AP Physics C: Mechanics FRQ Room

Analysis of a Ballistic Trajectory with Inaccurate Symmetry Assumption

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

Analysis of 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 a Two-Dimensional Collision

Two objects undergo a collision in a plane. Object A (mass m_A) experiences a force during the colli

Calculus-Based Kinematics Derivation

Consider an object moving along a straight line with constant acceleration. Use calculus to derive e

Car Acceleration on a Highway: Calculus Approach

A car's position along a straight highway is given by $$x(t)= 2*t^3 - 6*t^2 + 4*t$$, where $$x$$ is

Centripetal Acceleration in Circular Motion

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

Combined Translational and Rotational Motion Experiment

Design an experiment to study an object that exhibits both translational and rotational motion as it

Comparative Analysis of Kinematic Equations

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

Comparing Theoretical and Experimental Data in Uniform Acceleration

An experiment measures the velocity of an object under uniform acceleration, and the following table

Designing a Kinematics Lab Experiment

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

Determining Instantaneous Acceleration from a Displacement Graph

An experiment recorded the displacement of an object as a function of time using a high-precision se

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

Distance vs. Displacement Analysis in One-Dimensional Motion

An experiment recorded the motion of a car along a straight road where its distance traveled and dis

Evaluating Non-Uniform Acceleration from Experimental Data

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

Experimental Determination of g

In a free-fall experiment, a student fits the data to obtain the position function $$h(t)= 4.9*t^2$$

Free Fall under Gravity

A rock is dropped from the top of a 100-meter tall building. Neglect air resistance.

Free-Fall Motion Analysis

A rock is dropped (from rest) from the top of an 80 m cliff. Assume that the acceleration due to gra

FRQ 1: Calculus and One-Dimensional Kinematics (EASY)

An object's position is given by $$x(t)=\sin(t)$$. Answer the following: (a) Differentiate $$x(t)$$

FRQ 3: Displacement Data Analysis from a Position-Time Table

The table below provides the position (in meters) of an object at various times (in seconds): | Tim

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 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 8: Projectile Motion – Targeting a Moving Object

A researcher is tasked with designing a projectile launch system that accurately targets an object l

FRQ 9: Piecewise Acceleration Motion (HARD)

An object moves along a straight line with acceleration defined piecewise as follows: For $$0 \le t

FRQ 13: Average Speed vs. Average Velocity Analysis (EASY)

An object's position along the x-axis is given by $$x(t)=t^2-4*t+3$$ (in m) for $$0 \le t \le 5\,s$$

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

FRQ 19: Variable Acceleration – An Object Under Changing Forces

A researcher studies an object whose acceleration varies with time according to the function $$ a(t)

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

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

Impulse and Momentum with a Time-Dependent Force

A baseball (mass m = 0.145 kg) is struck by a bat. The force exerted by the bat is given by $$F(t)=

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

Motion in One Dimension: Variable Acceleration

An object moves along the x-axis with an acceleration given by $$a(t) = 3*t$$ (in m/s²). At time $$t

Motion on an Inclined Plane with Friction

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

Multi-Phase Vehicle Motion

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

Non-linear Position Function Analysis

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

Pendulum Motion and Kinematics

A pendulum of length 2 m oscillates with small amplitude. Its angular displacement is given by $$θ(t

Projectile Motion using Calculus

A projectile is launched from ground level at an angle of 30° with an initial speed of 40 m/s (negle

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

Terminal Velocity in Free Fall

Design an experiment to determine the terminal velocity of an object in free fall within a fluid med

Uniformly Accelerated Free Fall Analysis

In a free fall experiment, a rock is dropped from a height of 80 m. The following table presents mea

Uniformly Accelerated Motion on a Track

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

Variable Acceleration Analysis

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

Variable Acceleration Analysis Using Calculus

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

Work and Energy in Linear Motion

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

Analysis of Mechanical Advantage and Work in a Lever System

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

Ballistic Kinetic Energy with Air Resistance

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

Bouncing Ball Energy Loss Experiment

A ball is dropped from a known height and allowed to bounce repeatedly. The experiment calculates en

Calculus-based Integration of Work over a Variable Force

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 5*x$$ N. The particle

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 a Roller Coaster

A 500 kg roller coaster car is released from rest at the top of a hill 50 m above the bottom of a di

Derivation of the Work-Energy Theorem

Using calculus, derive the work–energy theorem starting from the definition of work and Newton's sec

Energy Analysis of a Bouncing Ball: Energy Loss and Power

A ball of mass $$m = 0.5 \;\text{kg}$$ is dropped from a height of $$10 \;\text{m}$$ and, after its

Energy Conservation 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 a Spring–Mass System

In this experiment, a 0.5 kg mass is attached to a vertical spring with a spring constant of 200 N/m

Energy Dissipation in an Oscillatory System

Consider a mass-spring oscillator with mass 1 kg and spring constant $$ k = 100 \;N/m $$, oscillatin

Energy in a Spring–Mass System

A mass is attached to a spring with a spring constant of $$k = 200\,N/m$$. The spring is compressed

Energy Transfer in a Bouncing Ball

A ball of mass 0.5 kg is dropped from a height of 10 m and, after hitting the ground, rebounds to a

Free‐Fall Impact Energy Experiment

In this experiment, a small cart is dropped from a known height and allowed to free-fall until it im

FRQ 3: Kinetic Energy Measurement in Free Fall

A researcher presents data claiming that objects dropped from rest convert all gravitational potenti

FRQ 5: Assessing the Independence of Power Output from Time Interval

A magazine article claims that two engines delivering the same work are equally powerful, regardless

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: Interpreting Drop Test Kinetic and Potential Energy Data

A study provides experimental data for a 3 kg ball dropped from various heights, with the measured s

FRQ 19: Equilibrium Points from a Nonlinear Potential Energy Function

A report presents the nonlinear potential energy function $$U(x) = (x - 2)^2 - (2 * x - 3)^3$$ and c

Instantaneous and Average Power of a Rocket Engine

A rocket engine produces a time-dependent force given by $$F(t) = 1000 + 200 * t$$ (N) for t in the

Kinetic Energy Gain in a Roller Coaster Ride

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

Measuring Work Done against Air Resistance in Free Fall

A researcher studies the motion of a 1 kg object falling under gravity with air resistance. The drag

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

Relationship Between Force and Potential Energy

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

Rotational Power in Gear Systems

An experiment measures the power output of a gear train by recording the torque and angular velocity

Variable Friction and Kinetic Energy Loss

A 5 kg block slides across a horizontal surface and comes to rest. The frictional force acting on th

Variable Mass Rocket Energy Analysis

A rocket with an initial mass of 1500 kg expels fuel and decreases its mass linearly to 1200 kg over

Vertical Lift Work Measurement Experiment

In this vertical lift experiment, an object is raised by a motor and its applied force and displacem

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 by Time-Dependent Force on a Car

A car of mass 1200 kg is subjected to a net force along a straight road given by $$F(t)=2000-50*t$$

Work Done by a Variable Force

An object is acted upon by a variable force given by $$F(x)=5\,x^2$$ (in newtons) along the x-axis.

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

Astronaut Recoil upon Throwing an Object

An astronaut of mass 90 kg, floating in space, throws a 1.5 kg tool directly away from her ship at 5

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

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 in a Coupled Mass-Spring System

Two masses (m1 = 1.5 kg and m2 = 1.5 kg) are connected by a light spring on a frictionless track. In

Center of Mass of a Nonuniform Circular Disk

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

Center of Mass of a System of Particles in 3D Space

Three point masses are located in 3D space with the following properties: - Mass 1: 2 kg at coordin

Composite Object: Rod with Attached Sphere

A uniform rod of length 1 m and mass 2 kg has a small sphere of mass 1 kg attached to its right end.

Conservation of 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

Determination of Collision Time from Impulse Data

In a crash-test experiment, the force on a car during impact is modeled by the equation $$F(t) = 100

Displacement from Variable Acceleration

A 2 kg particle experiences a force given by $$F(t)=10*t$$ N over the time interval from 0 s to 4 s.

Elastic Collision of Air Track Gliders

On a frictionless air track, Glider A (mass = 0.8 kg) moves to the right at 2.0 m/s while Glider B (

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: Investigating Collision Elasticity

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

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 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 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 14: Derivation of the Continuous Center of Mass Formula

Consider a one-dimensional object with a continuous mass distribution described by the density funct

FRQ 15: Center of Mass versus Center of Gravity

A popular science article claims that for a complex-shaped satellite orbiting Earth, the center of m

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

Glider Collision on an Air Track

Two gliders on a frictionless air track collide and stick together. Glider A (mass = $$0.50\,\text{k

Impulse and Center of Mass in a Soccer Kick

A soccer ball (mass $$0.45\,kg$$) initially at rest is kicked, resulting in a launch speed of $$25\,

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 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 Variable Thrust Rocket

A small model rocket experiences a thrust that varies with time as $$F(t)=50 - 10*t$$ (N) for $$0 \l

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 Coupled Freight Cars

Two freight cars are coupled and moving on a frictionless track at an initial speed of $$10\,m/s$$.

Impulse with Resistive Force

A 2-kg block on a frictionless surface is subjected to two forces simultaneously over a time interva

Impulsive Collision Involving Rotation

A 0.5 kg ball traveling at $$8\,m/s$$ horizontally strikes the end of a thin uniform rod of length $

Inelastic Collision Analysis with Rolling Carts

In a collision experiment, two carts on a frictionless track collide and their velocities are record

Inelastic Collision of a Pendulum Bob with a Block

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

Inelastic Collision: Combined Motion

A 0.6 kg ball moving at 4.0 m/s collides head-on with a 0.4 kg ball that is initially at rest. The b

Integrated Analysis of Momentum and Center of Mass in a Multi-Stage Experiment

In a complex experiment, a projectile traveling at 10 m/s with a mass of 5.0 kg breaks apart mid-fli

Mobile Robot Center of Gravity Analysis

A mobile robot features extendable arms that modify its mass distribution. With its arms retracted,

Motion of the Center of Mass under External Force

Consider a two-particle system consisting of masses $$m_1 = 4\,kg$$ and $$m_2 = 6\,kg$$. An external

Multi-Dimensional Inelastic Collision Analysis

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

Non-conservative Forces: Block on an Incline with Friction

A 2 kg block slides down a 30° incline that is 5 m long. The coefficient of kinetic friction between

Projectile Center-of-Mass Trajectory

A projectile is launched and its trajectory is recorded with emphasis on the motion of its center of

Projectile Motion with Drag Impulse Analysis

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

Rigid Body Dynamics: Torque and Rotation

A uniform meter stick (mass = 0.3 kg, length = 1 m) is pivoted at one end. A perpendicular force is

Rocket Propulsion and the Tsiolkovsky Rocket Equation

A rocket expels mass continuously and can be modeled as a variable mass system. Starting with a mass

Rotating Collision: Linear and Angular Momentum

A uniform disc of mass $$2$$ kg and radius $$0.5$$ m rotates with an angular velocity of $$10$$ rad/

Stability of a Suspended Mobile

A suspended mobile consists of three masses hanging from strings attached to a horizontal bar. The m

Tethered Satellites: Center of Mass and Thruster Impulse

Two satellites are connected by a 10-m long tether in space. Satellite A has a mass of 800 kg and Sa

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

Torque and Angular Motion of a Rigid Beam

A non-uniform beam of length 2 m is pivoted at one end. Its mass distribution is given by $$\lambda(

Two-Dimensional Collision Analysis

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

Two-Dimensional Collision and Momentum Conservation

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

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

Analyzing Rotational Equilibrium

A researcher is investigating conditions for rotational equilibrium in a beam subject to multiple fo

Angular Kinematics from Disk Data

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

Angular Kinematics from Experimental Data

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

Angular Kinematics of a Rotating Disk

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

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.

Calculus Derivation of the Moment of Inertia for a Disk

Using the integral formula $$I = \int r^2\,dm$$, derive the moment of inertia for a uniform circular

Calculus-Based Analysis of Angular Impulse

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

Centripetal Force and Angular Velocity Measurement

Design an experiment to measure the centripetal force acting on an object in circular motion and rel

Comparative Calculations for a Composite System

Consider a system of three beads, each of mass $$m$$, arranged along a rod of negligible mass and le

Comparative Study of Rotational Kinetic Energy in Different Shapes

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

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

Conservation of Angular Momentum in Rotational Collisions

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

Dynamic Stability of a Rotating Space Station

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

Energy Conservation in Rotational Motion

A hollow sphere of mass $$m = 5\,kg$$ and radius $$R = 0.3\,m$$ rolls without slipping down an incli

Energy Conversion in a Rolling Cylinder Experiment

A cylinder rolls without slipping down an inclined plane. The experiment examines how gravitational

Energy 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 =

Energy Transfer in Rolling Objects

Design an experiment to study the energy conversion in a rolling object down an incline, by measurin

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

Equilibrium Analysis in Rotational Systems

A seesaw, modeled as a uniform beam 4 m long pivoted exactly at its center, is in rotational equilib

Experimental Data: Angular Velocity vs Time Analysis

An experiment records the angular velocity of a rotating object over time. The provided graph shows

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 6: Angular Momentum Conservation on a Rotating Platform

A 50.0 kg person stands on a frictionless rotating platform that initially has a moment of inertia o

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 20: Time-Dependent Angular Acceleration with External Torque

A flywheel with moment of inertia \(I = 3.00\,kg\cdot m^2\) experiences an exponentially decaying ex

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 Continuous Rod

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

Moment of Inertia of a Hollow Cylinder with Thickness

Derive an expression for the moment of inertia of a hollow cylinder with inner radius $$R_1$$, outer

Parallel Axis Theorem in Rotational Systems

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

Parallel Axis Theorem: Composite Body Moment of Inertia

Consider a composite object consisting of a solid disk (mass $$M = 4.0 \text{ kg}$$, radius $$R = 0.

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 Energy Analysis

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

Rolling Motion: Energy Partition Analysis on an Inclined Plane

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

Rotational Inertia Measurement via Pulley Apparatus

A student sets up an experiment to measure the moment of inertia of a uniform disk using a pulley sy

Rotational Inertia of a Hollow Cylinder vs. a Solid Cylinder

Compare the rotational inertias of a solid cylinder and a thin-walled hollow cylinder, both of mass

Rotational Kinematics: Angular Displacement via Integration

A motor provides an angular acceleration given by $$\alpha(t) = 4 - 0.5*t$$ (in rad/s²) for $$0 \le

Torque and Angular Acceleration Relationship

An experiment measures the response of a rotating object to different applied torques. A graph is pl

Torque and Rotational Inertia: Uniform Rod

A uniform rod of length $$L = 2.0 \text{ m}$$ is pivoted about one end. A force of $$F = 10 \text{ N

Torque in a Multi-force System: Seesaw Equilibrium

A seesaw of length $$L = 3.0 \text{ m}$$ is balanced on a fulcrum located 1.0 m from the left end. T

Torque Measurement and Analysis

A recent experimental study claims that the relationship between force and torque is strictly linear

Torque on a Lever Arm

A lever arm is used to apply force at an angle. Consider a force of $$50*N$$ applied at an angle of

Torsion Pendulum and Restoring Torque Error

In a torsion pendulum experiment, a disk is suspended from a wire and its angular displacement due t

Torsion Pendulum Method for Irregular Objects' Inertia

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

Amplitude Dependence in a Nonlinear Oscillator

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

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

Analyzing Damped Oscillations in a Spring-Mass System

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

Calculus Approach to Energy Dissipation in a Damped Oscillator

Consider a damped oscillator described by the differential equation $$m\frac{d^2y}{dt^2} + b\frac{dy

Calculus-Derived Velocity and Acceleration in SHM

For a simple harmonic oscillator described by $$x(t)=A\sin(\omega t+\phi)$$, determine its velocity

Comparative Analysis: Energy Methods vs. Force Methods in SHM

In analyzing simple harmonic motion (SHM), two common approaches are the energy conservation method

Comparative Analysis: Spring-Mass System vs. Pendulum Oscillations

A researcher compares the oscillatory behavior of a horizontal spring-mass system with that of a sim

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

Conservation of Energy: Integral Approach in SHM

Utilize calculus to analyze energy conservation in a simple harmonic oscillator.

Coupled Oscillators: Normal Modes Analysis

Consider a system of two identical masses \(m\) placed on a frictionless surface and connected by th

Damped Harmonic Oscillator Dynamics

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

Damped Oscillations: Amplitude Decay Analysis

A mass-spring oscillator with $$m = 0.2\,kg$$ and damping constant $$b = 0.1\,kg/s$$ experiences dam

Damped Oscillations: Determining the Damping Coefficient

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

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

Derivation and Solution of the Differential Equation for SHM

Starting from Newton's second law, derive the differential equation governing the motion of a spring

Deriving Equations for a Damped Harmonic Oscillator

An experiment is designed to study the effects of damping in a spring-mass oscillator. This version

Deriving Velocity and Acceleration in SHM

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

Designing an SHM Experiment with Error Analysis

A researcher intends to study the simple harmonic motion of a pendulum using an optical sensor to re

Determination of Spring Constant from Oscillation Data

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

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 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

Energy Conservation in Vertical Spring Oscillations

A 1.5 kg block is attached to a vertical spring with force constant $$k = 300\,N/m$$. After reaching

Energy Conservation Verification Using Calculus

A mass-spring system oscillates with a displacement given by $$x(t) = 0.06 \cos(15t)$$. (a) Derive t

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 8: Energy Transformation in SHM

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

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 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

FRQ 17: Pendulum Nonlinear Effects

Consider a simple pendulum with length $$L = 1\ m$$. While for small angular displacements the perio

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

FRQ9: Energy Exchanges in a Mass-Spring Oscillator

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

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}}.

Investigation of Energy Conservation in SHM Using Calculus

A researcher analyses a mass-spring oscillator with its displacement given by $$y(t) = 0.1 * \cos(15

Kinematics of SHM: Period and Frequency Measurements

Analyze the kinematics of a simple harmonic oscillator using time measurements.

Nonlinear Characteristics of the Simple Pendulum

The standard formula for the period of a simple pendulum, \(T \approx 2\pi\sqrt{\frac{L}{g}}\), reli

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 on a Rotating Platform

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

Pendulum Oscillations: Small Angle Approximation

A simple pendulum of length $$L=1.5\,\text{m}$$ is set into small-amplitude oscillations. (a) Derive

Period Estimation Using Calculus in Simple Pendulum Experiments

An experimental study reports that integrating the motion equations of a simple pendulum leads to pe

Phase Shift Determination in SHM

A researcher studies a mass-spring oscillator and observes that at time $$t = 0$$ the block is at it

Phase Space Analysis of SHM

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

Resonance and Energy Amplification in Oscillatory Systems

In a driven, damped oscillator, the amplitude as a function of the driving frequency is given by $$

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 Analysis of SHM with Phase Shift

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

Vertical Oscillations: Energy and Force Analysis

Consider a block attached to a vertical spring. Analyze the system from both the force and energy pe

Vertical Spring Oscillator Investigation

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

Analysis of Tidal Forces Acting on an Orbiting Satellite

A researcher studies the tidal forces acting on a satellite orbiting a massive planet. Due to the fi

Analyzing Tidal Forces in a Two-Body System

Explain the origin of tidal forces in a gravitational two-body system and derive their expression us

Comparing Circular and Elliptical Orbits in a Lab Simulation

Different orbital shapes exhibit differing energy distributions. Answer the following: (a) A simula

Derivation of Escape Velocity from Earth's Surface Using Calculus

Using the principle of energy conservation and calculus, derive the expression for the escape veloci

Determination of Gravitational Parameter (GM) from Satellite Orbits

An observational study is undertaken to determine the gravitational parameter (GM) of a planet by an

Effective Gravitational Field on an Irregular Asteroid

An irregularly shaped asteroid has a non-uniform density distribution. To determine the effective gr

Effects of Non-Spherical Mass Distribution on Satellite Orbits

A planet is not perfectly spherical but exhibits a slight oblateness, introducing a perturbative ter

Elliptical Orbits and Eccentricity Calculation

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

Energy Conservation in a Swinging Mass Experiment

An experiment is performed wherein a mass is swung in a vertical circle to investigate conservation

FRQ 8: Elliptical Orbit – Perihelion and Aphelion Distances

An object travels in an elliptical orbit with a semimajor axis $$a$$ and eccentricity $$e$$. Answer

FRQ 19: Relativistic Corrections and Perihelion Precession

General relativity provides corrections to Newtonian gravity that can explain the observed perihelio

Gravitational Field Modeling for Extended Bodies

Compare the gravitational field produced by an extended, spherically symmetric body to that of a poi

Gravitational Field of a Uniform Ring

A researcher is investigating the gravitational field created by a thin uniform ring of mass $$M$$ a

Gravitational Force Calculation on a Satellite

A satellite with a mass of $$m = 500\,kg$$ orbits the Earth at a distance of $$r = 7.0 * 10^6\,m$$ (

Gravitational Parameter in Exoplanetary Systems

Using the provided exoplanetary data, analyze the consistency of the gravitational parameter (derive

Gravitational Potential Energy Measurement on a Roller Coaster

An amusement park ride features a roller coaster that reaches a maximum height of $$50 \;m$$ above t

Gravitational Slingshot and Energy Gain

A spacecraft uses a gravitational slingshot maneuver around a planet to gain additional speed. (a)

Gravitational Slingshot Maneuver Analysis

Gravitational slingshot (or flyby) maneuvers are used in space missions to change a spacecraft's vel

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

Kepler's Third Law and Orbital Analysis

A recent media report claims that the orbital period $$T$$ and the semi‐major axis $$a$$ of satellit

Newton's Law in Binary Star Systems

Using the provided data for a binary star system, analyze Newton's Law of Gravitation to determine t

Non-uniform Gravitational Fields in Planetary Interiors

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

Optimization of Orbital Maneuvers in Multi-Stage Rockets

A multi-stage rocket performs orbital maneuvers to transition from a low Earth orbit to a higher geo

Orbital Dynamics: Gravitational Force Variation

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

Role of Eccentricity in Orbital Dynamics

Orbital eccentricity (e) quantifies how much an orbit deviates from a circle. (a) Define orbital ec

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

Work Done in a Variable Gravitational Field

An object of mass $$m$$ is moved radially from a distance $$r_1$$ to $$r_2$$ in the gravitational fi