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 Signal in a Laboratory Experiment

In a laboratory experiment, the velocity of a moving cart is found to follow $$v(t)= 5 + 2*\cos(0.5*

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

Calculus-Based Analysis of a Car’s Accelerating Motion

A car traveling along a straight road accelerates from rest with an acceleration given by $$a(t)=2*t

Calculus-Based Kinematics Derivation

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

Coupled Motion: Translation and Rotation

A solid cylinder with mass $$m = 2.0$$ kg and radius $$R = 0.5$$ m is released from rest at the top

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

Determining Launch Angle from Experimental Data

A projectile is launched with an unknown initial speed and angle. In an experiment, the total flight

Experimental Evaluation of Vector Addition in Two Dimensions

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

FRQ 8: Circular Motion Kinematics (MEDIUM)

An object moves in uniform circular motion with its position given by $$\vec{r}(t)=(R\cos(\omega*t),

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 13: Comparative Analysis of Two Free Fall Experiments

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

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 15: Differentiation of a Cubic Displacement Function (EASY)

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

FRQ 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

Graphical Analysis of Distance and Displacement

A student records the position of a runner during a 10-second race, resulting in the following table

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

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 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

A block is released from rest on a frictionless incline with an angle of $$30^\circ$$. The accelerat

Motion with Time-Varying Acceleration (Drag Force Approximation)

An object in free fall experiences a time-dependent acceleration due to air resistance approximated

Oscillating Particle under Uniform Acceleration

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

Pendulum Motion and Kinematics

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

Projectile Motion from a Cliff

An object is launched from the edge of a 20-meter high cliff with an initial speed of 40 m/s at an a

Projectile Motion on an Inclined Plane

A ball is launched with an initial speed of $$30\,m/s$$ at an angle of $$40^\circ$$ above the horizo

Projectile Motion Revisited: Maximum Height and Impact Velocity

An object is projected vertically upward from ground level with an initial speed of $$50\,m/s$$. Ass

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

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 Analysis of Two Moving Objects

Two objects move along a straight track with positions given by $$x_A(t)= 3*t^2$$ and $$x_B(t)= 6*t

Relative Motion: Meeting of Two Objects

Two objects move along a straight line. Object A starts at x = 0 m with a constant velocity of 10 m/

Simultaneous Measurement of Velocity and Acceleration

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

Sinusoidal Motion

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

Variable Acceleration: Analyzing a Non-Uniformly Accelerated Motion

An object moves along a straight path with an acceleration given by $$a(t)=\frac{12}{(t+2)^2}$$ 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

Verification of Uniformly Accelerated Motion

A student conducts an experiment on a frictionless track using a cart. The student hypothesizes that

Analysis of Fall Dynamics with Air Resistance

An object with a mass of 0.5 kg is dropped from a height of 50 m. Air resistance is modeled as a dra

Calculus Analysis of a Ramp System

A 10 kg block is pushed up a frictionless ramp by an applied force given by $$F(x)=50 - 4\,x$$ (in n

Circular Motion with Tangential Work

An object is moving along a circular path of radius 3 m. While the centripetal force (directed towar

Composite System: Roller Coaster Energy Analysis

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

Compound Machine Energy Analysis Experiment

A compound machine consisting of a lever and a pulley is used to lift a load, and the work input is

Efficiency Analysis of a Mechanical System

A motor lifts a 100 kg mass by raising it 10 m in 20 seconds, using an electrical energy input of 15

Efficiency in Energy Conversion

A machine is used to convert electrical energy into mechanical work. It receives a constant electric

Energy Dissipation in Damped Oscillations

A damped harmonic oscillator consists of a 1 kg mass attached to a spring (k = 50 N/m) with a dampin

Energy Loss in a Ball with Air Resistance

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

Experiment on Electric Motor Power Output

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

Explosive Separation and Energy Distribution

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

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: Work Done on a Variable Mass System

A recent claim suggests that for systems with variable mass (e.g., rockets), the work done by a non-

FRQ 17: Energy Loss Analysis in a Frictional Pendulum

A pendulum bob with a mass of 0.8 kg is released from an initial height corresponding to a potential

FRQ 19: Analysis of Force–Time Data in a Crash Test

During a crash test, the force experienced by a dummy is recorded as a function of time. The force–t

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

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

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

Kinetic Energy Change Under a Variable Force

A 2-kg object is subjected to a variable force along a horizontal path given by $$F(x)= 4 + 0.2*x \;

Kinetic Energy Measurement in a Projectile Experiment

A researcher is studying the change in kinetic energy of a small projectile. A ball of mass $$m = 0.

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

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

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 Output from a Variable Force: Time-Dependent Problem

A particle is subjected to a time-dependent force given by $$F(t)= 5 \;\cos(0.5*t) \; (\text{N})$$.

Rotational Dynamics and Work-Energy in a Disk

A solid disk of mass 10 kg and radius 0.5 m starts from rest. A constant torque of 5 N·m is applied

Rotational Work and Energy Experiment

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

Solar Energy Mechanical Conversion Experiment

In a lab setup, a solar panel powers an electric motor that lifts a 10 kg mass vertically by 3 m at

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 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 Constant and Variable Forces

An object is acted upon by two different types of forces on separate occasions. In Part (a), a const

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

Angular Impulse and Rotation

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

Center of Mass Analysis in a Two-Mass Pulley System

In a two-mass pulley system, students aim to determine the center-of-mass motion by measuring accele

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 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 Triangular Plate

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

Center of Mass of a Variable Density Rod

A rod of length $$L = 1\,\text{m}$$ has a linear density given by $$\lambda(x) = 5 + 3*x$$ (in kg/m)

Center-of-Mass Motion Under an External Force

Consider a system composed of two masses, $$m_1=2.0\,kg$$ and $$m_2=3.0\,kg$$, connected by a light,

Collision with a Variable Coefficient of Restitution

Two carts on a frictionless track collide head-on. Cart A has a mass of 3 kg and Cart B a mass of 4

Complex Rotational and Translational Collision Involving Center of Mass

A uniform rod of length $$2$$ m and mass $$4$$ kg is pivoted frictionlessly about its center. A smal

Composite Body Center of Mass Calculation

A composite system consists of a uniform rectangular block (mass $$5\,kg$$, width $$0.4\,m$$) and 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

Data Analysis: Testing the Linear Relationship between Impulse and Momentum Change

An experiment is conducted where a cart is subjected to several impacts. For each trial, the measure

Derivation of the Rocket Equation Using Momentum Conservation

A rocket moving in space expels mass continuously. Assume that during an infinitesimal time interval

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 (

Experimental Design: Measuring Impulse with Force Sensors

Propose an experiment to measure the impulse delivered by a non-uniform force applied to a ball. You

Explosive Fragmentation: Momentum Transfer

A stationary object with a mass of 12 kg explodes into three fragments in a frictionless environment

Football Kick: Impulse and Average Force

A 0.4 kg football is punted and achieves a launch speed of 30 m/s as a result of a kick delivered ov

FRQ 6: Elastic Collision Analysis

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

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 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 Work: Discerning Differences

A particle of mass 3 kg is subjected to a force that varies with position according to $$F(x)=12*x^2

Impulse Calculation from Force-Time Graph

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

Impulse Delivered by 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 during a Controlled Fall onto an Airbag

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

Impulse from a Time-Varying Force with Graph Stimulus

A force sensor records the force applied to a hockey puck as a function of time while a player strik

Impulse from Force Sensor Data

In a collision experiment, a force sensor attached to a small car records the force applied during i

Impulse from Force-Time Graph

A soccer ball (mass = 0.43 kg) is kicked, and the force exerted by the kicker’s foot varies with tim

Impulse in a Non-Constant Force Field

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

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: Bullet-Block Interaction

A 0.02-kg bullet is fired at 400 m/s into a stationary 2-kg wooden block on a frictionless surface.

Momentum Analysis in an Asteroid Breakup

An asteroid of total mass 1000 kg fragments into two pieces in deep space. Fragment A (mass unknown)

Momentum and Energy in Elastic Collisions

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

Momentum Transfer in Off-Center Collisions on a Frictionless Track

In an experiment, a moving cart collides off-center with a stationary cart on a frictionless track,

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

Rotational Impulse and Angular Momentum

A rigid disc of mass 4 kg and radius 0.5 m rotates about its central axis with an initial angular sp

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

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

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 a Variable Moment of Inertia System

A rotating disk has a moment of inertia that decreases over time as its arms retract, following the

Analyzing Rotational Equilibrium

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

Angular Kinematics on a Rotating Platform

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

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 in Rotational Collisions

In this experiment, two disks with different moments of inertia and angular velocities are coupled t

Angular Momentum in a Variable Moment of Inertia System

A researcher examines the dynamics of a rotating system whose moment of inertia changes with time du

Angular Momentum Transfer in Coupled Rotating Disks

In an experiment, two disks are coupled so that they eventually rotate together without any external

Application of the Parallel Axis Theorem

An object with mass 5 kg has a moment of inertia about its center of mass $$I_{cm} = 0.2\,kg\,m^2$$.

Composite Object Rotational Dynamics Analysis

A researcher studies the rotational dynamics of a composite object composed of a uniform disk of mas

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

Conservation of Angular Momentum in Explosive Separation

A rotating disk of mass $$M = 5.0 \text{ kg}$$ and radius $$R = 0.5 \text{ m}$$, spinning at $$\omeg

Conservation of Angular Momentum in Rotational Collisions

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

Correlation Between Torque and Rotational Energy via Calculus

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

Designing a Rotational Experiment Using a Pulley System

A researcher designs an experiment to measure the rotational inertia of a pulley using a falling mas

Designing a Rotational Sensor

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

Designing a Rotational System with Specified Kinetic Energy

A researcher is tasked with designing a rotational system that must store a specified amount of kine

Determining the Effect of Friction on Rotational Motion

A flywheel with moment of inertia $$I = 1.0\,kg\,m^2$$ is initially spinning at an angular velocity

Effect of Changing Moment Arm on System Dynamics

Design an experiment where you systematically vary the moment arm (the distance from the pivot) in a

Effect of Force Angle on Measured Torque

An experiment is performed in which a force of constant magnitude $$F = 50\,N$$ is applied at a cons

Effect of Variable Applied Torque on Angular Acceleration

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

Energy Considerations in a Rotating Pendulum

A physical pendulum consisting of a rigid body with moment of inertia $$I$$ rotates about a pivot. T

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

Evaluating the Impact of Frictional Torque on Rotational Motion

A researcher studies how a constant frictional torque affects the rotational motion of a spinning ob

FRQ 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 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 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

Impact of Changing Radius on Rotational Motion

A rotating disk experiences a change in its effective radius from $$R_1$$ to $$R_2$$ due to deformat

Inelastic Collision of Rotating Disks

Two disks mounted on a common frictionless axle collide and stick together. Disk A has a moment of i

Investigating Non-uniform Density Effects on Moment of Inertia

Design an experiment to determine the moment of inertia for an object with a non-uniform mass distri

Mass Redistribution and Kinetic Energy in Rotating Systems

In a rotating system, a person on a rotating platform moves closer to the axis, reducing the system’

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

Non-Uniform Angular Acceleration

A disk has an angular acceleration described by the function $$\alpha(t) = 2 * t$$ (in rad/s²), and

Parallel Axis Theorem: Composite Body Moment of Inertia

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

Relation Between Linear and Angular Velocity on a Rotating Disk

In an experiment, a rotating disk is used to measure the linear speed of points located at different

Rotational Dynamics in a Non-Inertial Frame

In a rotating frame (such as on a merry-go-round), fictitious forces arise. Consider a situation whe

Rotational Dynamics of a Gyroscope

A gyroscope with a moment of inertia of $$I = 0.05\,kg\,m^2$$ is spinning with a spin angular veloci

Rotational Equilibrium Analysis of a Beam

A beam is in static equilibrium under the influence of several forces applied at different distances

Rotational Inertia Determination Using a Torsion Pendulum

You are provided with a torsion pendulum apparatus consisting of a rod suspended by a wire with a kn

Rotational Kinematics from Angular Velocity Graph

A rotating object's angular velocity increases linearly with time. The graph provided shows that $$\

Rotational Kinematics: Non-Uniform Angular Acceleration of a Disk

A disk rotates such that its angular velocity is given by $$\omega(t) = 3*t^2 - 2*t + 1$$ (in rad/s)

Rotational Kinetic Energy Storage in a Flywheel

An engineer is tasked with designing a flywheel energy storage system. The flywheel is required to s

Stability Analysis of a Block on a Rotating Platform (Tipping Experiment)

A block is placed on a rotating platform, and the conditions under which the block tips are investig

Time-Dependent Rotational Kinematics

A researcher is examining a system where the angular acceleration is not constant but depends linear

Time-varying Angular Acceleration in a Rotational System

A disk experiences an angular acceleration described by $$\alpha(t)=5\sin(2t) \text{ rad/s}^2$$.

Torque and the Right-Hand Rule Verification Experiment

Design an experiment to verify the direction of the torque vector as predicted by the right-hand rul

Torque Equilibrium in a Beam

A horizontal beam is pivoted at one end. On the beam, a force F1 = 100 N is applied 0.8 m from the p

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

Torsional Oscillator Analysis

A torsional pendulum consists of a disk suspended by a wire with torsion constant $$k$$. The system

Amplitude Decay in Damped Oscillations

A damped oscillator has its displacement described by the function $$y(t)=A_0*e^{-\frac{b}{2*m}t}*\c

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 $$y(t)= (0.05 + 0.01 * t) * \sin(8*t+0.2)$$

Calculating Damped SHM Energy Loss

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

Calculus Approach to Maximum Velocity in SHM

Consider an oscillator whose displacement is given by the sinusoidal function $$y(t) = A \sin(\omega

Comparative Analysis: Horizontal vs. Vertical Oscillations

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

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

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

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

Data Analysis from a Virtual SHM Experiment

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

Data Analysis of a Spring-Mass Experiment

A researcher experiments with a mass-spring system and records the period of oscillation for differe

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

Derivation of SHM Equations Using Calculus

Starting with Newton’s second law and Hooke’s law for a mass-spring system, derive the differential

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 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 the Phase Constant from Experimental Data

An experiment measuring the displacement of a simple harmonic oscillator produced the following data

Determining the Spring Constant from Oscillation Data

A student conducts an experiment to determine the spring constant $$k$$ of a spring by measuring 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

Differentiation of Sinusoidal Motion

Given the position function of a mass-spring oscillator as $$x(t) = 0.08\,\text{m} \sin(10\,t + 0.2)

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\

Effect of Amplitude on the Period of an Oscillator

An experiment is conducted to investigate if the period of a spring-mass oscillator depends on the a

Effects of Spring Constant Variation on Oscillatory Motion

A spring-mass system oscillates with motion given by $$y(t)=A*\cos(\omega*t)$$ where $$\omega=\sqrt{

Energy Analysis and Instantaneous Power in SHM

A block of mass $$m = 0.1\,kg$$ is attached to a spring with force constant $$k = 800\,N/m$$ and osc

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

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

Experimental Determination of Spring Constant

Utilize experimental data from a mass–spring oscillator to determine the spring constant.

Experimental Determination of Spring Constant via SHM

A physics lab report claims that the spring constant, k, of a mass-spring oscillator can be precisel

Frequency Response Analysis from Experimental Data

An experiment with a mass-spring oscillator produces displacement data over time as shown in the pro

FRQ 8: Energy Exchanges in SHM

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

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 14: Work Done by a Spring via Integration

Using calculus, derive an expression for the work done in stretching a spring from its natural lengt

FRQ 16: Maximum Speed in SHM via Energy Methods

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

FRQ 18: Comparing Oscillatory Systems

Compare the dynamics of a mass-spring system and a simple pendulum. Answer the following:

FRQ10: Damped Oscillations – Amplitude Decay and Velocity Derivation

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

FRQ13: Determining Damping Coefficient from Amplitude Decay

A damped oscillator with mass \(m = 0.1\,kg\) exhibits an exponential decay in its amplitude. Initia

FRQ17: Calculus-Based Derivation of Velocity and Acceleration in SHM

Consider a mass undergoing simple harmonic motion described by the displacement function: $$x(t)= A

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}}.

Graphical Analysis of Oscillatory Data

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

Graphical Analysis of SHM: Determining Phase and Frequency

A researcher records the oscillatory motion of a block on a spring and generates a position-vs.-time

Horizontal Spring Oscillator: Force and Energy Calculations

A mass is attached to a light spring on a frictionless horizontal surface. The spring has a force co

Kinematics of SHM and Calculus Differentiation

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

Kinematics of SHM: Period and Frequency Measurements

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

Mass Dependence in Oscillatory Motion

A spring with a force constant of $$k = 300 \; N/m$$ is used in two experiments. In the first, a blo

Nonlinear Effects in a Large-Amplitude Pendulum

A researcher studies the behavior of a simple pendulum at large amplitudes where the small-angle app

Nonlinear Pendulum Oscillations and Error Analysis

A pendulum with a length of $$L = 2.0\,m$$ is released from an initial angle of 45° (approximately 0

Oscillations in a Coupled Mass-Spring System

Two masses, $$m_1 = 0.1 \; kg$$ and $$m_2 = 0.2 \; kg$$, are coupled by a single spring with a force

Pendulum Motion Experimental Analysis

A simple pendulum experiment is used to measure the period of oscillation. A bob is attached to a ma

Period and Frequency of a Vertical Oscillator

A block of mass $$m = 1.5 \; kg$$ is suspended from a vertical spring with a force constant of $$k =

Sinusoidal Description and Phase Constant in SHM

A spring-mass oscillator has an amplitude $$A = 0.04 \; m$$ and a frequency $$f = 5.0 \; Hz$$. The d

Sinusoidal SHM with Phase Shift

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

Vertical Oscillations and Energy Analysis in a Spring–Mass System

Investigate the motion and energy conversion of a vertically oscillating mass–spring system.

Vertical Oscillations: Lab Data Analysis

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

Vertical Spring Oscillator Analysis

A vertical spring with a force constant of $$k = 400\,N/m$$ supports a mass of $$m = 2.0\,kg$$ and r

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

Vertical Spring-Mass Oscillator Dynamics

A block of mass $$m = 1.5 \; kg$$ is attached to the end of a vertical spring with force constant $$

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 a Two-Body Gravitational Interaction Using Calculus

Consider two objects of masses $$m_1$$ and $$m_2$$ that are initially at rest and start moving towar

Analyzing Hohmann Transfer Orbits for Satellite Maneuvers

Hohmann transfer orbits are used for efficient satellite maneuvers between two circular orbits. Answ

Calculus in Gravitational Work: Integration of Inverse Square Force

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

Center-of-Mass in the Sun-Earth System

Consider the Sun-Earth system, where the Sun has mass $$ M = 1.99 \times 10^{30} \text{ kg} $$, Eart

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

Comparing Circular and Elliptical Orbits in a Lab Simulation

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

Derivation of Gravitational Potential Energy Difference

A space probe's gravitational potential energy is given by $$U(r) = -\frac{G M m}{r}$$. Answer the f

Deriving Gravitational Potential from Gravitational Force

The gravitational potential \(V(r)\) is related to the gravitational force by calculus. (a) Show th

Deriving the Gravitational Potential Energy Function

Starting with Newton's law of gravitation expressed as $$F = - G * \frac{m_1 * m_2}{r^2}$$, derive t

Designing a Cavendish Experiment to Measure the Gravitational Constant

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

Designing a Modern Cavendish Experiment

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

Designing a Satellite Orbit Experiment

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

Designing a Satellite's Stable Orbit

A space agency plans to deploy a satellite into a stable circular orbit around a planet. The gravita

Determining the Center of Mass in a Celestial System

In studying the two-body problem, such as the Sun-planet system, the center of mass (or barycenter)

Dynamic Modeling of Planetary Motion in a Binary Star System

Consider a binary star system where two stars of comparable mass orbit their common center of mass i

Dynamics of a Binary Star System

Binary stars orbit around their common center of mass. Consider two stars with masses $$M_1$$ and $$

Effects of Eccentricity on Planetary Orbits

A series of simulations have been conducted for planetary orbits with varying eccentricity. The foll

Effects of Stellar Mass Variation in Binary Systems

In a binary star system, one star gradually loses mass. Analyze the impact on the orbital parameters

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 Conservation in Central Force Motion

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

Energy Conversion in an Asteroid's Elliptical Orbit

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

Experimental Analysis of Orbital Decay from a Satellite

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

FRQ 15: Gravitational Anomalies and Their Effects on Orbits

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

FRQ 19: Relativistic Corrections and Perihelion Precession

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

Gravitational Potential Energy Change in an Elliptical Orbit

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

Gravitational Potential Energy in a Non-Uniform Field

A spacecraft of mass m moves radially from a distance R₀ to a distance R from a planet of mass M. Th

Gravitational Slingshot and Energy Gain

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

Investigating Orbital Eccentricity Effects

Orbital eccentricity affects the dynamics of a planet's motion. Answer the following: (a) A graph i

Investigating Tidal Forces and Differential Gravity Effects

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

Mathematical Modeling of Tidal Forces

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

Modeling Tidal Forces with Calculus

Tidal forces arise due to the gradient in gravitational force across an object. These forces can cau

Nonlinear Gravitational Potential in a Drop-Test Apparatus

A drop-test apparatus experiment is designed to measure gravitational potential energy differences o

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

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

Orbital Perturbation due to Radial Impulse

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

Orbital Transfer Trajectories and Hohmann Transfers

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

Torsion Balance Gravitational Force Measurement

A research group performs an experiment using a torsion balance to measure the gravitational attract

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