AP Physics C: Mechanics FRQ Room
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AP Physics C: Mechanics Free Response Questions
The best way to get better at FRQs is practice. Browse through dozens of practice AP Physics C: Mechanics FRQs to get ready for the big day.
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
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
Calculus-Based Position Function Analysis
An object moves along a straight path with its position given by $$x(t)=t^3 - 6*t^2 + 9*t$$. A table
Car Acceleration on a Highway: Calculus Approach
A car's position along a straight highway is given by $$x(t)= 2*t^3 - 6*t^2 + 4*t$$, where $$x$$ is
Circular Motion: Centripetal Acceleration from Tangential Speed Function
An object moves in a circular path of constant radius $$R = 3.0\,m$$. Its tangential speed varies wi
Conservation of Energy in a Pendulum
Design an experiment to test the conservation of mechanical energy in a simple pendulum. Provide bot
Determining Acceleration Due to Gravity from Free Fall
A student conducted an experiment by dropping a metal ball from a height of 45 m. A digital timer co
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:
FRQ 5: Projectile from an Elevated Platform (HARD)
A ball is launched from the edge of a cliff 50 m high with an initial speed of $$20\,m/s$$ at an ang
FRQ 7: Projectile Trajectory Analysis
A projectile is fired from ground level with an initial speed of 50 m/s at an angle of 40° above the
FRQ 12: Investigating Terminal Velocity in Free Fall with Air Resistance
A researcher examines free fall motion by considering the influence of air resistance. The net force
FRQ 14: Analysis of a Parabolic Projectile Trajectory (MEDIUM)
The vertical motion of a projectile is described by the equation $$h(t)=-4.9*t^2+20*t+5$$ (in m). (a
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
Impact Analysis: Collision Avoidance
Two vehicles are moving along a straight road. Vehicle A travels with constant velocity described by
Inferring Acceleration from Velocity Data Using Calculus
The following table shows the time and corresponding velocity for an object moving in one dimension,
Integrating an Acceleration Function to Determine Motion
An object's acceleration is described by $$a(t)=6*t-5$$, with initial velocity $$v(0)=2$$ m/s and in
Kinematic Analysis of a Cyclist
A cyclist starts from rest and accelerates uniformly at 1.5 m/s² for 10 seconds, then rides at a con
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 Analysis Using Integrated Acceleration Data
Researchers used an accelerometer attached to a moving cart to record its acceleration over a period
Multi-Dimensional Motion Analysis and Vector Decomposition
An object moves in the plane and its position vector is given by $$\mathbf{r}(t)= (3t^2)\,\mathbf{i}
Multi-Phase Rocket Motion Analysis
A rocket is launched vertically. Its engines provide a constant acceleration for 10 seconds. After e
Newton's Second Law and Force Measurement on a Cart
Design an experiment using a cart on a frictionless track with a pulley system to verify Newton's se
Oscillating Particle under Uniform Acceleration
An object exhibits combined motion described by $$x(t)= 5*t^2 + 3*\sin(2*t)$$ (meters). Analyze its
Polynomial Position Function Analysis
A particle’s position along the x-axis is given by the function $$x(t)=t^3 - 6*t^2 + 9*t$$ (in meter
Projectile Motion with Calculus Integration
An object is launched from ground level with an initial speed of 50 m/s at an angle of 40° above the
Projectile Motion: Maximum Height and Range
An object is launched from ground level at an angle of 30° above the horizontal with an initial spee
Projectile Motion: Maximum Height and Range
A projectile is launched from ground level with an initial speed of $$v_0 = 60$$ m/s at an angle of
Projectile Range Analysis with Angular Misinterpretation
An experiment was conducted to analyze the range of a projectile launched at various angles. A fixed
Rotational Dynamics: Variable Torque
A solid disk with moment of inertia I is subject to a time-dependent torque given by $$\tau(t)= 5*t$
Terminal Velocity in Free Fall
Design an experiment to determine the terminal velocity of an object in free fall within a fluid med
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
Uniformly Accelerated Motion: Incorrect Baseline Velocity
A physics lab experiment tracked the displacement of a moving object using a laser sensor, measuring
Variable Acceleration Analysis Using Calculus
Design an experiment to investigate the motion of an object under a time-varying force that produces
Vector Addition in Two-Dimensional Projectile Motion
Design an experiment to test the principles of vector addition by analyzing the two-dimensional moti
Vector Decomposition in Projectile Motion
A projectile is launched from the ground with an initial speed of 40 m/s at an angle of 50° above th
Bouncing Ball Energy Loss Experiment
A ball is dropped from a known height and allowed to bounce repeatedly. The experiment calculates en
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
Determining Instantaneous Power from a Velocity-Time Graph
A car of mass 1200 kg has its velocity measured as a function of time. The graph provided represents
Elastic Potential Energy and Hooke’s Law
A spring with a spring constant of $$ k = 200 \;N/m $$ is compressed by 0.1 m from its equilibrium p
Energy Analysis in a Damped Spring-Mass Oscillator
A 1 kg mass attached to a spring with spring constant 100 N/m is initially compressed 0.1 m from its
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
FRQ 6: Work Done on a Crate on an Inclined Plane
A 20-kg crate is moved up a 30° inclined plane. Experimental measurements of the force component alo
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 12: Quantifying the Work Done by Friction
An experimental report claims that the negative work done by friction is constant regardless of the
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
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
Instantaneous Power in a Variable Force Scenario
An object is subjected to a time-dependent force given by $$F(t)=5*t$$ N, and its displacement is de
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
Investigating Work on an Inclined Plane
A researcher is investigating the work done on a 5 kg block being pushed up a frictionless inclined
Loop-the-Loop Roller Coaster: Work-Energy and Normal Force Analysis
A roller coaster car of mass 300 kg enters a vertical loop with a radius of 10 m.
Measuring Work Done against Air Resistance in Free Fall
A researcher studies the motion of a 1 kg object falling under gravity with air resistance. The drag
Numerical Integration of Work in a Variable Force Field
A researcher studies the work done on a particle moving along the x-axis under the influence of a va
Potential Energy Curves and Equilibrium Analysis
An object of mass 4 kg has a potential energy function given by $$U(x) = (x - 2)^2 - (2\,x - 3)^3$$.
Power and Efficiency in a Wind Turbine
A wind turbine with a rotor radius of 40 m extracts energy from wind. The wind speed varies with hei
Runner's Power Output Analysis
In a track experiment, a runner’s power output is calculated using the formula $$P = m*a*v$$ obtaine
Time-dependent Power and Differential Equations
A machine's power output, $$P(t)$$ in watts, is governed by the differential equation $$\frac{dP}{dt
Variable Gravitational Acceleration Over a Mountain
A hiker lifts a 10 kg backpack up a mountain where the gravitational acceleration decreases linearly
Work Done Against Friction
An 8 kg block slides on a horizontal surface with a kinetic friction coefficient of 0.25. It comes t
Work Done in a Resistive Medium
A particle of mass 1 kg moves in a straight line under the influence of two forces: a constant propu
Work Done on a Variable Inclined Plane
An object of mass $$m = 2 \;\text{kg}$$ is moved along an inclined plane whose angle of inclination
Work-Energy Analysis in Collisions
Two blocks with masses 2 kg and 3 kg undergo collisions under different circumstances. In an elastic
Work–Energy and Friction: Analyzing a Sliding Block
A researcher investigates the effect of friction on a sliding block. A 4 kg block is launched horizo
Work–Energy Experiment with a Spring Launch
A researcher studies a spring-launched projectile. A spring with a spring constant $$k = 500\,N/m$$
Astronaut Momentum Conservation
An astronaut with a total mass of 89 kg is floating in space near her shuttle. To reorient herself,
Ballistic Pendulum Analysis
A bullet with mass $$0.02$$ kg is fired horizontally into a pendulum bob of mass $$0.98$$ kg suspend
Center of Mass Determination for a Composite L-Shaped Object
Students perform an experiment to determine the center of mass of an L-shaped, composite board. The
Center of Mass of a Non-uniform Rod
A rod of length $$L = 2\,m$$ has a linear density given by $$\lambda(x) = 5 + 3*x$$ (in kg/m), where
Center of Mass of a Non-uniform Rod
Consider a rod of length L = 1 m with a linear density given by $$\lambda(x)=10+6*x$$, where x is me
Center of Mass of a Non-Uniform Rod
A thin rod of length $$0.8$$ m has a linear mass density given by $$\lambda(x) = 5 + 3*x^2$$ (kg/m),
Center of Mass of a Variable Density Rod
A rod of length $$L = 1\,\text{m}$$ has a linear density given by $$\lambda(x) = 5 + 3*x$$ (in kg/m)
Center-of-Mass of a Ladder with Varying Linear Density
A ladder of length $$L=2.0\,m$$ has a vertical linear mass density given by $$\lambda(y)=3.0(1+y)$$
Collision with a Variable Coefficient of Restitution
Two carts on a frictionless track collide head-on. Cart A has a mass of 3 kg and Cart B a mass of 4
Combined Translational and Rotational Motion Analysis
A uniform rod of length $$2\,m$$ and mass $$4\,kg$$ is pivoted at one end. Initially at rest, an imp
Composite Object: Rod with Attached Sphere
A uniform rod of length 1 m and mass 2 kg has a small sphere of mass 1 kg attached to its right end.
Conservation of Linear Momentum in a Glider Collision
On a frictionless air track, two gliders collide. The experimental data below list the masses and ve
Elastic and Inelastic Collision Analysis
Two balls collide head-on. The red ball (mass $$0.5\,kg$$, velocity $$+4\,m/s$$) and the green ball
Elastic Collision: Conservation of Momentum and Kinetic Energy
Two spheres A and B collide elastically. Sphere A has a mass of 0.6 kg and an initial velocity of $$
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
Explosive Fragmentation: Momentum Transfer
A stationary object with a mass of 12 kg explodes into three fragments in a frictionless environment
Football Kick: Impulse and Average Force
A 0.4 kg football is punted and achieves a launch speed of 30 m/s as a result of a kick delivered ov
Force from Potential Energy Graph
A potential energy function for a system is provided in the graph below, where the potential energy
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 $$\lambda(x) = 2 + 3*x$
FRQ 14: Derivation of the Continuous Center of Mass Formula
Consider a one-dimensional object with a continuous mass distribution described by the density funct
FRQ 15: Center of Mass versus Center of Gravity
A popular science article claims that for a complex-shaped satellite orbiting Earth, the center of m
Glider Collision on a Frictionless Air Track
Two gliders on an air track are involved in a head-on elastic collision. The data for the gliders is
Impulse Analysis in a Variable Mass Rocket
Consider a simplified rocket system where its mass decreases with time as it expels fuel. The mass i
Impulse and Kinetic Energy from a Time-Dependent Force on a Car
A 1200 kg car, initially at rest, is subjected to a time-dependent force given by $$F(t)=4000 - 500*
Impulse and Momentum 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 Calculation from a Force-Time Graph
A force acting on an object is depicted by a force-versus-time graph over the interval from t = 0 s
Impulse Delivered by a Variable Force on a Soccer Ball
A soccer ball of mass 0.45 kg is struck by a kick that applies a variable force over a brief time in
Impulse Measurement via Force-Time Graph Analysis
A student attaches a force sensor to a baseball bat to record the force exerted on a ball during imp
Impulsive Collision Involving Rotation
A 0.5 kg ball traveling at $$8\,m/s$$ horizontally strikes the end of a thin uniform rod of length $
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 Analysis of a Variable Mass Rocket
A rocket in space, free from external gravitational forces, expels fuel continuously. Its mass is gi
Motion of the Center of Mass Under External Force
Consider a system consisting of two point masses connected by a light rod. Mass m1 = 2 kg is located
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
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-conservative Forces: Block on an Incline with Friction
A 2 kg block slides down a 30° incline that is 5 m long. The coefficient of kinetic friction between
Nonuniform Circular Disk Center of Mass
A circular disk of radius $$R$$ has a nonuniform surface mass density given by $$\sigma(r,\theta)=\s
Oblique Collision of Ice-Hockey Pucks
Two ice-hockey pucks collide on frictionless ice. Puck A (mass = 0.2 kg) moves east at 5 m/s, and Pu
Off-Center Collision and Angular Momentum
A small ball (mass $$0.5\,kg$$) moving at $$4\,m/s$$ strikes a uniform rod (mass $$3\,kg$$, length $
Projectile Motion with Air Resistance Approximation
A 0.2 kg projectile is launched with an initial speed of 15 m/s at an angle of 40° above the horizon
Projectile Motion with Drag Impulse Analysis
A projectile is launched horizontally and experiences a drag force proportional to its velocity, giv
Ramp Push Experiment: Variable Force Integration
In a laboratory experiment, a cart on a frictionless ramp is pushed by a force that varies with time
Rigid Body Dynamics: Torque and Rotation
A uniform meter stick (mass = 0.3 kg, length = 1 m) is pivoted at one end. A perpendicular force is
Rocket Propulsion with Variable Mass
A rocket has an initial mass of $$M_0 = 50$$ kg (including fuel) and ejects fuel such that its mass
Rotational Dynamics Using Center of Mass
A uniform rod of length $$L=2.0\;m$$ and mass M rotates about a frictionless pivot at one end. (a)
Work Done by a Variable Force and Momentum Change
A 2-kg block on a frictionless surface is subjected to an external force described by $$F(t) = 15e^{
Analysis of Rolling Motion on an Incline
Consider a solid cylinder of mass $$M$$ and radius $$R$$ that rolls without slipping down an incline
Angular Kinematics from Experimental Data
A laboratory experiment measures the angular velocity $$\omega$$ of a rotating object as a function
Angular Kinematics of a Rotating Disk
Consider a rotating disk for which you want to measure angular displacement, angular velocity, and a
Angular Momentum Conservation in a Variable-Radius System
A student investigates angular momentum conservation on a rotating stool by attaching a weight to a
Angular Momentum Conservation on a Merry-Go-Round
A child of mass $$m = 30\,kg$$ stands at the edge of a merry-go-round that is modeled as a uniform d
Angular Momentum in Explosive Separation
A spinning disk with an initial angular velocity $$\omega_i$$ undergoes an explosion that breaks it
Application of the Parallel Axis Theorem
An object with mass 5 kg has a moment of inertia about its center of mass $$I_{cm} = 0.2\,kg\,m^2$$.
Calculus Derivation of the Moment of Inertia for a Disk
Using the integral formula $$I = \int r^2\,dm$$, derive the moment of inertia for a uniform circular
Calculus-Based Analysis of Angular Impulse
In rotational dynamics, angular impulse is defined as the integral of torque over a time interval, w
Comparative Dynamics of Rotational vs. Translational Motion in Rolling Objects
In a complex investigation, an object is rolled down an incline and both its angular and linear acce
Composite Body Rotation
A composite object is formed by welding a solid disk to a thin rod. The disk has mass $$M$$ and radi
Composite Rod and Point Masses Inertia Analysis
A uniform rod of length L and mass M is pivoted about its left end. Two small beads, each of mass m,
Coupled Rotational Dynamics of Two Disks
Two disks with different masses and radii are mounted on a common, frictionless axle. Disk A has mas
Dynamics of Coupled Rotational Systems
Two disks are coupled by a belt: Disk A has a moment of inertia $$I_A = 0.8 \text{ kg m}^2$$ and ini
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 Conversion in Rolling Motion Experiments
In an experiment, a sphere rolls without slipping down an inclined plane of height $$h$$. Measuremen
Energy Dissipation Due to Friction in a Spinning Disk
A disk is spun up to a high angular velocity and then allowed to slow down due to friction. The expe
Equilibrium Analysis in Rotational Systems
A seesaw, modeled as a uniform beam 4 m long pivoted exactly at its center, is in rotational equilib
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
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 12: Combined Translational and Rotational Motion with Slipping
A disk of mass M = 2.00 kg and radius R = 0.30 m is released on a 30° inclined plane with a kinetic
FRQ 14: Energy Loss in a Rotating Flywheel Due to Friction
A flywheel with moment of inertia \(I = 8.00\,kg\cdot m^2\) is initially spinning at \(\omega_0 = 15
Investigating Non-uniform Density Effects on Moment of Inertia
Design an experiment to determine the moment of inertia for an object with a non-uniform mass distri
Kinetic Energy Redistribution in Rotating Systems
A rotating disk initially has two weights attached at its rim, resulting in a moment of inertia $$I_
Lever and Torque Computations
This problem involves calculating torque in a lever system. A diagram is provided below.
Mass Redistribution and Kinetic Energy in Rotating Systems
In a rotating system, a person on a rotating platform moves closer to the axis, reducing the system’
Moment of Inertia of a Hollow Cylinder with Thickness
Derive an expression for the moment of inertia of a hollow cylinder with inner radius $$R_1$$, outer
Parallel Axis Theorem Application in Complex Systems
A composite object consists of a uniform rod of mass $$M = 4\,kg$$ and length $$L = 3\,m$$, and an a
Parallel Axis Theorem: Composite Body Moment of Inertia
Consider a composite object consisting of a solid disk (mass $$M = 4.0 \text{ kg}$$, radius $$R = 0.
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 Motion and Energy Conservation: Rolling Cylinder on an Incline
A solid cylinder of mass $$M = 5.0 \text{ kg}$$ and radius $$R = 0.4 \text{ m}$$ rolls without slipp
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 of a Sphere on an Incline
A solid sphere of mass M and radius R rolls without slipping down an inclined plane of height h star
Rolling Motion on an Inclined Plane
A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an inclined plane from a
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 Impulse and Change in Angular Momentum
A flywheel initially at rest receives a constant torque impulse over a brief time interval.
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
Seesaw Rotational Equilibrium
Two children are sitting on opposite ends of a seesaw (a uniform beam pivoting about its center). Ch
Simulation Analysis of Rotational Motion with Non-uniform Mass Distribution
A simulation of a rotating flexible system shows that the moment of inertia, $$I$$ (in kg m^2), chan
Torque in a Multi-force System: Seesaw Equilibrium
A seesaw of length $$L = 3.0 \text{ m}$$ is balanced on a fulcrum located 1.0 m from the left end. T
Torsion Pendulum Method for Irregular Objects' Inertia
This experiment uses a torsion pendulum to determine the moment of inertia of irregular objects. The
Variable Torque Function Integration
Consider a rotating body with constant moment of inertia I = 5 kg·m². The applied torque is time dep
Amplitude and Maximum Speed Relationship in SHM
A mass-spring oscillator undergoes simple harmonic motion with amplitude $$A$$ and angular frequency
Analysis of Phase Shift in Oscillator Data
An oscillator is described by the equation $$y = 0.03 \sin(8t + \phi_0)$$. It is experimentally meas
Calculus Derivation of Velocity and Acceleration in SHM
Educational materials claim that 'the instantaneous velocity and acceleration of a simple harmonic o
Calculus-Based Derivation of Work Done in Stretching a Spring
Investigate the work done in stretching a spring from its natural length using calculus.
Calculus-Based Prediction of Maximum Speed
A photogate system is set up to record the speed of a mass-spring oscillator. Answer the following p
Comparative Dynamics of Mass-Spring and Pendulum Oscillators
Analyze the motion of two oscillatory systems: a mass-spring oscillator and a simple pendulum (using
Comparative Energy Analysis: SHM vs. Pendulum
Compare the energy transformations in a spring-mass oscillator and a simple pendulum (operating unde
Comparing Sinusoidal and Non-Sinusoidal Oscillatory Motion
In some experiments, the oscillatory motion observed may deviate from a pure sinusoid. Consider a sy
Complex SHM: Superposition of Two Harmonic Motions
A block experiences two independent harmonic motions such that its displacement is given by $$x(t)=
Composite Oscillator: Two Springs in Series
A block with mass $$m = 1.0\,kg$$ is attached to two springs connected in series, with spring consta
Damped Oscillation: Logarithmic Decrement Analysis
A spring-mass oscillator exhibits damped oscillations, and the amplitudes of successive cycles are r
Damped Oscillations in a Spring-Mass System
In an experiment exploring damped oscillations, a block attached to a spring is set oscillating on a
Damped Oscillations: Amplitude Decay Analysis
A mass-spring oscillator with $$m = 0.2\,kg$$ and damping constant $$b = 0.1\,kg/s$$ experiences dam
Derivation of SHM Equations Using Calculus
Starting with Newton’s second law and Hooke’s law for a mass-spring system, derive the differential
Derivation of the SHM Differential Equation
Starting from basic principles, derive the differential equation that governs the motion of a mass a
Deriving Equations for a Damped Harmonic Oscillator
An experiment is designed to study the effects of damping in a spring-mass oscillator. This version
Determination of Gravitational Acceleration Using a Vertical Oscillator
A vertical spring-mass oscillator reaches equilibrium when the spring stretches by a distance $$d$$
Determination of Spring Constant Using SHM Data
An experiment on a mass-spring oscillator provides the following data for different masses and their
Determining Maximum Speed from Energy Considerations
An oscillator of mass $$m = 0.1 \; kg$$ is attached to a spring with a spring constant of $$k = 250
Determining Spring Constant from Force-Displacement Data
In a laboratory experiment, the force exerted by a spring is measured for various displacements. The
Differentiation in SHM: Velocity and Acceleration
An oscillator is described by the function $$y = A * \sin(\omega * t + \phi_0)$$. Investigate the ve
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 Mass–Spring Oscillator
Examine energy conservation in a mass–spring system and its experimental verification.
Energy Exchange in SHM
Consider a mass-spring oscillator with displacement given by: $$x(t) = A * \cos(\omega t)$$, with
Energy Transformation in SHM
A block of mass $$m = 0.2 \; kg$$ oscillates on a horizontal spring with a force constant of $$k = 1
Energy Transformations in a Mass-Spring System
A researcher investigates energy transformations in a mass-spring oscillator. The system consists of
Energy Transformations in SHM
Consider a block attached to a spring undergoing simple harmonic motion with displacement $$x(t)=A\s
Equation of Motion for a Simple Pendulum Beyond Small Angle Approximation
A researcher examines the motion of a simple pendulum without relying on the small-angle approximati
Evaluating Hooke's Law in Spring Oscillators
A recent media report claims that 'any spring, when compressed by 5 cm, always exerts the same resto
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 system is driven by an external force of the form \(F(t) = F_0 \cos(\omega_d t)\) and
Friction Effects in Horizontal Oscillatory Systems
A block attached to a horizontal spring oscillates with amplitude $$A$$, but friction is present. Th
FRQ 6: Pendulum Motion and the Small Angle Approximation
A simple pendulum is tested for various small angular displacements. The measured periods for oscill
FRQ 15: Determination of the Phase Constant
An oscillator with amplitude $$A = 0.05\ m$$ and angular frequency $$\omega = 8\ rad/s$$ is observed
FRQ 16: Maximum Speed in SHM via Energy Methods
Experimental data for a mass–spring oscillator, including amplitude, mass, spring constant, and meas
FRQ 19: Vertical Oscillator Dynamics
A mass is attached to a vertical spring. When displaced by a distance $$y$$ from its equilibrium pos
FRQ2: Maximum Speed of a Spring Oscillator via Energy Conservation
Consider a mass attached to a spring oscillating on a frictionless surface. The spring has a force c
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 $
FRQ18: Effect of Mass Variation on the Oscillation Period
The period \(T\) of a mass-spring oscillator is given by the formula: $$T = 2\pi\sqrt{\frac{m}{k}}.
FRQ20: Energy Dissipation in Damped Pendulum Oscillations
A damped pendulum oscillates with small angles such that its motion is approximately described by $
Graphical Analysis of SHM: Determining Phase and Frequency
A researcher records the oscillatory motion of a block on a spring and generates a position-vs.-time
Influence of Initial Phase on Oscillator Motion
Consider an oscillator described by $$y = A\sin(\omega t + \phi_0)$$. Explore how variations in the
Investigation of Energy Conservation in SHM Using Calculus
A researcher analyses a mass-spring oscillator with its displacement given by $$y(t) = 0.1 * \cos(15
Kinematics and Phase Angle Determination
An oscillator is described by the equation $$y = A\sin(\omega t + \phi_0)$$ with an initial conditio
Kinematics of SHM: Period and Frequency Measurements
Analyze the kinematics of a simple harmonic oscillator using time measurements.
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
Nonlinear Pendulum Oscillations and Error Analysis
A pendulum with a length of $$L = 2.0\,m$$ is released from an initial angle of 45° (approximately 0
Period and Frequency Determination
A mass on a spring is observed to take $$0.20\,s$$ to move from its maximum displacement on one side
Period and Frequency Determination from Half Cycle Data
A mass-spring oscillator completes half of a full cycle (i.e. moving from maximum displacement on on
SHM with Phase Shift: Initial Conditions Analysis
An oscillator is described by the equation: $$y(t) = A * \sin(2\pi * f * t + \phi)$$ In a particul
Sinusoidal Analysis of SHM with Phase Shift
Examine a sinusoidally described simple harmonic oscillator with a phase shift.
Sinusoidal Description and Phase Constant in SHM
A spring-mass oscillator has an amplitude $$A = 0.04 \; m$$ and a frequency $$f = 5.0 \; Hz$$. The d
Sinusoidal SHM with Phase Shift
An oscillator undergoes simple harmonic motion described by the equation $$y(t) = A\sin(\omega t + \
Spring Force and Elastic Potential Energy
A spring with a force constant $$k = 300\,N/m$$ and a natural length of 0.50 m is stretched to a len
Spring Force and Energy Analysis
A researcher is studying the behavior of a horizontal spring. The spring has a natural length of 12
Vertical Oscillations on a Spring
A block of mass $$m = 1.5\,kg$$ is attached to a vertical spring with a force constant of $$k = 300\
Vertical Oscillator with Offset Equilibrium
A vertical mass-spring system has a mass of $$m = 1.0\,kg$$ attached to a spring with force constant
Vertical Spring-Mass Oscillator Analysis
In this experiment, a block of mass is attached to a vertical spring. After the block reaches equili
Analysis of a Gravitational Potential Energy Graph
A graph representing gravitational potential energy as a function of distance is provided below. The
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 Orbits
Analyze the relationship between orbital radius and angular momentum as depicted in the provided gra
Barycenter of the Sun-Earth System
A researcher investigates the center of mass (barycenter) of the Sun-Earth system. Given that the ma
Cometary Orbits: Analyzing Highly Eccentric Trajectories
Comets often exhibit highly eccentric orbits. Their dynamics can provide insight into gravitational
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
Designing a Modern Cavendish Experiment
A researcher designs an experiment modeled after the Cavendish torsion balance to determine the grav
Determination of Gravitational Constant Using a Compound Pendulum
A compound pendulum experiment is conducted to determine the gravitational constant (G) by measuring
Determining the Gravitational Constant using a Torsion Balance
An experimental apparatus utilizing a torsion balance is used to measure the gravitational forces be
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
Eccentricity and Elliptical Orbits
Discuss the role of eccentricity in elliptical orbits and derive the orbit equation in polar coordin
Effects of Eccentricity on Planetary Orbits
A series of simulations have been conducted for planetary orbits with varying eccentricity. The foll
Elliptical Orbits and Angular Motion
A planet follows an elliptical orbit around its star. Investigate how variations in orbital distance
Elliptical Orbits and Eccentricity Calculation
An object is in an elliptical orbit around a star. (a) Define the eccentricity $$e$$ in relation to
Experimental Design for Measuring Gravitational Constants
Design an experiment using a torsion balance to measure the gravitational constant $$G$$.
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 13: Orbital Transfer Maneuver – Hohmann Transfer
A spacecraft in a circular orbit of radius $$r_1$$ wishes to transfer to a higher circular orbit of
FRQ 15: Gravitational Anomalies and Their Effects on Orbits
A satellite experiences a small perturbation in the gravitational potential due to a local mass anom
Gravitational Assist Maneuver Simulation
Gravitational assist maneuvers, which use the gravity of a planet to alter a spacecraft’s trajectory
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 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 Tidal Forces
Consider a spacecraft of linear size L located at a distance R from the center of a massive body of
Gravity Assist in Three-Body Dynamics
In a gravitational slingshot (gravity assist) maneuver, a spacecraft can change its velocity by inte
Modeling Tidal Forces with Calculus
Tidal forces arise due to the gradient in gravitational force across an object. These forces can cau
Newton vs. Einstein: Conceptual Analysis of Gravity
Compare and contrast Newton's Law of Gravitation with Einstein's theory of General Relativity. Answe
Newtonian Approximation of Gravitational Lensing
Although gravitational lensing is accurately described by General Relativity, a simplified Newtonian
Orbit Stability from Potential Energy Diagrams
Analyze the provided potential energy diagram and determine the regions corresponding to stable and
Orbital Decay due to Atmospheric Drag
A satellite in low Earth orbit experiences atmospheric drag, which can be modeled as a force $$F_d =
Orbital Decay Due to Atmospheric Drag
A satellite in low Earth orbit experiences atmospheric drag, resulting in a gradual decrease of its
Orbital Transfer and the Hohmann Maneuver
A spacecraft performs a Hohmann transfer to move from a circular orbit of radius $$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
Perturbation Analysis in Elliptical Orbits
An elliptical orbit is subject to a small, constant perturbing force that causes a slow change in th
Planetary Orbits and Energy Considerations
Consider a planet in a highly eccentric orbit. The total orbital energy (kinetic plus potential) is
Satellite Orbit Simulation: Finite Burn and Hohmann Transfer Error
A research team develops a computer simulation to model a satellite's orbital transfer using a Hohma
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