AP Physics C: Mechanics FRQ Room

Acceleration from a Given Velocity Function

An object moves along a straight line with its velocity described by $$v(t)= 5*t^2 - 3*t + 2$$ (m/s)

Analysis of a Velocity-Time Graph

A velocity vs. time graph for an object moving in one dimension displays a linear increase in veloci

Analyzing a Parabolic Trajectory

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

Average vs. Instantaneous Quantities

A particle’s displacement is given by the integral function $$x(t)= \int_0^t e^{-\tau} \cos(\tau)\,d

Conservation of Momentum in Collisions

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

Designing a Kinematics Lab Experiment

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

Distance vs. Displacement Analysis in One-Dimensional Motion

An object moves along a straight path and its motion is described by the velocity function $$v(t) =

Experimental Data and Constant Acceleration

A ball rolling down a ramp has its displacement measured at various times as shown in the table belo

Free Fall Analysis with Terminal Velocity Consideration

A rock is dropped from a cliff 80 meters high. A researcher claims that the measured fall time of th

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

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

FRQ 3: 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 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 6: Relative Motion in Two Dimensions (HARD)

In the xy-plane, two particles have position functions given by: $$\vec{r}_A(t)=(2*t, t^2)$$ and $$\

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 9: Application of the Big Five Equations

An object starts with an initial velocity of 8 m/s and reaches a final velocity of 20 m/s after trav

FRQ 10: Experimental Analysis of Free Fall

Below is experimental data from free fall tests for objects dropped from various heights: | Height

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 13: Analyzing a Two-Dimensional Collision with Projectiles

A researcher conducts an experiment with two projectiles launched simultaneously from different posi

FRQ 13: Comparative Analysis of Two Free Fall Experiments

The following data summarizes two experiments where objects were dropped from different heights: |

FRQ 14: Differentiation of a Position Function

An object’s position is given by the function $$x(t)= t^3 - 6*t^2 + 9*t$$ (with x in meters and t in

FRQ 14: Work and Energy in Kinematics – Rolling Ball on an Incline

A researcher is studying a ball rolling down an inclined plane with friction. In addition to the kin

FRQ 15: Investigating Uniformly Accelerated Motion Using Integrals

A researcher records the acceleration of an object with a sensor, finding that the acceleration vari

Graphical Analysis of Kinematic Data

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

Impulse and Momentum with a Variable Force

A cart of mass $$m = 5.0\,kg$$ is subjected to a time-dependent force described by $$F(t)=5*t²$$ (in

Investigating Lab Data: Graph Interpretation and Improvements

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

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

Kinematic Analysis of Circular Motion

A particle moves along a circular path of constant radius R. Its speed increases according to the fu

Motion on an Inclined Plane with Friction

Design an experiment to measure the acceleration of an object sliding down an inclined plane with fr

Motion with Air Resistance: Approximating Terminal Velocity

A small sphere falling through a medium experiences air resistance proportional to its velocity. Its

Non-linear Position Function Analysis

A particle moves along the x-axis with a non-linear, decaying oscillatory position given by $$x(t)=e

Oscillating Particle under Uniform Acceleration

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

Oscillatory Motion: Mass-Spring System

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

Predicting Motion in a Resistive Medium

An object in free fall experiences a time-dependent acceleration given by $$a(t)=g\left(1-\frac{t}{T

Projectile Motion and Calculus Analysis

A ball is thrown from a platform. Its horizontal and vertical positions are given by $$x(t)=20*t$$ a

Projectile Motion: Launch from a Moving Platform

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

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

Terminal Velocity Experiment

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

Uniformly Accelerated Motion: Derivation and Application

A car accelerates uniformly from rest. Its velocity as a function of time can be expressed as $$v(t)

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).

Calculus-Based Examination of a Spring System

A spring with a spring constant $$k = 200\,N/m$$ is compressed by a distance x and then released. An

Composite System: Roller Coaster Energy Analysis

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

Damped Oscillations and Energy Dissipation in a Mass-Spring System

A damped harmonic oscillator with mass m = 2 kg, spring constant k = 50 N/m, and damping coefficient

Energy Analysis in a Mass-Spring Oscillator

A mass-spring system consists of a 1 kg mass attached to a spring with a spring constant of 100 N/m.

Energy Analysis in Circular Motion

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

Energy Dissipation in an Oscillatory System

An oscillatory system with damping has its energy decay described by $$E(t) = E_0*e^{-\gamma*t}$$.

Energy Loss in Inelastic Collisions

Two objects collide and stick together. Object 1 has a mass of 2 kg and is traveling at 4 m/s, while

Equilibrium Points from a Potential Energy Function

A particle of mass 4 kg experiences a potential energy given by $$U(x) = (x - 2)^2 - (2*x - 3)^3$$ (

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

Experiment on Electric Motor Power Output

Design an experiment to measure the power output of an electric motor used in a small robotic car.

FRQ 1: Vertical Lifting Experiment – Work Calculation

A lab student lifts a 1.5 kg mass vertically at constant velocity and records the force applied alon

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

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

FRQ 10: Conservation of Energy in a Pendulum Experiment

A simple pendulum with a length of 2.0 m is released from an angle of 30° with respect to the vertic

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

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

FRQ 15: 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: Evaluating Power Output Measurements in a Rocket Launch

A media report asserts that the power output of a rocket engine can be approximated by the formula $

FRQ 20: Evaluating Efficiency in a Conveyor Belt System

In a conveyor belt system used for transporting goods, the work input and the corresponding measured

Impulse and Energy Transfer via Calculus

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

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),

Investigation of Non-Conservative Forces in a Roller Coaster Model

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

Nonlinear Spring Energy Experiment

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

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

Oscillatory Motion Energy Exchange Experiment

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

Potential Energy Curve Analysis

A particle moves in a one-dimensional potential given by $$U(x) = (x-2)^2 - (2x-3)^3$$. A graph of t

Potential Energy Curve of a Diatomic Molecule

The interatomic potential energy of a diatomic molecule can be approximated by the function $$U(r) =

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 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 Output Measurement in an Elevator Experiment

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

Projectile Motion Energy Analysis

A 1-kg projectile is launched with an initial speed of 20 m/s at an angle of 60° above the horizonta

Rotational Kinetic Energy in a Rolling Object

A solid sphere of mass 3 kg and radius 0.2 m rolls without slipping down an incline from a height of

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

Variable Force and Work on a Block

A 4 kg block is pushed along a horizontal, frictionless surface by a variable force given by $$F(x)

Variable Gravitational Acceleration Over a Mountain

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

Work and Energy on an Inclined Plane with Variable Friction

A 4 kg block slides down a 10 m long incline set at 30° from the horizontal. The frictional force al

Work Done by a Time-Dependent Force

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

Work Done in a Variable Gravitational Field

A satellite of mass 500 kg is to be raised from an altitude of 200 km to 400 km above Earth's surfac

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 Theorem Application

A 0.5 kg cart increases its speed from 2 m/s to 8 m/s on a frictionless track. Using the work-energy

Angular Impulse and Rotation

A uniform disk (mass = 4 kg, radius = 0.5 m) rotates initially at 20 rad/s. A constant tangential fo

Angular Momentum Transfer in a Collision

A 0.5 kg ball moving horizontally at 10 m/s strikes a stationary 0.3 kg rod (length 2 m) pivoted at

Ballistic Pendulum Analysis

A projectile of mass $$0.2\,kg$$ is fired horizontally and embeds into a pendulum bob of mass $$2.8\

Block on an Incline: Collision and Momentum

A 2-kg block slides down a frictionless incline of length 4 m, which makes an angle of 30° with the

Center of Gravity vs. Center of Mass in a Non-Uniform Rod

A vertical rod of length 1.0 m has a linear density given by $$\lambda(x)=5+2*x$$ (kg/m), where x is

Center of Mass Acceleration under Variable Force

Two blocks with masses $$m_1 = 3\,\text{kg}$$ and $$m_2 = 2\,\text{kg}$$ are connected by a light ro

Center of Mass Calculation for a Curved, Variable Density Wire

Students attempt to determine the center of mass of a flexible wire whose density varies along its l

Center of Mass Calculation of a Non-Homogeneous Beam

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

Center of Mass for Discrete Particles in the Plane

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

Center of Mass of a 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 Variable Density Disk

A circular disk of radius R = 0.5 m has a surface mass density that varies with the radial distance

Center-of-Mass Shift in an Internal Explosion

Three masses are arranged on a frictionless horizontal surface: m1 = 1 kg at (0,0), m2 = 2 kg at (1,

Conservation of Angular Momentum on a Rotating Platform

An ice skater of mass 50 kg spins with arms extended, having a moment of inertia of 3 kg·m² and an a

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 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 in Two Dimensions

Two particles collide elastically on a frictionless plane. Particle 1 (mass $$m_1=1.0\,kg$$) initial

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 $$

Evaluating Energy Dissipation in an Inelastic Collision

Two vehicles collide and stick together in an inelastic collision. The experimental data below provi

Fragmentation and Impulse

A stationary explosive device of total mass $$5\,\text{kg}$$ breaks into two fragments. One fragment

FRQ 2: Center of Mass of a Composite Lamina

Consider a composite lamina composed of two rectangular regions in the xy-plane. Region A is 0.8 m b

FRQ 20: Two-Dimensional Collision Analysis

In the xy-plane, Object 1 (mass = 1.5 kg) moves with velocity $$\vec{v}_1 = (3\hat{i} + 2\hat{j})\ m

Glancing Collision of Billiard Balls

Two billiard balls, each of mass 0.17 kg, undergo an elastic glancing collision. Initially, Ball 1 m

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 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 a cart is recorded by a sensor and is represented by the following graph: the forc

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 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 on a Pendulum Bob

A pendulum bob of mass $$1.0$$ kg is initially at rest hanging from a string. An impulsive force is

Impulse on a Rolling Soccer Ball with Piecewise Force

A soccer ball of mass $$0.43\,kg$$ is rolling and experiences friction that acts in two stages: a co

Impulse-Momentum Theorem with a Non-constant Force

A 0.2 kg hockey puck is struck by a mallet. The force applied by the mallet varies with time and is

Inelastic Collision Energy Loss Analysis

Two carts on a frictionless track undergo a completely inelastic collision. Cart A has a mass of $$1

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

Momentum Analysis of a Variable-Density Moving Rod

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

Momentum and Energy in Elastic Collisions

Two gliders on a frictionless air track undergo an elastic head-on collision. Glider A (mass $$0.8\,

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 for a Two-Block System with External Force

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

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

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

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

Rocket Propulsion: Variable Mass System

A rocket with an initial mass of 500 kg (including fuel) expels gas at a constant exhaust velocity o

Spring-Loaded Collision with Impulsive Force

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

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

Three-Body Collision on a Frictionless Table

Three particles with masses 1 kg, 2 kg, and 3 kg are initially placed along the x-axis at x = 0 m, 4

Two-Dimensional Collision of Ice Skaters

Two ice skaters initially at rest push off from each other on frictionless ice. Skater A (50 kg) mov

Variable Density Rod: Mass and Center of Mass Calculation

A thin rod of length $$L = 2$$ m has a linear density given by $$\lambda(x) = 2 + 3 * x$$ (kg/m), wh

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^{

Acceleration of a Rotating Rigid Body with Frictional Torque

A disk with moment of inertia $$I=2\text{ kg\cdot m}^2$$ experiences a frictional torque proportiona

Analysis of Angular Displacement in a Rotating Disk

In this experiment, several dots are marked along the radius of a rotating disk. The students record

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.

Analysis of Rotational Equilibrium in a Complex System

A hanging sign is suspended by two cables attached at different points. The sign rotates about a piv

Angular Kinematics Analysis Using Graphical Data

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

Angular Kinematics of a Rotating Disk

Consider a rotating disk for which you want to measure angular displacement, angular velocity, and a

Angular Momentum Conservation in Figure Skating

A figure skater spins with an initial angular velocity $$\omega_0$$ and moment of inertia $$I_0$$. W

Angular Momentum Conservation on a Rotating Platform

A child stands on a circular merry-go-round. Initially, the child is at 2.5 m from the center and th

Angular Momentum in Explosive Separation

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

Calculus Analysis of Angular Velocity in a Variable Moment of Inertia System

A figure skater’s moment of inertia changes as she pulls her arms in. Assume her moment of inertia v

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

Conservation of Angular Momentum in a Merry-Go-Round Experiment

In this experiment, a child stands on the edge of a rotating merry-go-round. The child then walks to

Conservation of Angular Momentum in a Merry-Go-Round System

A researcher investigates the conservation of angular momentum in a system consisting of a rotating

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

Derivation of the Moment of Inertia for a Thin Rod

A uniform thin rod of length $$L$$ and mass $$M$$ rotates about an axis perpendicular to the rod thr

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

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

Experimental Investigation of Rolling Without Slipping

An experimental apparatus is used to study rolling without slipping for various cylindrical objects.

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

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

Influence of Friction on Rolling Without Slipping

An experiment investigates the effect of surface friction on rolling objects. The angular velocity o

Investigating the Big Five Equations for Rotational Motion

A researcher is verifying the 'Big Five' equations of rotational kinematics under constant angular a

Mathematical Modeling of Brake Systems

A braking system applies a constant torque of $$\tau = 15 \text{ Nm}$$ on a flywheel with moment of

Moment of Inertia of a Continuous Rod

Consider a uniform thin rod of length $$L$$ and total mass $$M$$. The rod rotates about an axis perp

Optimization of Torque in a Rotating Drill

An engineer is designing a rotating drill system in which a variable force is applied along a lever

Rolling Motion Energy Analysis

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

Rotational Inertia of a Non-Uniform Disk

A disk of radius R has a surface mass density given by $$\sigma(r)= \sigma_0 \left(1+\frac{r}{R}\rig

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

Time-dependent Torque and Angular Momentum Change

A disk is subjected to a time-dependent torque described by $$\tau(t) = 10 * \cos(t)$$ N*m. The disk

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 Rotational Inertia in Engine Mechanisms

You are exploring the behavior of torque in a small engine. Design an experiment to measure the engi

Torsion Pendulum Method for Irregular Objects' Inertia

This experiment uses a torsion pendulum to determine the moment of inertia of irregular objects. The

Calculating Damped SHM Energy Loss

A student records the amplitude decay of a damped oscillator and calculates the energy using $$U=\fr

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 Analysis of Horizontal and Vertical SHM Systems

A researcher compares the oscillations of a mass attached to a spring in horizontal and vertical con

Comparative Analysis: Spring-Mass vs. Pendulum Oscillators

An experiment compares the oscillatory behavior of a spring-mass system and a simple pendulum. Answe

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)=

Coupled Oscillators: Two Springs in Parallel

A block is attached to two springs arranged in parallel with spring constants $$k_1 = 250\,\text{N/m

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 System

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

Damped Oscillations in a Spring-Mass System

In an experiment exploring damped oscillations, a block attached to a spring is set oscillating on a

Data Analysis and Calculus Estimation in SHM

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

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 Equation of Motion Using Calculus

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

Deriving Velocity and Acceleration in SHM

A particle oscillates according to the equation $$y(t) = 0.03\sin(12t + \frac{\pi}{4})$$, where \(t\

Determination of Maximum Elastic Potential Energy

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

Determination of Spring Constant Using SHM Data

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

Determining Oscillation Frequency from Acceleration Data

An accelerometer attached to a mass-spring system records acceleration data during oscillations. The

Determining the Spring Constant from SHM Measurements

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

Differential Equation of Coupled Oscillators

A more advanced experiment involves studying two masses attached by springs (coupled oscillators) to

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

Driven Oscillations and Resonance in a Spring Oscillator

A mass-spring oscillator is subjected to an external sinusoidal driving force given by $$F_d(t)=F_0\

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

Energy Analysis in Simple Harmonic Motion

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

Energy Conservation in a Spring Oscillator

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

Energy Conservation in Vertical Oscillators

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

Energy Transformations in a Mass-Spring System

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

Energy Transformations in a Spring Oscillator

A block attached to a horizontal spring oscillates with amplitude $$A$$. Consider the energy transfo

Energy Transformations in SHM

Consider a block attached to a spring undergoing simple harmonic motion with displacement $$x(t)=A\s

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

FRQ 2: Maximum Speed in SHM

A block of mass $$m = 0.05\ kg$$ oscillates on a spring with a force constant of $$k = 500\ N/m$$ an

FRQ 4: Vertical Oscillations of a Spring-Block System

A block of mass $$m = 1.5\ kg$$ is attached to a vertical spring with force constant $$k = 300\ N/m$

FRQ 8: Energy Transformation in SHM

Consider a mass-spring system undergoing simple harmonic motion where mechanical energy is conserved

FRQ 12: Deriving Velocity and Acceleration Functions

Starting with the position function for a simple harmonic oscillator: $$y = A \sin(\omega t + \phi_0

FRQ 13: Experimental Design for Hooke's Law Verification

Design an experiment to verify Hooke's law using a spring and a set of known masses. Answer the foll

FRQ 16: Maximum Speed in SHM via Energy Methods

Experimental data for a mass–spring oscillator, including amplitude, mass, spring constant, and meas

FRQ6: Calculus Derivation of Velocity and Acceleration in SHM

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

FRQ8: Comparing Spring-Mass and Pendulum Oscillators

Compare two classic oscillatory systems: a horizontal spring-mass oscillator (with restoring force $

Integral Calculation of Work Done in SHM

An experiment is devised to measure the work done on a spring during a complete compression-extensio

Interpretation of a Lab Setup Diagram for a Spring-Mass Oscillator

Examine the provided schematic diagram of a spring-mass oscillator experimental setup. (a) Describe

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 and Calculus Differentiation

A block-spring oscillator is observed to undergo simple harmonic motion. In an experiment, students

Non-linear Effects in Simple Pendulum Motion

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

Pendulum Approximation and Small-Angle Motion

A simple pendulum of length $$L$$ oscillates with small amplitude. Using the small-angle approximati

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

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 Shift Determination in SHM

In an experiment to determine the phase shift of a simple harmonic oscillator, a block attached to a

Phase Space Analysis of SHM

For a mass-spring oscillator exhibiting simple harmonic motion with a solution $$x(t) = A\sin(\omega

Sinusoidal Description and Phase Shift in SHM

A spring oscillator has an amplitude of $$A = 0.05\,m$$ and oscillates with a frequency of $$f = 5.0

Sinusoidal SHM with Phase Shift

An oscillator undergoes simple harmonic motion described by the equation $$y(t) = A\sin(\omega t + \

Superposition and Beats in Oscillatory Motion

Two simple harmonic motions are given by $$y_1(t)=A\,\sin(2\pi f_1 t)$$ and $$y_2(t)=A\,\sin(2\pi f_

Systematic Error Analysis in SHM Experiments

The table below shows measured time intervals and displacements from several trials in an experiment

Transit Time of a Simple Pendulum in Different Gravitational Fields

A simple pendulum with a length of $$L=2.0\,\text{m}$$ oscillates on Earth (with \(g=9.81\,\text{m/s

Vertical Spring Oscillator Investigation

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

Analysis of Gravitational Anomalies: Local Variations in g

Local measurements of gravitational acceleration $$g$$ exhibit small variations due to underlying de

Application of Kepler's Third Law

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

Areal Velocity and Angular Momentum in Planetary Motion

A planet of mass $$m$$ orbits a star while conserving its angular momentum $$L$$. Answer the followi

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

Calculus Analysis of Gravitational Potential Energy

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

Cometary Orbits: Analyzing Highly Eccentric Trajectories

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

Derivation of Gravitational Potential Energy

Starting from Newton's law of gravitation given by $$F(r) = -G * \frac{M * m}{r^2}$$, derive the exp

Designing a Modern Cavendish Experiment

A researcher designs an experiment modeled after the Cavendish torsion balance to determine the grav

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

Energy Conversion in an Asteroid's Elliptical Orbit

A computer simulation is conducted to study the energy conversion in an asteroid's elliptical orbit

Escape Velocity and Energy Conservation

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

Escape Velocity Derivation

A spacecraft of mass m is located on the surface of a planet with mass M and radius R. Using energy

Experimental Analysis of Gravitational Acceleration

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

FRQ 1: Gravitational Force between Two Masses

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

FRQ 7: Escape Velocity Calculation

Escape velocity is the minimum speed required for an object to escape from the gravitational influen

FRQ 10: Gravitational Interactions in a Three-Body System

Consider a simplified system with three masses, $$m_1$$, $$m_2$$, and $$m_3$$, located at fixed posi

FRQ 16: Graphical Analysis of Gravitational Potential Energy vs. Height

The graph provided shows experimental data for gravitational potential energy (in joules) versus hei

Gravitational Field Strength Variation

Derive the gravitational field strength as a function of distance from a point mass and analyze how

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 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 around a planet to gain additional speed. Answe

Impact of Mass Loss on a Comet's Orbit

A comet loses mass due to sublimation as it approaches the Sun. This variable mass affects its orbit

Investigating Tidal Forces and Differential Gravity Effects

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

Laboratory Test of Newton's Law of Gravitation using a Torsion Balance

Newton's Law of Gravitation can be tested in a laboratory setting using sensitive apparatus such as

Mathematical Modeling of Tidal Forces

Using the provided data on tidal forces measured at different distances, analyze how the tidal force

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 Transfer Trajectories and Hohmann Transfers

A spacecraft in low Earth orbit (LEO) is planning a Hohmann transfer to reach a geosynchronous orbit

Predicting Orbital Decay Due to Atmospheric Drag

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

Satellite Orbital Decay with Atmospheric Drag Consideration

An experiment is designed to measure the decay of a satellite's orbit by tracking its altitude over

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

Stability Analysis of a Satellite in Low Earth Orbit

A satellite is in a circular low Earth orbit at an altitude of $$h = 400 \ \text{km}$$. Answer the f

Tidal Forces in Gravitational Fields

An extended object in a gravitational field experiences a differential force across its length, know

Variable Gravitational Field and Calculated Potential Energy

Consider an object moving radially in a gravitational field described by $$F(r) = -\frac{G * M * m}{

Variation of Gravitational Force with Distance

Consider the gravitational force given by $$F(r) = \frac{G m_1 m_2}{r^2}$$. Answer the following par