AP Physics C: Mechanics FRQ Room

Analysis of Motion from a Position Function

A particle moving along a line has its position described by $$x(t)=t^4 - 8t^2 + 16$$ (in meters) wh

Analyzing a Parabolic Trajectory

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

Centripetal Acceleration in Circular Motion

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

Critical Evaluation of Experimental Kinematics Data

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

Deriving Velocity and Acceleration from a Position Function

Consider an object moving along a straight line with its position given by $$x(t)=\sin(t)$$, where x

Differential Equation of Motion Under Gravity and Drag

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

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

Experimental Evaluation of Vector Addition in Two Dimensions

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

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 4: Velocity-Time Graph Analysis (EASY)

A velocity versus time graph for an object shows the following behavior: • From $$t=0$$ to $$t=2\,s$

FRQ 5: Derivation of Motion Equations from Calculus

A researcher aims to derive the standard kinematic equations using calculus for an object moving wit

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: Air Resistance and Terminal Velocity

An object of mass 2 kg is falling under gravity while experiencing air resistance proportional to it

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 11: Modeling Motion with Differential Equations

A researcher is modeling the motion of a ski jumper. The dynamics of the jumper's flight can be expr

FRQ 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 18: Motion of a Robot – Programming and Modeling Its Movement

A robotics research laboratory programs a robot to move on a flat plane. The robot's position is mod

Impulse and Variable Force Analysis

Design an experiment to measure the impulse delivered to a ball by a bat, given that the force varie

Instantaneous vs. Average Velocity

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

Investigating Lab Data: Graph Interpretation and Improvements

In a motion sensor experiment, an object’s displacement as a function of time was recorded, resultin

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 with Variable Acceleration

An object has a time-dependent acceleration given by $$a(t)= 6*t - 4$$ (in m/s^2) and starts from re

One-Dimensional Uniform Acceleration Analysis

An object moves along a straight line with its position given by $$x(t) = -2*t^3 + 5*t^2 - 3*t + 1$$

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

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

Relative Motion: Two Trains on Parallel Tracks

Two trains move on parallel tracks. Train A has its position given by $$x_A(t)=25t$$ and Train B by

Rotational Kinematics of a Spinning Disk

Design an experiment to measure the angular acceleration of a spinning disk using a photogate sensor

Rotational Motion: Angular Kinematics

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

Simultaneous Measurement of Velocity and Acceleration

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

Slope Analysis in a Velocity-Time Graph

A physics lab recorded an object’s velocity over time using an electronic sensor, and the resulting

Time vs. Position Data Analysis: Initial Conditions Overlooked

A student conducted an experiment to study an object’s motion by recording its position over time us

Trajectory Optimization of an Accelerating Car

A car accelerates according to $$a(t)= 6 - 0.5*t$$ (m/s²) with an initial velocity of 10 m/s at $$t=

Variable Acceleration Analysis Using Calculus

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

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).

Vector Addition in Two-Dimensional Projectile Motion

Design an experiment to test the principles of vector addition by analyzing the two-dimensional moti

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 a Potential Energy Curve

A particle of mass 4 kg moves along the x-axis under the influence of a potential energy function gi

Ballistic Kinetic Energy with Air Resistance

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

Block Under a Varying Force

A 2 kg block moves along a frictionless horizontal surface under the influence of a variable force g

Comparative Analysis of Constant and Variable Force Work

Two experiments are performed on a 3 kg object. In Experiment 1, the object is moved under a constan

Composite System: Roller Coaster Energy Analysis

A roller coaster car of mass 600 kg travels along a frictionless track. The following table provides

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

Determining Speed of a Roller Coaster Considering Friction

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

Elastic Potential Energy in a Spring

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.1 m. Analyze the energy store

Elastic Potential Energy in a Spring System

A spring with a spring constant of $$k = 200\,N/m$$ is compressed by 0.25 m from its equilibrium pos

Energy Analysis of a Damped Pendulum

A pendulum of length 2 m and mass 1 kg oscillates in air. The damping force due to air resistance is

Energy Analysis of a Damped Spring-Mass Oscillator

A spring-mass system consists of a mass $$m = 2 \;\text{kg}$$ attached to a spring with force consta

Energy Loss in an Inelastic Collision

A 2 kg object moving at 4 m/s collides and sticks to a 3 kg object initially at rest.

FRQ 2: Work Done by a Variable Force on a Crate Along a Horizontal Floor

A crate is pulled along a horizontal surface with an applied force that varies with displacement. Th

FRQ 2: Work-Energy Theorem in Lifting

A news article claims that the work done in lifting an object is independent of the velocity at whic

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 7: Energy Loss Due to Friction on a Sliding Object

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

FRQ 11: Deriving Force from a Potential Energy Function

A study posits that the force acting on a particle can be obtained via $$F(x) = - \frac{dU}{dx}$$. E

FRQ 12: Quantifying the Work Done by Friction

An experimental report claims that the negative work done by friction is constant regardless of the

High-Power Engine Performance Test

An engine is tested on a dynamometer. Its instantaneous force is given by $$F(t)=1000 + 200*t$$ N fo

Impulse and Energy Transfer via Calculus

A 2-kg object experiences a time-dependent force given by $$F(t)= 4*t$$ N for $$0 \leq t \leq 5\,s$$

Inelastic Collision and Energy Dissipation

Two blocks undergo a perfectly inelastic collision. A 3 kg block moving at 5 m/s collides with a 2 k

Integration of Varying Forces in a Sled Motion

A 20 kg sled is pushed along a snowy hill by an applied force given by $$F(s)=80\,e^{-0.2\,s}$$ (N),

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

Kinetic Energy and Work-Energy Theorem Application

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

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

Nonlinear Spring Energy Experiment

In an experiment with a nonlinear spring, a mass is attached and the force is measured as a function

Oscillations in a Mass-Spring System

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

Power and Energy in High-Speed Systems: Rocket Launch Analysis

A researcher is analyzing the power requirements of a rocket engine during a launch phase. A rocket

Power in a Repeated Jumping Robot

A robot of mass 50 kg repeatedly jumps vertically. In each jump, its engine does work to convert kin

Power Output in a Variable Force Scenario

A force acting on an object causes work to be done such that the work as a function of time is given

Power Output in Elevator Lifting

A motor lifts an elevator of mass 500 kg at a constant speed of 2.5 m/s over a vertical displacement

Power Output in Mechanical Systems

Consider a scenario where a worker lifts a load while delivering power that varies over time.

Power Output Variation in a Machine

An object is being moved along a straight path by a constant force of 10 N. The displacement as a fu

Rolling Sphere Energy Experiment

A solid sphere is rolled without slipping down a tilted ramp, and its kinetic energy is measured at

Rotational Work and Energy in a Falling Rod

A uniform thin rod of length $$L = 2.0\,m$$ and mass $$m = 4.0\,kg$$ is initially held horizontally

Time-Varying Velocity and Instantaneous Power Measurement

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

Variable Force and Velocity: Power and Work Analysis

A machine applies a time-dependent force given by $$F(t)=50+10\,t$$ (N) while the displacement of an

Variable Force with Angular Displacement

A 15 kg crate is pulled along a horizontal floor by a rope. The tension in the rope varies with the

Variable Force Work Calculation

An object moves along the x-axis under the influence of a variable force given by $$ F(x) = 3 * x^2

Variable Gravitational Acceleration Over a Mountain

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

Work by Non-Conservative Forces in a Loop

A block experiences a variable nonconservative force along a path given by $$ F(x)= 20 * \sin(x) $$

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 and Friction: Analyzing a Sliding Block

A researcher investigates the effect of friction on a sliding block. A 4 kg block is launched horizo

Balancing a Composite System's Center of Mass

A thin uniform rod of length $$3$$ m (mass $$1$$ kg) has two point masses attached to it: a $$2$$ kg

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 Composite Object

A composite object is constructed by rigidly attaching two uniform rectangular plates. Plate A has a

Center of Mass of a Rectangular Plate with Variable Density

A thin rectangular plate extends from $$x=0$$ to $$x=2$$ m and from $$y=0$$ to $$y=3$$ m. Its surfac

Center of Mass of a Triangular Plate

A thin, homogeneous triangular plate has vertices at (0,0), (4,0), and (0,3) in meters. Determine it

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

Crash Test Analysis in a Vehicle Collision

In a crash test, a car with a mass of 1200 kg collides with a barrier and comes to a complete stop i

Elastic Collision: Two Gliders on an Air Track

Two gliders on an air track experience a head-on elastic collision. Glider X (mass = 1 kg) initially

Elastic vs. Inelastic Collision Analysis of Carts

Two carts on a low-friction track undergo a collision. The masses and velocities before and after co

Experiment Design: Spring-Loaded Impulse Mechanism

A spring-loaded mechanism is used to deliver an impulse to a 0.5 kg cart. The spring has a constant

Experimental Design: 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

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 3: Motion of the Center of Mass under External Force

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

FRQ 4: Impulse from a Time-Dependent Force

A 0.8 kg ball experiences a force given by $$F(t)=10*\sin((\pi*t)/2)$$ (N) for $$0 \le t \le 2\ s$$.

FRQ 6: Elastic Collision Analysis

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

FRQ 8: Center of Mass and Stability

A uniform beam of length $$5 \ m$$ and mass $$50 \ kg$$ is hinged at one end and held horizontally b

FRQ 9: Rocket Propulsion and Momentum Conservation

A rocket in space initially has a mass of $$500 \ kg$$ and is traveling at $$20 \ m/s$$. It ejects $

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

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

Impulse and Swing Angle in a Pendulum

A pendulum bob of mass $$1.0\,kg$$ initially at rest is given a horizontal push by a time-dependent

Impulse Calculation from a Force-Time Graph

A force acting on a cart is recorded by a sensor and is represented by the following graph: the forc

Impulse Delivered by a Time-Dependent Damping Force

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

Impulse in a Non-Constant Force Field

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

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

Impulse Transfer on a Rotating Rod

A uniform rod of length $$4\,\text{m}$$ and mass $$8\,\text{kg}$$ is initially at rest, pivoted fric

Inclined Plane: Center of Mass and Impulse Analysis

A block of mass $$3$$ kg rests on a frictionless inclined plane with an angle of $$30^\circ$$. The i

Measuring Center of Mass of an Irregular Lamina

A lab is designed to measure the center of mass (COM) of an irregular, thin lamina of (ideally) unif

Momentum Change under a Non-Constant Force

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

Momentum Conservation in Glider Collisions on an Air Track

In a laboratory experiment, students investigate conservation of momentum by allowing two gliders to

Motion of Center of Mass Under External Force

Consider a system with total mass $$M=5\,kg$$. An external force acts on the system given by $$F_{ex

Motion of the Center of Mass with External Force

Consider two blocks (masses 3 kg and 5 kg) connected by a light spring on a frictionless surface. In

Nonuniform Rod Center of Mass

Consider a rod of length $$L = 1.0\,m$$ whose linear density is given by $$\lambda(x)=6+4*x$$ (in kg

Oblique Collision of Two Billiard Balls

Two billiard balls, each of mass $$0.17\,\text{kg}$$, undergo an oblique collision on a frictionless

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 Drag Impulse Analysis

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

Rocket Propulsion Momentum Problem

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

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

Variable Force Collision Analysis from Graph Data

A steel ball (mass = 0.2 kg) collides with a wall, and the force versus time data during the collisi

Vibrational Motion: Coupled Oscillators

Two masses (m1 = 0.5 kg and m2 = 0.5 kg) are connected by a spring with spring constant k = 200 N/m

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.

Analyzing Variable Torque and Angular Acceleration

A rotor with constant moment of inertia $$I = 4 \text{ kg\cdot m}^2$$ is subjected to a time-varying

Angular Acceleration from Variable Torque

Design an experiment where you apply a time-dependent (variable) torque to a rotating object and mea

Angular Kinematics from Disk Data

A rotating disk’s angular displacement is recorded over time during a period of uniform angular acce

Angular Kinematics on a Rotating Platform

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

Angular Momentum Conservation: Ice Skater

An ice skater is spinning with an initial angular velocity $$\omega_i$$ and an initial moment of ine

Calculation of Rotational Inertia for Composite System

A uniform disk of mass $$M = 2\,kg$$ and radius $$R = 0.5\,m$$ has two small beads, each of mass $$m

Calculus-Based Analysis of Angular Impulse

In rotational dynamics, angular impulse is defined as the integral of torque over a time interval, w

Comparative Analysis of Rotational and Translational Dynamics

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

Comparative Angular Momentum in Different Systems

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

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,

Computational Modeling of a Spinning Disk with Variable Torque

A spinning disk with a constant moment of inertia of $$I = 0.3\,kg\,m^2$$ is subjected to a time-var

Cylinder Rolling on an Incline

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

Designing a Rotational Sensor

An engineer is designing a sensor to measure rotational torque in a mechanical system. The sensor ou

Determining the Moment of Inertia of a Compound Pendulum

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

Differentiating Between Contact and Rolling Without Slip

An experiment is conducted to analyze the motions of objects rolling down an inclined plane. Both th

Dynamic Analysis of a Gyroscope: Precession

A gyroscope consists of a spinning disk mounted on a pivot. The center of mass is offset from the pi

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

Energy Dissipation in a Rotating System with Friction

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

FRQ 4: Rotational Kinematics of a Disk

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

FRQ 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

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 16: Composite Rotational Inertia via Integration

A thin rod of length L = 3.00 m has a linear mass density that varies along its length according to

Graphical Analysis of Rotational Kinematics

A graph of angular velocity $$\omega$$ (in rad/s) versus time $$t$$ (in s) for a rotating wheel is p

Investigation of Angular Acceleration from Experimental Data

In an experiment, the angular displacement (in radians) of a rotating object was recorded at various

Lever Torque Application

A mechanic uses a long lever to lift a heavy object. A force of $$F = 100 \text{ N}$$ is applied at

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

Moments of Inertia for Point Masses on a Rod

Three beads, each of mass 2 kg, are fixed on a massless rod of length 1.2 m. In one configuration, o

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 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 in Rotational Systems

A uniform disk of mass $$M$$ and radius $$R$$ has a moment of inertia about its central axis given b

Quantitative Analysis of Rolling Down an Incline

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

Rolling Motion Dynamics Down an Incline

A solid cylinder with mass $$M = 3\,kg$$ and radius $$R = 0.2\,m$$ rolls without slipping down an in

Rolling Motion on an Inclined Plane

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

Rolling Motion with Slipping Transition

A cylinder initially rolls without slipping down an incline. At a certain point, due to a change in

Time-Resolved Analysis of Angular Acceleration

A wheel's angular velocity is given by $$\omega(t)= 3t^2 + 2t\,rad/s$$. Calculate its angular accele

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 Measurement and Angular Acceleration Experiment

In this experiment, you will investigate the relationship between applied force, moment arm, and the

Variable Mass Distribution and Moment of Inertia

A rod of length $$L = 3.0 \text{ m}$$ has a linear mass density given by $$\lambda(x) = \lambda_0*(1

Variable Torque Function Integration

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

Verification of the Parallel Axis Theorem

Students test the parallel axis theorem by measuring the moment of inertia of a uniform rod about se

Amplitude Dependence in a Nonlinear Oscillator

Consider an oscillator whose restoring force is not perfectly linear but is given by: $$F = -k * x

Analyzing a Mass-Spring System on an Inclined Plane

A block of mass $$m = 1.0\,\text{kg}$$ is attached to a spring with spring constant $$k = 100\,\text

Analyzing Phase Shift and Amplitude Modulation in SHM

An oscillator’s displacement is given by the equation $$y(t)= (0.05 + 0.01 * t) * \sin(8*t+0.2)$$

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

Calculus Application in SHM: Derivatives and Acceleration

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

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 Oscillator Velocity and Acceleration

For an oscillator described by $$y = A\sin(\omega t + \phi_0)$$, derive its velocity and acceleratio

Comparison of Horizontal and Vertical Oscillations

Compare a mass-spring oscillator on a frictionless horizontal surface with a vertical spring-block s

Coupled Oscillations in a Two-Mass Spring System

Consider two masses, $$m_1$$ and $$m_2$$, connected by a spring with spring constant $$k$$. Addition

Damped Harmonic Oscillator Dynamics

A mass-spring oscillator with damping is modeled by a damping force proportional to the velocity. Co

Damped Oscillatory Motion Analysis

A mass-spring system undergoing damped oscillations has a damping force given by \(F_d = - b * v\),

Data Analysis and Calculus Estimation in SHM

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

Data Analysis from a Virtual SHM Experiment

A virtual experiment on simple harmonic motion produces the following data for the displacement of a

Deriving the Equation of Motion Using Calculus

An undamped harmonic oscillator is governed by the differential equation $$\ddot{x} + \omega^2 x = 0

Determination of Gravitational Acceleration Using a Vertical Oscillator

A vertical spring-mass oscillator reaches equilibrium when the spring stretches by a distance $$d$$

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 Initial Phase in SHM

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

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

The position of an oscillator is given by the function $$y(t)=0.05 * \sin(10*t+0.3)$$ (with $$y$$ in

Effect of a Variable Spring Constant on SHM

A specially engineered spring has a force constant that varies with displacement, given by $$k(x)=k_

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

Elastic Potential Energy and Maximum Speed Calculation

A block of mass $$m = 0.10\,\text{kg}$$ is attached to a spring with a spring constant of $$k = 200\

Energy Conservation via Calculus Integration

In a spring oscillator experiment, energy is exchanged between elastic potential energy and kinetic

Energy Exchange in SHM

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

Evaluating Damped Oscillatory Motion Effects

A mass-spring oscillator with mass $$m = 0.5 \; kg$$, spring constant $$k = 100 \; N/m$$, and dampin

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

Evaluating the Role of Calculus in Predicting Oscillator Dynamics

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

Forced Oscillations and Resonance

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

Fourier Analysis of Oscillation Data

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

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 17: Determination of Damping Coefficient

An oscillatory system exhibits damping, with its amplitude decreasing over successive cycles. Analyz

Graphical Analysis of Oscillatory Data

A researcher records and graphs the displacement of a mass-spring oscillator as a function of time.

Hooke's Law and Spring Force Calculation

Consider a spring that obeys Hooke’s law, $$F = -k * x$$, where k is the spring constant and x is th

Hooke’s Law and Work in Spring Systems

A spring with a spring constant $$k = 250\,N/m$$ is initially at its natural length. (a) Using Hooke

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

Measuring g with a Simple Pendulum

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

Mechanical Energy in SHM

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

Momentum and Impulse Analysis in Oscillatory Motion

A block of mass $$m = 0.4 \; kg$$ oscillates on a spring, and its displacement is given by $$x(t)=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 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 Oscillation under Small-Angle Approximation

A simple pendulum consists of a bob of negligible size suspended from a pivot by a massless string o

Pendulum Period and Data Analysis

Explore the period of a simple pendulum and compare experimental data with theoretical predictions.

Period of a Physical Pendulum: A Calculus Approach

A physical pendulum consists of a uniform thin rod of length $$L$$ and mass $$m$$, pivoted at one en

Phase Angle Determination from Initial Conditions

An oscillator exhibits motion described by $$y(t) = A * \sin(\omega*t + \phi_0)$$ with amplitude $$A

Phase Shift and Time Determination in SHM

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

Phase Space Analysis of SHM

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

Resonance in a Driven Harmonic Oscillator

Analyze a damped, driven harmonic oscillator and explore the conditions for resonance.

Resonance in Forced Oscillations

A researcher sets up a forced oscillation experiment with a mass-spring system subject to a sinusoid

Restoring Force in a Non-Ideal Pendulum

For a pendulum with a bob of mass $$m$$ swinging through large angles, the exact restoring force is

Sinusoidal Analysis of SHM with Phase Shift

Examine a sinusoidally described simple harmonic oscillator with a phase shift.

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

Vertical Mass-Spring Oscillator: Equilibrium and Oscillations

A 2.0-kg mass is attached to the end of a vertical spring with a force constant $$k = 250\,N/m$$. Th

Vertical Oscillations: Lab Data Analysis

A mass-spring system is arranged vertically in a laboratory setup. The oscillatory motion of the blo

Vertical Spring-Block Oscillator: Equilibrium and Oscillations

A block of mass $$m = 1.80\,kg$$ is attached to a vertical spring with force constant $$k = 400\,N/m

Analyzing Multi-body Interactions in a Three-Body Problem

Consider a simplified three-body system consisting of two stars, each of mass M, and a planet of mas

Analyzing Three-Body Gravitational Interactions

Consider a system comprising a star, a planet, and a moon. In such a three-body system, gravitationa

Barycenter of the Sun-Planet System

Consider the Sun-Earth system, where the Sun's mass is M = 1.99×10^30 kg, the Earth's mass is m = 5.

Cannonball Trajectory in a Non-Uniform Gravitational Field

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

Cometary Orbits: Analyzing Highly Eccentric Trajectories

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

Comparative Analysis of Orbital Periods for Different Planets

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

Derivation of Kepler’s Second Law via Calculus

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

Designing a Satellite Orbit Experiment

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

Determination of Gravitational Constant Using a Compound Pendulum

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

Determining the Gravitational Constant using a Torsion Balance

An experimental apparatus utilizing a torsion balance is used to measure the gravitational forces be

Determining the L1 Lagrange Point

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

Elliptical Orbit Dynamics: Speed Variation Analysis

For a planet or satellite in an elliptical orbit, the speed varies along the orbit due to conservati

Energy Conservation in Central Force Motion

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

Escape Velocity and Energy Conservation

Consider an object of mass m escaping from a planet with mass M, where gravitational potential energ

Experimental Analysis of Gravitational Acceleration

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

Experimental Verification of Conservation of Energy in a Gravitational Field

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

FRQ 1: Gravitational Force between Two Masses

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

Gravitational Energy in a Binary Star System

Binary star systems are bound by gravity. The total mechanical energy of such a system includes kine

Gravitational Potential Energy Change for a Satellite

A satellite is moved from a lower orbit to a higher orbit around a planet. Gravitational potential e

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 to Rotational Kinetic Energy Conversion

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

Gravitational Slingshot Maneuver

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

Impact of Relativistic Effects on Orbital Motion

Discuss how relativistic effects modify the orbital motion of a planet when it orbits close to a ver

Investigating Tidal Forces in a Binary Star System

Tidal forces in binary star systems can have significant effects on the stars’ structures. Answer th

Modeling Orbital Decay with Differential Equations

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

Orbital Energy and Conservation Laws

For a satellite in a circular orbit of radius $$r$$ around a planet, the kinetic energy is given by

Orbital Precession Analysis

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

Pendulum Orbital Analog and Kepler's Third Law

In a classroom experiment, a simple pendulum is used as an analog to planetary orbits in order to va

Perturbation Analysis in Elliptical Orbits

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

Tidal Heating and Energy Dissipation

Tidal forces in planetary systems can lead to energy dissipation in satellites, resulting in tidal h