where $$k = 200 \; N/m

Investigating Damping Effects in a Spring-Mass Oscillator

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

Investigating Nonlinear Oscillations in a Large-Amplitude Pendulum

Students perform an experiment to analyze the period of a pendulum swinging at large amplitudes (up

Investigating the Effect of an External Driving Force

An experiment is conducted where a spring-mass system is subjected to an external periodic driving f

Lagrangian Mechanics of the Simple Harmonic Oscillator

A researcher employs Lagrangian mechanics to analyze a mass-spring oscillator. Consider a mass $$m$$

Maximum Speed and Energy Conservation in SHM

A mass-spring oscillator undergoes simple harmonic motion with displacement given by $$x(t)=A \sin(\

Non-linear Effects in Simple Pendulum Motion

Examine the non-linear behavior of a pendulum when the small-angle approximation is not valid.

Oscillatory Motion of a Block on a Horizontal Spring

A block of mass $$m = 0.8 \; kg$$ is attached to a horizontal spring with a spring constant of $$k =

Pendulum Motion: Small Angle Approximation and Beyond

A simple pendulum consists of a mass $$m = 0.3 \; kg$$ attached to a massless rod of length $$L = 1.

Pendulum Oscillations for Large Angles

For a simple pendulum with length \(L\) oscillating with a maximum angle \(\theta_{\text{max}}\) tha

Period and Frequency Determination from Time Measurements

A block oscillates on a spring. It takes 0.25 s for the block to move from its maximum displacement

Phase Shift and Time Determination in SHM

Analyze the effects of phase shift in a sinusoidal oscillator and determine specific times correspon

SHM: Spring Force and Energy Derivation

A spring with force constant $$k = 200 \;\text{N/m}$$ is fixed at one end. When the other end is dis

Small-Angle Pendulum Experiment

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

Spring Force and Energy Analysis

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

Spring-Block Oscillator: Phase Angle and Motion Description

A block attached to a horizontal spring oscillates without friction. The motion of the block is desc

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 of a Mass-Spring System

A vertical spring with a spring constant of $$k = 150\,\text{N/m}$$ supports a block of mass $$m = 2

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\

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

Angular Momentum Conservation in Orbital Motion

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

Application of Kepler's Third Law

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

Calculating Gravitational Potential in a Non-Uniform Planet

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

Calculus Derivation of Gravitational Potential Energy

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

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

Calculus in Orbital Motion: Area Sweep in an Elliptical Orbit

Kepler's Second Law implies that the rate of area sweep (dA/dt) is constant for an orbiting body. In

Cannonball Trajectory in a Non-Uniform Gravitational Field

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

Derivation of Equations of Motion in a Gravitational Field Using Lagrangian Mechanics

A researcher analyzes the motion of a particle of mass $$m$$ moving radially under the influence of

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

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 Gravitational Potential from Gravitational Force

The gravitational potential \(V(r)\) is related to the gravitational force by calculus. (a) Show th

Deriving the Gravitational Field from a Potential Function

Given the gravitational potential function $$V(r)= -\frac{G*m*M}{r}$$, you are to derive the gravita

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 a Falling Object in a Gravitational Field

A mass is dropped from a height in a gravitational field and its motion is tracked to study energy c

Effects of Eccentricity on Planetary Orbits

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

Energy Analysis in Multi-Body Systems

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

Energy Conservation in Central Force Motion

A particle of mass $$m$$ moves under the gravitational influence of a large mass $$M$$. Analyze its

Experimental Analysis of Gravitational Acceleration

An experiment was conducted to measure the acceleration due to gravity $$g$$ at various altitudes. (

Gravitational Analysis of a Composite Mass Distribution

A researcher studies the gravitational field of an irregular object composed of two connected sphere

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

Gravity Assist in Three-Body Dynamics

In a gravitational slingshot (gravity assist) maneuver, a spacecraft can change its velocity by inte

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

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 Laws and Orbital Dynamics

A researcher investigates several near-circular planetary orbits around a distant star. Observationa

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

Non-uniform Gravitational Fields in Planetary Interiors

Investigate how gravitational acceleration varies within a planet assuming it has a uniform density.

Nonlinear Gravitational Potential in a Drop-Test Apparatus

A drop-test apparatus experiment is designed to measure gravitational potential energy differences o

Orbit Stability from Potential Energy Diagrams

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

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

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

Perturbation in Orbital Motion

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

Predicting Orbital Decay Due to Atmospheric Drag

A low Earth orbit satellite experiences orbital decay due to atmospheric drag. Assume that the drag

Torsion Balance Gravitational Force Measurement

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

Verifying Kepler's Second Law and Angular Momentum Conservation

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

Work Done by Gravitational Force on a Falling Object

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