AP Physics C: Mechanics FRQ Room

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 a Velocity-Vs-Time Graph

An object’s velocity is recorded along a straight path. The velocity-time graph indicates that the o

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

Analyzing Circular Motion: Speed and Acceleration

A particle moves with a constant speed of 15 m/s along a circular path of radius 10 m.

Analyzing Two-Dimensional Motion Using a High-Speed Camera

In an experiment on projectile motion, a high-speed camera was used to record the two-dimensional mo

Average vs. Instantaneous Quantities

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

Block on an Inclined Plane: Kinematic Analysis

A block of mass m is released from rest at the top of a frictionless inclined plane of length L = 10

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

Circular Motion: Centripetal Acceleration from Tangential Speed Function

An object moves in a circular path of constant radius $$R = 3.0\,m$$. Its tangential speed varies wi

Conservation of Energy in a Pendulum

Design an experiment to test the conservation of mechanical energy in a simple pendulum. Provide bot

Critical Evaluation of Experimental Kinematics Data

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

Deriving Velocity and Acceleration from a Position Function

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

Distance vs. Displacement Analysis in One-Dimensional Motion

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

Experimental Determination of g

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

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 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: Calculus-Based Displacement Calculation

An object has a velocity given by the function $$v(t) = 3*t^2 - 2*t + 5$$ (with t in seconds and v i

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 8: Vector Addition in Two-Dimensional Motion

An object moves in a plane following these displacements in sequence: 4 m east, 3 m north, 5 m west,

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 10: Experimental Analysis of Free Fall

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

FRQ 12: Graphical Analysis of Vertical Motion (MEDIUM)

A graph of vertical displacement for a projectile is modeled by the function $$y(t)=5*t-4.9*t^2$$ (i

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 17: Analyzing Motion from a Cubic Position Function

An object’s position is given by $$x(t)= 2*t^3 - 9*t^2 + 4*t$$ (in meters, with time in seconds). An

FRQ 17: Vector Field Analysis – Wind Impact on Projectile Motion

A researcher examines how a time-varying wind affects the horizontal motion of a projectile. In this

Graphical Analysis of Distance and Displacement

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

Investigating Motion on an Inclined Plane

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

Lab Investigation: Effects of Launch Angle on Projectile Range

In a controlled laboratory experiment, a student launches a projectile with a fixed initial speed of

Motion Along a Curved Track

A roller coaster car moves along a curved track. Its displacement along the track is given by $$s(t)

Motion of a Bus on a Curved Track

A bus moves along a curved track with its acceleration given by $$a(t)= 0.2*t + 0.5*\sin(t)$$ (m/s²)

Motion on an Inclined Plane with Friction

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

Non-linear Position Function Analysis

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

Optimizing the Launch Angle for Maximum Horizontal Displacement on Uneven Terrain

A projectile is launched with an initial speed of 40 m/s from a platform that is 10 m above the grou

Parametric Trajectory Analysis

A particle moves in the plane with its position given by: $$x(t)=2t^2$$ and $$y(t)=20t - 4.9t^2$$, w

Projectile Motion on Level Ground

An object is launched from ground level at a 45° angle with an initial speed of 30 m/s (neglect air

Projectile Motion with Calculus Integration

An object is launched from ground level with an initial speed of 50 m/s at an angle of 40° above the

Projectile Motion with Drag

Design an experiment to analyze the trajectory of a projectile when including the effects of drag fo

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

Simple Harmonic Motion in a Spring-Mass System

Design an experiment to investigate simple harmonic motion (SHM) using a spring-mass system. Describ

Uniformly Accelerated Motion: Incorrect Baseline Velocity

A physics lab experiment tracked the displacement of a moving object using a laser sensor, measuring

Variable Acceleration Analysis Using Calculus

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

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

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

Assessing Energy Conversion in a Pendulum Experiment

A researcher conducts an experiment with a simple pendulum of length L = 2 m and a bob of mass 0.5 k

Calculating Kinetic Energy from a Velocity Function

A particle of mass $$m = 1 \;\text{kg}$$ moves along the x-axis with a velocity given by $$v(t)= 3*t

Calculus Application of a Variable Force

A force acting on an object is given by $$F(x) = 5*x^2$$ N. Consider the displacement of the object

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

Dissipative Work under Variable Friction

A 5 kg block is sliding on a horizontal surface with an initial speed of 10 m/s. The coefficient of

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

Elastic Potential Energy in a Spring System

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

Energy Loss in an Inelastic Collision

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

Energy Loss in Inelastic Collisions

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

Experiment on Electric Motor Power Output

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

FRQ 4: Conservation of Mechanical Energy in a Roller Coaster

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

FRQ 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 10: Conservation of Energy in a Pendulum Experiment

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

FRQ 12: Analysis of Cyclist Power Output During a Ride

A cyclist’s performance is monitored by a power meter during a ride. The measured power output over

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

Gravitational Potential Energy Conversion

A 10 kg object is dropped from a height of 15 m in a vacuum. Using conservation of energy methods, a

Horizontal Pulling Work Experiment

A crate is pulled along a horizontal floor by a worker using a rope that makes a constant angle with

Interpreting a Diagram of Work–Energy Processes

A detailed diagram is provided that illustrates a block sliding down an inclined plane with friction

Kinetic Energy and Work-Energy Theorem Application

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

Model Rocket Power Measurement Experiment

In this experiment, a model rocket’s engine power output is determined by measuring its constant spe

Potential Energy Curve Analysis

An object of mass 4 kg moves in a potential energy field described by $$U(x) = (x-1)^2 - 0.5*(x-2)^3

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

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

Power Output Measurement in an Elevator Experiment

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

Projectile Motion Energy Analysis

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

Roller Coaster Energy Analysis

A roller coaster car of mass 500 kg is released from rest at a height of 50 m and descends along a t

Rotational Power in Gear Systems

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

Variable Force Work Calculation and Kinetic Energy Analysis

Consider an object moving along the x-axis under the influence of a variable force given by $$F(x) =

Work and Energy in Circular Motion

A 2-kg mass is attached to a string and constrained to move on a frictionless vertical circular path

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 Against Friction on an Inclined Plane

A 5-kg box slides down a 10-m long inclined plane that makes an angle of 30° with the horizontal. Th

Work Done by a Variable Exponential Force

A piston is subjected to a variable force described by $$F(x)=500*\exp(-0.5*x)$$ N, where x is in me

Work Done in a Variable Gravitational Field

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

Work Done on an Object by a Central Force

An object moves under a central attractive force given by $$F(r)= -\frac{A}{r^2}$$ with $$A = 50 \;\

Work–Energy Experiment with a Spring Launch

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

Analyzing a Multi-Peak Force-Time Graph

A cart of mass 2 kg is subjected to a force that varies with time according to a piecewise function:

Billiard Ball Collision and Impulse Analysis

In a game of billiards, a moving ball (mass $$0.17\,kg$$, speed $$2\,m/s$$) collides with a stationa

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

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

Center of Gravity vs. Center of Mass in a Tilted Rod

A uniform rod is supported on one end by a fulcrum. In an experiment, additional weights were placed

Center of Mass Acceleration under Variable Force

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

Center of Mass for Discrete Particles

Consider a system of three particles in the xy-plane with the following properties: • Particle A: m

Center of Mass of a Non-uniform Circular Disk

A thin circular disk of radius $$R$$ has a surface mass density given by $$\sigma(\theta)= k\,(1+\co

Center of Mass of a Non-Uniform Rod

A thin rod of length $$0.8$$ m has a linear mass density given by $$\lambda(x) = 5 + 3*x^2$$ (kg/m),

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 an Irregular Lamina in Polar Coordinates

Consider a lamina occupying the region defined in polar coordinates by $$0 \le r \le 2(1+\cos(θ))$$

Center-of-Mass Shift in an Internal Explosion

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

Conservation of Linear Momentum in Colliding Carts

Two carts on a frictionless track collide. Cart A (mass = 2 kg) moves to the right with a speed of 3

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

Dynamics of Center of Mass under a Time-Varying External Force

A system consists of two blocks with masses of 3 kg and 5 kg. A time-varying external force given by

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

Explosive Separation of Particle System

A 10 kg object at rest explodes into three fragments with masses 3 kg, 4 kg, and 3 kg. After the exp

Fragmentation and Impulse

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

FRQ 9: Rocket Propulsion and Momentum Conservation

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

FRQ 10: Collision with Rotational Motion

A uniform disk of mass $$2 \ kg$$ and radius $$0.5 \ m$$ is rolling without slipping at $$4 \ m/s$$

FRQ 11: Experimental Evaluation: Measurement of Center of Mass

A media report claims that a new laser-based method can determine the center of mass of irregular ob

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 16: Momentum Conservation in a Multi-Particle System

Three particles are aligned along the x-axis with masses $$m_1 = 1 \ kg$$, $$m_2 = 2 \ kg$$, and $$m

Glider Collision on a Frictionless Air Track

Two gliders on an air track are involved in a head-on elastic collision. The data for the gliders is

Impulse and Momentum with Variable Force Function

A ball is struck by a kick where the force exerted on it varies with time according to the function

Impulse Calculation from a Force-Time Graph

A force acting on an object is depicted by a force-versus-time graph over the interval from t = 0 s

Impulse Delivered by a Variable Force

A particle experiences a time-dependent force along the x-axis given by $$F(t)=4*t^2 - 12*t + 9$$ N

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 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 with a Movable Platform

A 0.3 kg ball moving horizontally at 8 m/s collides with and sticks to a 1.2 kg movable platform tha

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.

Nonuniform Circular Disk Center of Mass

A circular disk of radius $$R$$ has a nonuniform surface mass density given by $$\sigma(r,\theta)=\s

Projectile Motion with Air Resistance Approximation

A 0.2 kg projectile is launched with an initial speed of 15 m/s at an angle of 40° above the horizon

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

Sequential Collisions in One Dimension

A 1-kg cart moving at $$4\,\text{m/s}$$ collides elastically with a stationary 0.5-kg cart. Immediat

Time-Varying Force on a Block

A 2 kg block on a frictionless surface is subjected to a time-varying force given by $$F(t) = 15*\si

Two-Ball Collision Dynamics

Two balls collide head-on in a controlled experiment. The red ball (mass = 0.5 kg) moves to the righ

Two-Dimensional Collision of Ice Skaters

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

Variable Density Rod: Mass and Center of Mass Calculation

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

Vibrational Motion: Coupled Oscillators

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

Analysis of Angular Displacement in a Rotating Disk

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

Analysis of Gyroscopic Precession

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

Angular Kinematics from Experimental Data

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

Angular Kinematics: Modeling a Rotating Spring System

A disk is attached to a torsional spring that exerts a restoring torque given by $$T = -k \times \th

Angular Momentum Conservation in a Variable-Radius System

A student investigates angular momentum conservation on a rotating stool by attaching a weight to a

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

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 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

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 Explosive Separation

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

Critical Analysis of Torque in Mechanical Systems

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

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

Dynamics of a Rotating Rod with Sliding Masses

In this advanced experiment, masses are allowed to slide along a horizontal rod attached to a pivot.

Energy Considerations in a Rotating Pendulum

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

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 10: Comparison of Rotational and Translational Kinetic Energy

A solid sphere of mass M = 4.00 kg and radius R = 0.10 m rolls without slipping down a ramp of heigh

FRQ 11: Impact of Mass Distribution on Angular Acceleration

Two wheels have identical mass M and radius R. Wheel A has all its mass concentrated at the rim (\(I

FRQ 13: Dynamics of a Variable Torque System

A rod with moment of inertia \(I = 4.00\,kg\cdot m^2\) is subjected to an angular-dependent torque g

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

Investigation of Torque on a Rotating Pulley

In an experiment, a student applied a constant force of $$F = 40\,N$$ at varying distances (moment a

Lever and Torque Computations

This problem involves calculating torque in a lever system. A diagram is provided below.

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’

Non-uniform Rotational Acceleration: Differentiation from Graph

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

Rolling Cylinder Down an Incline

A solid cylinder rolls without slipping down an incline. A set of measurements were made at differen

Rolling Motion Energy Conversion Experiment

A researcher investigates the energy conversion in rolling motion without slipping. A solid cylinder

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 Impulse and Change in Angular Momentum

A flywheel initially at rest receives a constant torque impulse over a brief time interval.

Rotational Inertia of a Composite Bead System

A researcher is investigating the effect of discrete mass distribution on the rotational inertia of

Rotational Inertia of a Non-Uniform Disk

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

Rotational Kinematics on a Spinning Disk

A rotating disk's angular displacement is measured by the function $$\theta(t) = 0.2\,t^2 + 2\,t$$ (

Seesaw Rotational Equilibrium

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

Static Equilibrium of a Beam

A uniform beam of length 4 m and weight 200 N is supported at one end by a wall and held horizontal

Testing the Parallel Axis Theorem

An experiment is conducted on a uniform disk with mass $$M$$ and radius $$R$$. The disk's moment of

Torque and Rotational Inertia in Engine Mechanisms

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

Torque from a Distributed Load

A uniform beam of length $$L$$ has a constant linear weight density $$w$$ (in N/m).

Torsion Pendulum Method for Irregular Objects' Inertia

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

Analysis of Phase Shift in Oscillator Data

An oscillator is described by the equation $$y = 0.03 \sin(8t + \phi_0)$$. It is experimentally meas

Analyzing Damped Oscillations in a Spring-Mass System

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

Analyzing the Half-Cycle Method in Oscillation Experiments

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

Calculus Approach to Maximum Velocity in SHM

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

Calculus Derivation of Velocity and Acceleration in SHM

Educational materials claim that 'the instantaneous velocity and acceleration of a simple harmonic o

Calculus-Based Analysis of Velocity and Acceleration

Consider an oscillator whose displacement is defined by: $$x(t) = 0.1 * \sin(6*t)$$ (a) Differenti

Comparative Analysis of Horizontal and Vertical SHM Systems

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

Comparative Analysis: Horizontal vs. Vertical Oscillations

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

Comparing Sinusoidal and Non-Sinusoidal Oscillatory Motion

In some experiments, the oscillatory motion observed may deviate from a pure sinusoid. Consider a sy

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: Normal Modes and Energy Transfer

Consider a system of two identical masses coupled by springs and attached to fixed supports. Analyze

Coupled Oscillators: Two Springs in Parallel

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

Damped Oscillation: Logarithmic Decrement Analysis

A spring-mass oscillator exhibits damped oscillations, and the amplitudes of successive cycles are r

Damped Oscillations in a Spring-Mass System

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

Damped Oscillations: Determining the Damping Coefficient

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

Determining Oscillation Frequency from Acceleration Data

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

Differentiating SHM: Velocity and Acceleration

A block attached to a spring oscillates on a frictionless track and its position is recorded by a se

Effect of a Nonlinear Restoring Force on Oscillation

A modified oscillator has a restoring force given by $$F = -k x - \alpha x^3$$, where $$\alpha$$ is

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

Effect of Mass Variation on SHM

A block attached to a spring oscillates, and the period of oscillation is given by $$T = 2\pi\sqrt{\

Effects of Mass Variation on Oscillation Frequency

A student investigates how the oscillation frequency of a spring-block system varies with the mass a

Elastic Potential Energy and Maximum Speed Calculation

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

Energy Conservation in a Simple Pendulum

A simple pendulum of length $$L$$ and mass $$m$$ is displaced by a small angle $$\theta$$ from the v

Energy Exchange in SHM

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

Evaluating Damped Oscillatory Motion Effects

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

Evaluating Experimental Uncertainties in SHM Measurements

Accurate measurement of the oscillation period is crucial in SHM experiments. In this context, uncer

Experimental Determination of Spring Constant

In a lab experiment, students measure the displacement of a spring under various applied forces. The

Forced Oscillations and Resonance

A mass-spring system is driven by an external force of the form \(F(t) = F_0 \cos(\omega_d t)\) and

FRQ 1: Hooke’s Law Experiment

In a laboratory experiment, the restoring force of a spring was measured for various displacements f

FRQ 3: Determining Period and Frequency

An oscillating block moves from its position of maximum stretch to its maximum compression in $$0.25

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 17: Pendulum Nonlinear Effects

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

FRQ 19: Vertical Oscillator Dynamics

A mass is attached to a vertical spring. When displaced by a distance $$y$$ from its equilibrium pos

FRQ 20: Oscillator with Time-Varying Mass

Consider a spring-mass system in which the mass varies with time according to $$m(t) = m_0 + \alpha

FRQ12: Phase Shift and Time Translation in SHM

An oscillator is described by the equation $$x(t) = A\sin(\omega t+\phi_0)$$. Answer the following:

FRQ16: Resonance in a Driven, Damped Oscillator

A damped oscillator is subjected to an external periodic driving force of the form $$F_d \cos(\omega

FRQ19: Coupled Oscillators – Normal Modes of a Two-Mass, Three-Spring System

Consider a system in which two identical masses \(m\) are connected in series with three identical s

Hooke's Law Force Calculation

A spring at its natural length is stretched by a displacement $$x$$. (a) Derive the expression that

Influence of Initial Phase on Oscillator Motion

Consider an oscillator described by $$y = A\sin(\omega t + \phi_0)$$. Explore how variations in the

Integral Calculation of Work Done in SHM

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

Investigating Nonlinear Oscillations in a Large-Amplitude Pendulum

Students perform an experiment to analyze the period of a pendulum swinging at large amplitudes (up

Kinematics and Phase Angle Determination

An oscillator is described by the equation $$y = A\sin(\omega t + \phi_0)$$ with an initial conditio

Mass Variation and Frequency in SHM

Consider a spring oscillator with a constant spring constant of $$k = 200\,N/m$$. The frequency of o

Momentum and Impulse Analysis in Oscillatory Motion

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

Nonlinear Characteristics of the Simple Pendulum

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

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 Angle Dependence and the Small Angle Approximation

A recent news article claims that 'the period of a pendulum is completely independent of the amplitu

Pendulum Approximation and Small-Angle Motion

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

Pendulum Motion and the Small Angle Approximation

A simple pendulum of length $$L$$ oscillates with small angular displacements. Analyze its motion us

Pendulum Oscillation Experiment: Frequency and Energy Analysis

A simple pendulum consists of a bob of mass $$m = 0.2\;kg$$ attached to a massless string of length

Pendulum Oscillations: Small Angle Approximation

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

Pendulum Period Measurement Experiment

A group of students measure the period of a simple pendulum by timing multiple oscillations using a

Phase Space Analysis of SHM

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

SHM with a Varying Force Constant

In an experiment on a spring-mass system, a student investigates oscillations at various amplitudes.

Spring Force Investigation

A researcher investigates the force exerted by a spring using Hooke's law. The aim is to verify the

Spring-Mass Oscillator on an Inclined Plane

A mass $$m = 0.5\,kg$$ is attached to a spring with force constant $$k = 150\,N/m$$, and the assembl

Uncertainty Analysis in SHM Period Measurements

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

Vertical Oscillations and Energy Analysis in a Spring–Mass System

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

Angular Momentum Conservation during Gravitational Collapse

An interstellar cloud of gas with initial radius R and angular velocity ω undergoes gravitational co

Angular Momentum Conservation in Orbital Motion

Angular momentum conservation plays a critical role in determining the properties of orbital motion.

Application of Kepler's Second Law to Orbital Motion

Kepler's Second Law states that a line joining a planet and the Sun sweeps out equal areas in equal

Application of Kepler's Third Law

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

Assessment of Newton's Second Law Along a Gravitational Incline

A small cart is placed on a frictionless inclined plane that makes an angle $$\theta$$ with the hori

Calculus in Determining Work Against Gravity over Altitude Change

A spacecraft gradually moves from an initial orbital radius r₁ to a higher radius r₂. The work done

Cannonball Trajectory in a Non-Uniform Gravitational Field

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

Center of Mass in the Sun-Earth System

Determine the location of the barycenter for the Sun-Earth system. Use $$M = 1.99 \times 10^{30} \ \

Center of Mass of the Sun-Earth System

Consider the Sun-Earth system where the mass of the Earth is $$m = 5.98 \times 10^{24} \text{ kg}$$,

Comparative Analysis of Gravitational Forces

Using the data provided, compare the gravitational forces between various pairs of celestial bodies.

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

Comparative Analysis of Planetary Orbits

Two planets orbit the same star with different semimajor axes. Use Kepler's Third Law to analyze and

Comparison of Gravitational and Centripetal Forces

For a satellite in a stable circular orbit, investigate the balance between gravitational and centri

Comparison of Orbital Dynamics: Moon vs. Artificial Satellites

A researcher compares the gravitational forces and orbital characteristics of the Moon and an artifi

Derivation of Escape Velocity Using Calculus

Using the concept of gravitational potential energy, derive the escape velocity from a planet of mas

Derivation of Gravitational Field due to a Spherical Shell

A thin spherical shell of uniform density with total mass M and radius R produces a gravitational fi

Derivation of Gravitational Potential Energy

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

Derivation of Orbital Period from Gravitational Force

Using calculus, derive the expression for the orbital period of a planet in circular orbit from Newt

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

Effects of Eccentricity on Planetary Orbits

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

Energy Conversion in a Gravitational Slingshot Maneuver

A spacecraft performing a gravitational slingshot maneuver around a planet converts gravitational po

FRQ 11: Time-Dependent Gravitational Force in Radial Motion

A spaceship travels radially away from a planet under the influence of gravity. Consider the gravita

FRQ 12: Designing a Geosynchronous Satellite Orbit

A geosynchronous satellite must orbit the Earth with an orbital period equal to one sidereal day (\(

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

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

Gravitational Energy in a Three-Body System

Consider three point masses m1, m2, and m3 placed at the vertices of a triangle. Analyze the gravita

Gravitational Field Produced by a Thin Uniform Disk

A researcher calculates the gravitational field produced by a thin circular disk of total mass $$M$$

Gravitational Field Strength Variation

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

Gravitational Parameter in Exoplanetary Systems

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

Gravitational Potential Energy Change for a Satellite

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

Gravitational Potential Energy Change in an Elliptical Orbit

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

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

Integration of Variable Gravitational Force over an Extended Body

Consider a uniform rod of length L and total mass m, oriented radially away from the center of a pla

Kepler's Third Law and Satellite Orbits

Using the provided data for satellite orbits, analyze Kepler’s Third Law. Determine the relationship

Orbital Energy and Conservation Laws

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

Perturbation Analysis of Satellite Orbits

Study the graph showing small deviations from a nominal circular orbit of a satellite. Analyze the p

Verifying Kepler's Second Law and Angular Momentum Conservation

Kepler’s Second Law states that a line joining a planet and the Sun sweeps out equal areas during eq