AP Physics C: Mechanics FRQ Room

Analysis of Air Resistance Effects on Free Fall

In an experiment on free fall, an object is dropped and its velocity is measured over time. The calc

Calculus Analysis of a Parabolic Trajectory

A projectile is launched with the equations of motion given by $$x(t)=10*t$$ and $$y(t)=50*t-4.9*t^2

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

Conservation of Energy in Projectile Motion

A projectile is launched from the ground with an initial speed $$v_0 = 50$$ m/s at an angle of $$45°

Designing an Experiment: Motion on an Inclined Air Track

You are asked to design an experiment to determine the coefficient of kinetic friction on an incline

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

FRQ 2: Projectile Motion – Launch Experiment

A researcher uses a projectile launcher to study the flight of a ball launched at an angle. The ball

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: Motion with Non-Uniform Acceleration

An object experiences a time-dependent acceleration given by $$a(t) = 4*t - 2$$ (in m/s²) and an ini

FRQ 10: Comparative Analysis of Two Cars with Different Acceleration Profiles

A researcher compares the motion of two cars starting from rest. Car A accelerates at a constant rat

FRQ 13: Analyzing a Two-Dimensional Collision with Projectiles

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

FRQ 15: Investigating Uniformly Accelerated Motion Using Integrals

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

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

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

FRQ 20: Evaluating Data Uncertainty from a Velocity-Time Graph (EXTREME)

A velocity vs. time graph obtained from an experiment includes error bars on each data point. The be

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

FRQ 20: Real-World Application – Car Braking Analysis

A car traveling at 30 m/s begins braking uniformly until it comes to a complete stop. Sensors record

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)=

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

Integrating an Acceleration Function to Determine Motion

An object's acceleration is described by $$a(t)=6*t-5$$, with initial velocity $$v(0)=2$$ m/s and 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

Investigating Motion on an Inclined Plane

A lab experiment was set up to study the motion of a cart on an inclined air track. The cart’s displ

Kinematic Analysis of Circular Motion

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

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

Kinematics with Non-Constant Acceleration

An object moves along a straight line with a time-dependent acceleration given by $$a(t)=3t - 2\,m/s

Motion Analysis Using Integrated Acceleration Data

Researchers used an accelerometer attached to a moving cart to record its acceleration over a period

Motion on an Inclined Plane

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

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 Variable Acceleration

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

Non-linear Position Function Analysis

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

One-Dimensional Uniform Acceleration Analysis

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

Pendulum Energy Conservation Experiment

Design an experiment to test the conservation of mechanical energy in a simple pendulum system. Your

Polynomial Position Function Analysis

A particle’s position along the x-axis is given by the function $$x(t)=t^3 - 6*t^2 + 9*t$$ (in meter

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 with Timing Error

In an outdoor lab experiment, a projectile launcher was used to fire a ball at a 45° angle relative

Projectile Motion: Height and Flight Time Analysis

A projectile is launched with an initial speed of 50 m/s at a 45° angle. The following table shows o

Round Trip Motion Analysis

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

Slope Analysis in a Velocity-Time Graph

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

Time vs. Position Data Analysis: Initial Conditions Overlooked

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

Uniformly Accelerated Motion on an Incline

A block starts from rest and slides down a frictionless incline of angle 30° and length 5.0 m.

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).

Variable Net Force Experiment

A cart on a frictionless track is subjected to a variable net force given by $$F(t)= 10*t$$ (N). The

Vector Addition in Two-Dimensional Motion

An object is launched with a horizontal velocity of $$30\,m/s$$ and a vertical velocity of $$40\,m/s

Vector Decomposition in Displacement Measurements

A team conducts an experiment where a cart's displacement in two perpendicular directions is given b

Vector Displacement and Total Distance

An object moves along a straight line in two phases. First, it moves 10 m to the east, then it moves

Work and Energy in Linear Motion

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

Analysis of a Potential Energy Curve

An object of mass 3 kg moves along the x-axis in a potential energy field given by $$U(x) = (x-1)(x-

Analysis of a Potential Energy Curve

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

Analysis of Force and Velocity Data

An object is subjected to a variable force recorded by a sensor, given by $$F(t)=100-5*t$$ (in newto

Analysis of Potential Energy Curves

Consider the provided graph representing the potential energy function $$U(x)$$ for a diatomic molec

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

Collision and Energy Loss Analysis

Two objects collide inelastically and stick together. Object A has mass 3 kg moving at 4 m/s, and Ob

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 Mechanical Energy in a Pendulum

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

Energy Conservation on a Frictional Ramp with Calculus Approach

A 2-kg block slides down a straight inclined ramp from a height of $$h = 2 \;\text{m}$$. The ramp is

Energy Dissipation due to Friction

A 10 kg block is pushed along a horizontal surface with a coefficient of kinetic friction $$\mu = 0.

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 Transformation in a Roller Coaster

A roller coaster car of mass $$m = 500 \;\text{kg}$$ starts from rest at a height $$H = 50 \;\text{m

FRQ 2: Work-Energy Theorem in Lifting

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

FRQ 4: Potential Energy Curve Analysis for a Diatomic Molecule

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

FRQ 5: 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: Energy Loss Due to Friction on a Sliding Object

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

FRQ 8: Pendulum Energy Transformations with Damping

An experimental study on a pendulum claims that its mechanical energy is conserved, assuming only gr

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 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 15: Energy Conservation in an Oscillating Spring–Mass System

A 2-kg mass attached to a spring (with spring constant k = 200 N/m) oscillates horizontally. A displ

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

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

Inclined Plane Energy Transfer Experiment

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

Multi‐Phase Cart Energy Experiment

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

Oscillatory Motion Energy Exchange Experiment

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

Particle Dynamics in a Variable Force Field

A particle of mass 2 kg moves along the x-axis under a force given by $$F(x) = 12 - 2*x$$ (in newton

Potential Energy Curve Analysis

An object of mass m is subject to a potential energy function given by $$U(x) = x^3 - 6*x^2 + 9*x +

Potential Energy Curve Analysis

An object has a potential energy given by $$ U(x) = (x-2)^2 - (2*x-3)^3 $$, where U is in joules and

Power and Energy Efficiency in a Conveyor Belt Experiment

A researcher investigates the energy efficiency of a conveyor belt system in a manufacturing facilit

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 from a Variable Force: Time-Dependent Problem

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

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 Energy Analysis with Air Resistance Correction

A 0.5 kg ball is thrown vertically upward with an initial speed of 20 m/s. Air resistance does negat

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 Power in Gear Systems

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

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

Time-dependent Power and Differential Equations

A machine's power output, $$P(t)$$ in watts, is governed by the differential equation $$\frac{dP}{dt

Variable Force and Velocity: Power and Work Analysis

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

Variable Force Robotic Arm Power Experiment

In this experiment, a robotic arm exerts a time-varying force on an object as it moves along a horiz

Wind Tunnel Analysis of Mechanical Energy Extraction

In a wind tunnel experiment, a miniature wind turbine is tested. The force exerted by the wind on th

Work and Energy in a Pulley System

A researcher investigates a two-mass system connected by a massless, frictionless pulley. Mass m1 =

Work–Energy Experiment with a Spring Launch

A researcher studies a spring-launched projectile. A spring with a spring constant $$k = 500\,N/m$$

Work–Energy Theorem Verification in Projectile Motion

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

Analysis of an Oblique Collision

Two ice skaters are initially at rest on a frictionless ice surface. They push off each other; skate

Angular Momentum Change in a Disc–Rod Collision Experiment

In a rotational collision experiment, a spinning disc collides with a stationary rod. Motion sensors

Astronaut Momentum Conservation

An astronaut with a total mass of 89 kg is floating in space near her shuttle. To reorient herself,

Astronaut Recoil in Space

An astronaut with a total mass of 90 kg, initially at rest relative to her shuttle, throws a 2 kg to

Balancing a Composite System's Center of Mass

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

Center of Mass 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 of a Lamina with Nonuniform Density

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

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

Conservation of Linear Momentum in a Glider Collision

On a frictionless air track, two gliders collide. The experimental data below list the masses and ve

Determination of an Unknown Mass via Collision Data

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

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

Dynamics of a Falling Object with Air Resistance

An object of mass 0.1 kg is dropped from a height and experiences air resistance modeled as $$F_{air

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 Collision with Time-Dependent Pre-Collision Motion

Particle A (mass = 1 kg) has a velocity given by $$v_A(t)=4-t$$ (m/s) for $$0 \leq t \leq 2$$ s. It

Elastic Collision: Two Gliders on an Air Track

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

Explosive Separation and Momentum Conservation

A 2 kg projectile traveling at 15 m/s explodes into two equal fragments (1 kg each). One fragment mo

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 1: Center of Mass of a Non-Uniform Rod

Consider a rod of length $$L = 1.2 \ m$$ with a linear mass density given by $$\lambda(x) = 2 + 3*x$

FRQ 17: Impulse from a Functional Force

A particle experiences a force given by $$F(t) = 4*t$$ (N) over the time interval $$0 \le t \le 5\ s

Impulse and Kinetic Energy Loss in a Perfectly Inelastic Collision with a Spring

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

Impulse Calculation from Force-Time Graph

A force sensor records a time-dependent force acting on an object. The force is modeled by the equat

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 from Force Sensor Data

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

Impulse Transfer on a Rotating Rod

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

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

Meteor Impact: Conservation of Momentum and Energy Dissipation

A meteor with a mass of 5000 kg is traveling at 20 km/s (20000 m/s) and impacts the Earth, breaking

Motion of Center of Mass Under External Force

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

Motion of the Center of Mass under Applied Force

Two blocks, with masses 3 kg and 5 kg, are connected by a massless rope on a frictionless surface. A

Multi-Stage Rocket Propulsion using Momentum Conservation

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

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

Nonuniform Rod: Total Mass and Center of Mass

A rod of length $$1.0$$ m has a linear density given by $$\lambda(x) = 10 + 6*x$$ (kg/m), where $$x$

Oblique Collision of Ice Pucks

Two ice pucks are sliding on a frictionless ice surface. Puck A (mass = 0.2 kg) is moving with a vel

Recoil of an Astronaut after Throwing a Tool

An astronaut with a total mass of 90 kg, initially stationary in space, throws a 2 kg tool at a spee

Rocket Propulsion and Center of Mass Dynamics

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

Rocket Propulsion: Variable Mass System

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

Stability Analysis Using Center of Mass on a Pivoted Beam

A uniform beam of length 2.0 m and mass 10 kg is pivoted at one end. A 20 kg mass is suspended from

Stability of a Suspended Mobile

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

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 and Momentum Conservation

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

Two-Stage Collision in Coupled Carts

Two carts on a frictionless track undergo a two-stage event. Initially, cart A (mass $$2\,kg$$, velo

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 Gyroscopic Precession

A spinning gyroscope of moment of inertia $$I$$ has an angular momentum $$L$$ and is subject to a gr

Angular Impulse Analysis

A flywheel is subjected to a time-dependent torque given by $$\tau(t) = 50 * e^{-2*t}$$ N*m for $$t

Angular Kinematics from Disk Data

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

Angular Kinematics on a Rotating Platform

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

Angular Kinematics with Variable Angular Acceleration

A disk rotates with a non-uniform angular acceleration given by $$\alpha(t) = 4*t$$ (in rad/s²). The

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: Ice Skater

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

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 a Dual-Wheel System

Two wheels, A and B, are coupled so that friction between them eventually brings them to a common an

Calculation of Rotational Inertia for Composite System

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

Calculus 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 in Determining the Moment of Inertia of a Continuous Object

A researcher is investigating how non-uniform mass distribution affects the moment of inertia of a t

Calculus-Based Determination of Angular Displacement

A rotating object's angular velocity is recorded as a function of time, and a graph of angular veloc

Comparative Study of Rotational Kinetic Energy in Different Shapes

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

Composite Body Rotation

A composite object is formed by welding a solid disk to a thin rod. The disk has mass $$M$$ and radi

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

Critical Analysis of Torque in Mechanical Systems

A media report on engine performance claims that a 10% increase in the applied force always results

Cylinder Rolling Down an Incline

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

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

Determining the Moment of Inertia of a Non-Uniform Rod

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

Effect of Friction on Rotational Motion

Design an experiment to quantify the torque losses due to friction in a rotating apparatus. Your goa

Energy Dissipation Due to Friction in a Spinning Disk

A disk is spun up to a high angular velocity and then allowed to slow down due to friction. The expe

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 9: Experimental Determination of Moment of Inertia

A student performs an experiment to determine the moment of inertia of a uniform disk by measuring i

FRQ 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 17: Moment of Inertia of a Non-Uniform Rod

A rod of length L = 2.00 m has a density that varies with position according to $$\rho(x) = \rho_0 *

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 Mass Distribution on Angular Acceleration

An experiment varies the mass distribution of a rotating rod under a constant applied torque. The ta

Investigation of Angular Acceleration from Experimental Data

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

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 Hollow Cylinder with Thickness

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

Net Torque and Angular Acceleration Calculation

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

Non-Uniform Angular Acceleration

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

Non-uniform Rotational Acceleration: Differentiation from Graph

A rotating disk exhibits a non-uniform angular velocity as a function of time. The experimental grap

Physical Pendulum with Offset Mass Distribution

A physical pendulum is constructed from a rigid body of mass $$M$$ with its center of mass located a

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 with Slipping Transition

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

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 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 Equilibrium of a Beam with Distributed Load

A uniform beam of length $$L = 4.0 \text{ m}$$ and mass 10 kg is hinged at one end. A variable distr

Seesaw Rotational Equilibrium

Two children are sitting on opposite ends of a seesaw (a uniform beam pivoting about its center). Ch

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 Measurement and Analysis

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

Torque on a Uniform Rod with Distributed Force

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

Torsional Oscillator Analysis

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

Wrench Torque Analysis

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

Advanced Pendulum Oscillator: Beyond the Small-Angle Approximation

For a simple pendulum with a large amplitude, the period deviates from the small-angle approximation

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 the Half-Cycle Method in Oscillation Experiments

A media report asserts that 'timing just half a cycle of a pendulum is sufficient to determine its f

Calculating Damped SHM Energy Loss

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

Calculus Derivative Analysis in SHM

Given an oscillator with a position function $$y = 0.05 \sin(12t + \frac{\pi}{6})$$, where $$y$$ is

Comparative Analysis of Oscillator Systems

Consider two oscillator systems: a horizontal spring-block oscillator with mass \(m\) and spring con

Damped Harmonic Oscillator Dynamics

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

Damped Oscillations: Determining the Damping Coefficient

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

Designing an Experiment on the Inverse Relationship between Mass and Period

A researcher designs an experiment to study the relationship $$T = 2\pi * \sqrt{\frac{m}{k}}$$ in a

Determination of Spring Constant from Oscillation Data

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

Determining Maximum Speed from Energy Considerations

An oscillator of mass $$m = 0.1 \; kg$$ is attached to a spring with a spring constant of $$k = 250

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

Differentiation in SHM: Velocity and Acceleration

An oscillator is described by the function $$y = A * \sin(\omega * t + \phi_0)$$. Investigate the ve

Energy Conservation Verification Using Calculus

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

Energy Exchange in Oscillatory Systems

A new research article claims that 'the maximum speed of a block on a spring is invariant with respe

Energy Transformation in SHM

A block of mass $$m = 0.2 \; kg$$ oscillates on a horizontal spring with a force constant of $$k = 1

Forced Oscillations and Resonance

An oscillator is driven by an external force and is modeled by the equation $$m\ddot{x} + kx = F_0 \

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 9: Effect of Spring Constant on Frequency

For a mass-spring oscillator, the frequency is given by $$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$. An

FRQ3: Kinematics of SHM – Period and Frequency

A block-spring oscillator is observed to take 0.4 s to travel from its maximum displacement in one d

FRQ5: Sinusoidal Description of Oscillatory Motion and Phase Determination

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

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

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

Integration of Variable Force to Derive Potential Energy

A non-linear spring exerts a force given by $$F(x)= - k * x - \alpha * x^3$$, where $$k = 200 \; N/m

Investigating Nonlinearity in Large-Amplitude Oscillations

A recent experimental paper claims that 'at large amplitudes, the assumption of simple harmonic moti

Measuring the Spring Constant: An Experimental Investigation

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

Mechanical Energy in SHM

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

Modeling Amplitude Reduction Due to Non-Conservative Forces

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

Momentum and Impulse Analysis in Oscillatory Motion

A block of mass $$m = 0.4 \; kg$$ oscillates on a spring, and its displacement is given by $$x(t)=0

Momentum Transfer in a Spring-Mass Collision

A block of mass $$m = 0.25\,kg$$ moving at a speed of $$v = 2.0\,m/s$$ collides with the free end of

Non-linear Effects in Simple Pendulum Motion

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

Oscillation Frequency's Dependence on Mass and Spring Constant

A research claim suggests that 'doubling the mass of an oscillating system will always decrease the

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

Oscillations of a Liquid Column in a U-tube

A U-tube containing a liquid with density $$\rho$$ exhibits oscillatory motion when the liquid is di

Oscillatory Motion on an Inclined Plane

A block of mass $$m$$ is attached to a spring (constant $$k$$) on a frictionless inclined plane with

Pendulum Energy Dynamics

Analyze the energy dynamics of a simple pendulum using both theoretical derivations and numerical ca

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

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

SHM with Phase Shift: Initial Conditions Analysis

An oscillator is described by the equation: $$y(t) = A * \sin(2\pi * f * t + \phi)$$ In a particul

Small-Angle Pendulum Experiment

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

Spring Force and Energy Analysis

A spring with a force constant $$k = 350 \; N/m$$ and a natural (unstretched) length of 20 cm is str

Spring Oscillator on an Inclined Plane

A block of mass \(m = 2\,kg\) is attached to a spring with spring constant \(k = 150\,N/m\) on an in

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

Uncertainty Analysis in SHM Period Measurements

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

Vertical Spring–Block Oscillator Dynamics

Investigate the dynamics of a block oscillating vertically on a spring.

Work Done in Spring Oscillation via Calculus

A spring with a constant of $$k = 150\,N/m$$ is stretched from its natural length to a displacement

Analysis of Orbital Transfer Maneuvers Using Calculus

A spacecraft is initially in a circular orbit of radius $$ r_1 $$ and is to be transferred to a circ

Analyzing Three-Body Gravitational Interactions

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

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 in Orbital Motion: Area Sweep in an Elliptical Orbit

Kepler's Second Law implies that the rate of area sweep (dA/dt) is constant for an orbiting body. In

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

Determining Orbital Eccentricity from Observational Data

Astronomers collect data of a planet's distance from its star at various times and wish to determine

Dynamics of a Binary Star System

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

Dynamics of a Falling Object in a Gravitational Field

A mass is dropped from a height in a gravitational field and its motion is tracked to study energy c

Dynamics of Binary Star Systems

Two stars with masses $$M_1$$ and $$M_2$$ orbit their common barycenter. (a) Derive expressions for

Eccentricity and Elliptical Orbits

Discuss the role of eccentricity in elliptical orbits and derive the orbit equation in polar coordin

Elliptical Orbits and Angular Motion

A planet follows an elliptical orbit around its star. Investigate how variations in orbital distance

Energy Conservation in Elliptical Orbits

Energy conservation is a key principle in orbital dynamics, particularly in elliptical orbits where

Escape Velocity Derivation

The escape velocity is the minimum speed required for an object to escape from the gravitational fie

Gravitational Parameter in Exoplanetary Systems

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

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 Potential Energy Measurement on a Ramp

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

Gravitational Slingshot Maneuver

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

Gravitational Slingshot Maneuver in Space Missions

A spacecraft uses a gravitational slingshot maneuver to gain speed by passing close to a planet. Ass

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

Orbital Mechanics: Applying Kepler's Third Law

A satellite orbits Earth in an elliptical orbit, which for the sake of this problem can be approxima

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

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 by Gravitational Force in Radial Motion

A spacecraft of mass $$m$$ moves radially under the gravitational influence of a mass $$M$$. Answer