AP Physics C: Mechanics FRQ Room

Analysis of Air Resistance on a Falling Object

An object falling under gravity experiences a net acceleration modeled by $$a(t)= 9.8 - 0.5*t$$ (m/s

Analyzing Motion with a Nonlinear Acceleration Function

A particle moves along the x-axis with an acceleration given by $$a(t)=4\cos(t)$$ (m/s²). It has an

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

Comparative Analysis of Kinematic Equations

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

Conservation of Momentum in Collisions

Design an experiment using an air track to test the conservation of momentum in elastic collisions.

Dynamic Motion Analysis: Cubic Position Function

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

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

Free-Fall Experiment Analysis

A rock is dropped from a 50-meter high cliff. Neglecting air resistance and using $$g= 9.8\; m/s^2$$

FRQ 1: One‐Dimensional Constant Acceleration

An object moves along a straight line with constant acceleration. Its initial velocity is 5 m/s and

FRQ 3: Projectile Motion with Calculus Analysis (MEDIUM)

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

FRQ 15: Differentiation of a Cubic Displacement Function (EASY)

An object's displacement is given by $$s(t)=3*t^3-5*t^2+2*t$$ (in m). (a) Find the velocity function

FRQ 16: Integration of a Decaying Velocity Function (HARD)

An object has a velocity function given by $$v(t)=4*e^{-t}-2$$ (in m/s) for $$t\ge0$$, and its initi

FRQ 18: Determining Launch Parameters from Projectile Data (EXTREME)

A projectile’s path was experimentally measured, and the following graph (provided) shows its parabo

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

FRQ 19: Comparative Kinematics – Two Launch Angles

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

Graphical Analysis of Kinematic Data

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

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

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

Kinematics in a SmartLab Setup: Integration Error

In a SmartLab experiment, sensors measured both the acceleration and displacement of an object movin

Kinematics of a Decelerating Vehicle

A car traveling at 30 m/s starts braking and comes to a stop after covering a distance of 120 m unde

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

Multi-Phase Vehicle Motion

A vehicle undergoes three consecutive phases of motion: - Phase 1: It accelerates uniformly from res

Non-Uniform Acceleration Analysis

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

Optimization of Projectile Range

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

Oscillating Particle under Uniform Acceleration

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

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 Range Analysis with Angular Misinterpretation

An experiment was conducted to analyze the range of a projectile launched at various angles. A fixed

Round Trip Motion Analysis

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

Simple Harmonic Motion in a Spring-Mass System

Design an experiment to investigate simple harmonic motion (SHM) using a spring-mass system. Describ

Sinusoidal Motion

A particle’s position along the x-axis is given by $$x(t)=8\sin(t)$$, where $$0\leq t\leq2\pi$$ (x i

Terminal Velocity Experiment

An experiment involves dropping objects of varying shapes from a tall building to study terminal vel

Time-Dependent Acceleration and Displacement

A particle’s acceleration is given by the function $$a(t)=6-2*t$$ (in $$m/s^2$$) for $$0 \le t \le 4

Variable Acceleration Analysis

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

Calculating Kinetic Energy from a Velocity Function

A particle of mass $$m = 1 \;\text{kg}$$ moves along the x-axis with a velocity given by $$v(t)= 3*t

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

Comparative Analysis of Constant vs. Variable Gravitational Work

An object of mass $$m= 10 \;\text{kg}$$ is lifted from the ground (assume $$x=0$$) to a height of $$

Conservation of Energy in Free Fall

Consider a ball of mass 3 kg that is dropped from a height of 10 m above the ground. Air resistance

Conservation of Mechanical Energy in a Pendulum

A simple pendulum consists of a bob attached to a massless string of length 2 m. The bob is pulled a

Conservation of Mechanical Energy in a Pendulum

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

Elastic Collision and Energy Transfer

Two blocks, A (2 kg) and B (3 kg), slide without friction on a horizontal surface. Initially, block

Energy Conservation in a Pendulum

A simple pendulum has a length $$L = 2 \text{ m}$$ and is released from rest at an initial angle of

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 Dissipation in a Bouncing Ball

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

Energy Loss in a Ball with Air Resistance

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

Evaluation of Elastic Potential Energy in a Spring-Mass System

A researcher studies energy storage in a spring–mass system. A spring with a spring constant $$k = 2

Explosive Separation and Energy Distribution

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

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 4: Conservation of Mechanical Energy in a Roller Coaster

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

FRQ 4: Potential Energy Curve Analysis for a Diatomic Molecule

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

FRQ 5: Analysis of Work and Potential Energy in a Spring–Mass System

A mass–spring system is tested in a laboratory. A spring attached to a mass is stretched from its eq

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 9: Calculus-Based Work Determination in a Braking Scenario

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

Inclined Plane Energy Transfer Experiment

In an experiment, a 2 kg block is released from rest and slides down a frictionless inclined plane o

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

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

Multi‐Phase Cart Energy Experiment

A small cart is sent along a track that includes a flat section, an inclined ramp, and a loop-the-lo

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 Fluctuations in a Jogger

A 70 kg jogger runs along a track with an instantaneous velocity given by $$v(t) = 2 + 0.5\,t$$ (in

Projectile Launch: Energy and Air Resistance Considerations

A 0.2 kg projectile is launched vertically upward in a vacuum with an initial speed of 30 m/s.

Projectile Motion and Energy Conservation

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

Pulley System Work–Energy Verification

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

Rocket Engine Power Output Analysis

A rocket of mass 1000 kg is traveling horizontally at a constant speed of 8.0 m/s under an engine th

Rotational Work and Energy Experiment

A uniform disk with a moment of inertia $$I = 0.5\,kg\cdot m^2$$ is subjected to a variable torque d

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 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

Spring Elastic Potential Energy

A spring with a force constant of $$k = 800\,N/m$$ is compressed by 0.1 m.

Spring Energy Experiment: Measuring Nonlinear Work

A spring exhibits a force-displacement relationship given by $$F(x) = k*x + a*x^3$$, where $$k = 50\

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 and Power in an Engine

A 1500 kg car is accelerated from rest by an engine whose power output varies with time according to

Work Done by a Variable Exponential Force

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

Work Done by Non‐Conservative Forces with Variable Friction

A 10-kg block slides along a horizontal surface. The coefficient of kinetic friction varies with pos

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 an Object by a Central Force

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

Work with a Variable Force on a Straight Path

A particle experiences a variable force along the x-axis given by $$F(x)= 10 + 3*x \; (\text{N})$$.

Work-Energy Analysis on an Inclined Plane

A 20 kg block slides down an inclined plane of length 8 m, which is inclined at an angle of 30° rela

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 Inelastic Collisions

A 3-kg cart moving at 4 m/s collides inelastically with a stationary 2-kg cart on a frictionless tra

Work–Energy Theorem Verification in Projectile Motion

A projectile of mass 0.2 kg is launched horizontally from a platform 4 m high. High-speed cameras me

Analyzing a Multi-Peak Force-Time Graph

A cart of mass 2 kg is subjected to a force that varies with time according to a piecewise function:

Calculus-Based Analysis of a Variable Density Disk

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

Center of Mass 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 of a Composite Three-Dimensional Object

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

Center of Mass of a Lamina with Nonuniform Density

A thin, triangular lamina has vertices at (0,0), (4,0), and (0,3). Its surface mass density is given

Center of Mass of a Non-uniform Circular Disk

A thin circular disk of radius $$R$$ has a surface mass density given by $$\sigma(\theta)= k\,(1+\co

Center of Mass of a Non-uniform Rod

A thin rod of length 1.0 m has a linear mass density given by $$\lambda(x) = 5 + 3*x$$ (kg/m), where

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 an Irregular Lamina in Polar Coordinates

Consider a lamina occupying the region defined in polar coordinates by $$0 \le r \le 2(1+\cos(θ))$$

Circular Motion: Banked Curve Analysis

A car of mass 1200 kg negotiates a banked curve of radius 50 m with no friction required. The curve

Conservation of Linear Momentum in Colliding Carts

Two carts on a frictionless track collide. Cart A (mass = 2 kg) moves to the right with a speed of 3

Determination of an Unknown Mass via Collision Data

A moving ball of known mass collides with a stationary ball of unknown mass. The experimental data a

Elastic and Inelastic Collision Analysis

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

Elastic Collision on Air Track

Two gliders are on a near-frictionless air track. Glider A (mass = 0.8 kg) is traveling to the right

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 11: Experimental Evaluation: Measurement of Center of Mass

A media report claims that a new laser-based method can determine the center of mass of irregular ob

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

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

Impulse Analysis with Error Bars

In an experiment, a variable force acting on a cart is measured and described by the equation $$F(t)

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 Kinetic Energy Loss in a Perfectly Inelastic Collision with a Spring

A 0.7 kg ball moving at $$6\,m/s$$ collides inelastically with a 2.3 kg block at rest. After collisi

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 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 Calculation from Force-Time Graph

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

Impulse Delivered by a Time-Dependent Damping Force

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

Impulse Delivered by a Variable Force

A particle experiences a time-dependent force along the x-axis given by $$F(t)=4*t^2 - 12*t + 9$$ N

Impulse during a Controlled Fall onto an Airbag

A stuntman with a mass of 80 kg falls and lands on an airbag, which decelerates him uniformly from 8

Impulse in a Non-Constant Force Field

A particle moves through a region where the force depends on position, described by $$F(x) = 3 * x$$

Inelastic Collision: Two Blocks on a Frictionless Surface

Block A (mass = 5 kg, initial velocity = 2 m/s) and Block B (mass = 3 kg, initially at rest) collide

Multi-Dimensional Inelastic Collision Analysis

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

Multi-Stage Rocket Propulsion using Momentum Conservation

A rocket with an initial total mass of 1000 kg expels propellant at a constant exhaust velocity of $

Recoil Dynamics in a Firearm Event

A 5.0 kg rifle fires a 0.025 kg bullet horizontally with a speed of 400 m/s. Experimental measuremen

Rocket Propulsion and Center of Mass Dynamics

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

Rolling Cylinder on an Incline

A uniform solid cylinder (mass = 4 kg, radius = 0.5 m) rolls without slipping down a 30° incline. An

Stability and Center of Mass of a Structure

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

Structural Stability: Crane Center of Mass Analysis

An engineer is analyzing a crane consisting of a 500-kg horizontal beam whose center of mass is at 5

Two-Dimensional Collision and Momentum Conservation

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

Two-Dimensional Elastic Collision Analysis

A 1 kg particle moving horizontally at 4 m/s collides elastically with a 2 kg particle initially at

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 from Experimental Data

A laboratory experiment measures the angular velocity $$\omega$$ of a rotating object as a function

Angular Kinematics on a Rotating Platform

A rotating platform starts from rest and accelerates with a constant angular acceleration $$\alpha$$

Calculus Derivation of Moment of Inertia for a Thin Ring

Derive the moment of inertia for a thin ring of radius $$R$$ and mass $$m$$ using calculus.

Combined Translational and Rotational Dynamics

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

Comparative Study of Angular Kinematics at Different Radii

In a lab experiment, students measure the angular displacement and corresponding linear displacement

Comparison of Rolling Objects with Different Mass Distributions

Consider two cylinders, one solid and one hollow, each of mass $$M$$ and radius $$R$$, that roll wit

Correlation Between Torque and Rotational Energy via Calculus

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

Cylinder Rolling Down an Incline

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an incline of angle $$\t

Derivation of Angular Kinematics Equations

A set of experiments records the angular displacement of a rotating wheel over time, yielding the fo

Determining the Moment of Inertia of a Non-Uniform Rod

A non-uniform rod of length $$L = 2\,m$$ is analyzed to determine its moment of inertia about one en

Dynamic Stability of a Rotating Space Station

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

Effect of Variable Applied Torque on Angular Acceleration

In a controlled experiment, a variable torque is applied to a rotating disk. The resulting change in

Equilibrium Analysis in a Rotating Beam System

In a lab experiment, a student investigates the equilibrium of a beam pivoted at its center with var

FRQ 1: Torque Analysis on a Wrench

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

FRQ 15: Angular Momentum in an Inelastic Collision on a Rotating Platform

A figure skater stands on a frictionless rotating platform with a moment of inertia of \(10.00\,kg\c

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

Gyroscopic Precession

A spinning gyroscope with an angular momentum $$L$$ experiences an external torque $$\tau$$ causing

Impact of Mass Distribution on Rotational Kinetic Energy

This experiment investigates how different mass distributions affect the rotational kinetic energy o

Impulse and Angular Momentum: Impact on a Rotating Disk

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

Kinetic Energy Redistribution in Rotating Systems

A rotating disk initially has two weights attached at its rim, resulting in a moment of inertia $$I_

Measuring Frictional Torque in a Rotating Apparatus

In this experiment, a rotating apparatus is allowed to decelerate freely due only to friction. By re

Net Torque and Angular Acceleration Calculation

A disc with a moment of inertia of 0.5 kg m^2 is acted upon by two tangential forces: 10 N at a radi

Parallel Axis Theorem in Compound Systems

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

Quantitative Analysis of Rolling Down an Incline

An object rolls without slipping down an inclined plane. Measurements are taken at different incline

Rolling Cylinder Down an Incline

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

Rolling Motion 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 Energy Conversion Experiment

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

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 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 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 on a Uniform Rod with Distributed Force

A researcher is studying the effect of a distributed force along a uniform rod of length $$L$$ pivot

Verification of the Parallel Axis Theorem

A solid disk of mass M = 4 kg and radius R = 0.5 m is considered. The moment of inertia about its ce

Wrench Torque Analysis

A mechanic uses a wrench to loosen a bolt. Consider a wrench of length L = 0.3 m. In part (a), the m

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

Comparative Analysis: Horizontal vs. Vertical Oscillations

Compare and contrast the dynamics of a horizontal spring-mass oscillator (on a frictionless surface)

Comparative Dynamics of Mass-Spring and Pendulum Oscillators

Analyze the motion of two oscillatory systems: a mass-spring oscillator and a simple pendulum (using

Coupled Oscillators Investigation

A researcher investigates two masses, $$m_1$$ and $$m_2$$, connected in series by two identical spri

Damped Oscillation: Logarithmic Decrement Analysis

A spring-mass oscillator exhibits damped oscillations, and the amplitudes of successive cycles are r

Data Analysis of Damped Oscillations

A damped oscillator is described by the function: $$y(t) = 0.1 * e^{-\frac{b * t}{2m}} * \cos(\omeg

Dependence of Maximum Speed on Amplitude

For a spring-mass oscillator undergoing simple harmonic motion, analyze how the maximum speed $$v_{m

Derivation and Solution of SHM Differential Equation

A mass-spring system exhibits simple harmonic motion. Derive the differential equation governing the

Derivation of Total Mechanical Energy Conservation in SHM

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

Deriving the General Solution of SHM

Derive and analyze the general solution for simple harmonic motion from the governing differential e

Determination of Spring Constant from Oscillation Data

A researcher collects oscillation data for different masses attached to a spring. The data is summar

Determination of Spring Constant Using SHM Data

An experiment on a mass-spring oscillator provides the following data for different masses and their

Determination of the Damping Coefficient from Amplitude Decay

Consider an oscillator with damping such that its displacement is given by: $$x(t) = A e^{-\frac{b}

Determining Spring Constant from Experimental Data

An experiment on a spring produced the following data relating displacement $$x$$ (in meters) to for

Driven Oscillations and Resonant Response

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

Effect of Amplitude on Acceleration in SHM

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

Effects of Mass Variation on Oscillation Frequency

A student investigates how the oscillation frequency of a spring-block system varies with the mass a

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 Conservation in Pendulum Motion

A pendulum bob of mass $$m=0.5\,\text{kg}$$ is released from rest at an angle of $$20^\circ$$ from t

Evaluating Experimental Uncertainties in SHM Measurements

Accurate measurement of the oscillation period is crucial in SHM experiments. In this context, uncer

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

Fourier Analysis of Oscillation Data

In an advanced experiment, the oscillation of a spring-mass system is recorded and analyzed using Fo

Friction Effects in Horizontal Oscillatory Systems

A block attached to a horizontal spring oscillates with amplitude $$A$$, but friction is present. Th

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

FRQ 7: Differentiation of SHM to Obtain Velocity and Acceleration

Consider an oscillator described by $$y = A \sin(\omega t + \phi_0)$$. A set of experimental velocit

FRQ 8: Energy Exchanges in SHM

A graph of elastic potential energy vs. displacement for a spring is provided. Use calculus to deriv

FRQ 10: Differential Equation of a Horizontal Mass-Spring System

Consider a mass attached to a horizontal spring on a frictionless surface. Answer the following:

FRQ 15: Graphical Analysis of Restoring Force

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

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

FRQ4: Vertical Spring-Block Oscillator – Equilibrium and Oscillations

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

FRQ5: Sinusoidal Description of Oscillatory Motion and Phase Determination

A mass-spring system oscillates with an amplitude of $$A = 0.06\,m$$ and a frequency of $$f = 2.0\,H

FRQ9: Energy Exchanges in a Mass-Spring Oscillator

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

FRQ12: Phase Shift and Time Translation in SHM

An oscillator is described by the equation $$x(t) = A\sin(\omega t+\phi_0)$$. Answer the following:

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}}.

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

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 Approach to SHM: From Acceleration to Displacement

A mass undergoing simple harmonic motion has an acceleration given by $$a(t) = -\omega^2 * A * \sin(

Investigating Nonlinear Oscillations in a Large-Amplitude Pendulum

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

Measuring the Spring Constant: An Experimental Investigation

A student performs an experiment to determine the spring constant of a coil spring. The following da

Mechanical Energy in SHM

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

Modeling Amplitude Reduction Due to Non-Conservative Forces

In a real oscillatory system, non-conservative forces (like friction) result in an energy loss per c

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

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

Pendulum Dynamics Beyond the Small-Angle Approximation

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

Pendulum Experiment Analysis

A researcher uses a simple pendulum to measure gravitational acceleration. The pendulum has a length

Pendulum Motion: Small-Angle Approximation

A simple pendulum of length $$L = 0.80\,m$$ is released from a small angle. (a) Using the small-angl

Pendulum on a Rotating Platform

A simple pendulum of length $$L$$ is mounted on a platform that rotates with constant angular speed

Phase Constant and Sinusoidal Motion

A mass-spring system oscillates according to $$x(t) = A \sin(\omega t + \phi_0)$$ with an amplitude

SHM with a Varying Force Constant

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

Simple Pendulum Energy Analysis

Consider a simple pendulum of length $$L$$ and mass $$m$$ undergoing small oscillations. Answer the

Sinusoidal Oscillator and Phase Constant

A mass attached to a spring oscillates horizontally on a frictionless surface, and its displacement

Small-Angle Pendulum Experiment

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

Time-Dependent Analysis of Oscillatory Motion

An oscillator's displacement is given by the function $$x(t)=0.03 * \cos(12*t)$$ (with $$x$$ in mete

Time-Derivative Analysis of Displacement in SHM

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

Uncertainty Analysis in SHM Period Measurements

In an experiment designed to measure the period of a spring-mass oscillator, several sources of unce

Barycenter in a Two-Body System

In a two-body system consisting of masses m₁ and m₂ separated by a distance R, the barycenter (cente

Calculating Gravitational Potential in a Non-Uniform Planet

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

Calculus Analysis of Gravitational Potential Energy

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

Calculus in Determining Work Against Gravity over Altitude Change

A spacecraft gradually moves from an initial orbital radius r₁ to a higher radius r₂. The work done

Calculus in Gravitational Work: Integration of Inverse Square Force

Using Newton's Law of Gravitation, the force on an object is given by $$F(r) = -\frac{G * M * m}{r^2

Comparative Analysis of Gravitational Forces

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

Comparison of Gravitational and Centripetal Forces

For a satellite in a stable circular orbit, investigate the balance between gravitational and centri

Comparison of Orbital Dynamics: Moon vs. Artificial Satellites

A researcher compares the gravitational forces and orbital characteristics of the Moon and an artifi

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

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

Designing a Cavendish Experiment to Measure the Gravitational Constant

A student plans to design a version of the Cavendish experiment to measure the gravitational constan

Determining Planetary Mass from Satellite Orbital Data

Satellite orbital data can be used to determine the mass of the planet they orbit. Answer the follow

Elliptical Orbit Simulation Error in Barycenter Consideration

A computer simulation is developed to mimic the elliptical orbits of planets using Newton's law of g

Elliptical Orbits and Eccentricity Calculation

An object is in an elliptical orbit around a star. (a) Define the eccentricity $$e$$ in relation to

Energy Dissipation in Orbital Decay

A satellite experiences a tangential drag force given by $$F_{drag} = -b * v$$, where $$b$$ is a con

Experimental Analysis of Orbital Decay from a Satellite

A satellite in low Earth orbit is gradually losing altitude due to atmospheric drag. Experimental da

Free-Fall Measurement on a Curved Incline

An experiment is designed to measure gravitational acceleration by allowing a small ball to roll dow

FRQ 12: Designing a Geosynchronous Satellite Orbit

A geosynchronous satellite must orbit the Earth with an orbital period equal to one sidereal day (\(

FRQ 15: Gravitational Anomalies and Their Effects on Orbits

A satellite experiences a small perturbation in the gravitational potential due to a local mass anom

FRQ 18: Non-Uniform Circular Motion in a Varying Gravitational Field

An object in orbit around a planet experiences non-uniform circular motion due to variations in the

Gravitational Acceleration Variation with Altitude

Examine the data on gravitational acceleration at various altitudes and analyze how gravitational ac

Gravitational Assist Maneuver Simulation

Gravitational assist maneuvers, which use the gravity of a planet to alter a spacecraft’s trajectory

Gravitational Potential Energy Change in an Elliptical Orbit

A satellite moves in an elliptical orbit around Earth. Its gravitational potential energy is given b

Gravitational Potential Energy Differences in Multi-Body Systems

Consider a test mass moving in the gravitational field of two bodies, $$M_1$$ and $$M_2$$. Answer th

Gravitational Potential Energy Measurement on a Ramp

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

Gravitational Potential to Rotational Kinetic Energy Conversion

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

Gravity Assist in Three-Body Dynamics

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

Modeling Orbital Decay due to Atmospheric Drag

A low Earth orbit satellite experiences orbital decay primarily due to atmospheric drag. The drag fo

Modeling Orbital Decay with Differential Equations

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

Optimizing Orbital Transfer Maneuvers: Hohmann Transfer

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

Orbital Dynamics: Gravitational Force Variation

Examine the following experimental evidence on the gravitational force as a function of distance for

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

Orbital Period and Semimajor Axis Relationship Using Kepler's Third Law

A researcher collects observational data for various moons orbiting a giant planet. The table below

Orbital Perturbation due to Radial Impulse

A satellite in a circular orbit of radius R receives a small radial impulse, altering its orbit into

Orbital Perturbations and Precession

Investigate how small perturbative forces lead to the precession of a planet's orbit.

Orbital Speed Variation in Elliptical Orbits

Analyze the provided data on orbital speeds at different points in an elliptical orbit. Discuss how

Planetary Orbits and Energy Considerations

Consider a planet in a highly eccentric orbit. The total orbital energy (kinetic plus potential) is

Speed Variation in Elliptical Orbits via Angular Momentum Conservation

In an elliptical orbit, a satellite’s speed varies along its path. Using the conservation of angular

Tidal Forces and their Impact on Orbital Dynamics

A moon orbits a planet and experiences tidal forces. Analyze how these forces are derived and their