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Physics For Engineers Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in physics for engineers.

Measurement and Vectors - Theory & Concepts - Significant Figures

Interactive simulation.

Interactive Physics Simulation

Significant Figures & Measurement Lab

Learn the fundamental rules of significant digits. Compare real vs. measured values in the Lab, or parse complex arithmetic in the Sig-Fig Engine.

Interactive Expressions
Significant Figure Guidelines
  • Non-zero digits are always significant ($4.7$ has $2$ sig figs).
  • Captive zeros between non-zeros are significant ($503$ has $3$ sig figs).
  • Leading zeros are never significant ($0.02$ has $1$ sig fig).
  • Trailing zeros are significant only if a decimal point is explicitly shown ($15.0$ has $3$ sig figs; $150$ is ambiguous).
Step-By-Step Parse Logic
Operand A (14.50):

Decimal present. Non-zero digits and trailing zeros are significant. Zeros before the first non-zero digit () are leading zeros and not significant. Total significant figures: 4. Decimal places: 2.

Operand B (3.2):

Decimal present. Non-zero digits and trailing zeros are significant. Zeros before the first non-zero digit () are leading zeros and not significant. Total significant figures: 2. Decimal places: 1.

Multiplication/Division Rule: Round to the least number of significant figures.

Operand A has 4 sig figs. Operand B has 2 sig figs. The least is 2 sig figs. Rounding the raw result (46.400000000000006) to 2 sig figs yields: 46.

Operand A Sig Figs
4
Operand B Sig Figs
2
Least decimals
1
Raw Mathematical Output
46.400000000000006
Significant Figure Rounded
46

Kinematics - Theory & Concepts - Kinematics Graph

Interactive simulation.

Interactive Physics Simulation

Kinematics Motion & Graph Simulator

Adjust initial velocity and acceleration, and scrub time. Analyze how position, velocity, and acceleration curves behave, and observe that the area under the v-t curve equals displacement.

Select Active Graph
6.0 m/s
1.5 m/s²
8.0 s
4.0 s
Governing Formulas
Position & Velocityx(t)=v0t+12at2,v(t)=v0+atx(t) = v_0 t + \frac{1}{2}a t^2, \quad v(t) = v_0 + a t
Calculus Theoremv(t)=dxdt,x(t)=∫0tv(τ)dτ(Area)v(t) = \frac{dx}{dt}, \quad x(t) = \int_{0}^{t} v(\tau) d\tau \quad (\text{Area})
Moving cart synchronized with graphsValueTimePosition vs. Time (x-t)
Position (x)
36.00 m
Velocity (v)
12.00 m/s
Acceleration
1.50 m/s²

Newton's Laws of Motion - Theory & Concepts - Newtons Second Law

Interactive simulation.

Interactive Physics Simulation

Newton's Second Law & Free-Body Diagram Simulator

Apply a constant horizontal pulling force to a rolling cart. Observe how mass and kinetic friction create counteracting forces in a complete Free-Body Diagram (FBD).

8 kg
35 N
0.25
Newtonian Dynamics Rules
Net Pull Force:Fnet=Fa−fkF_{net} = F_a - f_k
Acceleration:a=Fnet/Ma = F_{net} / M
Friction Resistance:fk=μk⋅M⋅gf_k = \mu_k \cdot M \cdot g

If the applied pulling force ($F_a$) is lower than the static friction limit ($f_k$), the cart remains completely at rest.

Friction Resistance (fk)
19.62 N
Net Pulling Force (Fnet)
15.38 N
Acceleration (a)
1.92 m/s²

Work, Energy, and Power - Theory & Concepts - Conservation Energy

Interactive simulation.

Interactive Physics Simulation

Conservation of Energy

Study the conversion between potential and kinetic energy on a frictionless track. The total mechanical energy remains conserved at all points.

2 kg
10 m

Energy Diagnostic Chart

Kinetic Energy (K)0 J
Potential Energy (U)196 J
Total Mechanical Energy (E)196 J
Governing Formulas
Conservation of EnergyE=K+U=constantE = K + U = \text{constant}
Kinetic & Potential EnergyK=12mv2,U=mghK = \frac{1}{2}mv^2, \quad U = mgh
Time0.0 s
Height (h)10.0 m
Speed (v)0.0 m/s
Position (x)0.0 m

Work, Energy, and Power - Theory & Concepts - Power Work Rate

Interactive simulation.

Interactive Physics Simulation

Mechanical Work & Power Vector Simulator

Move the block along the path and rotate the force vector. Visually decompose the force into its parallel and perpendicular components to prove that only forces along the displacement perform work.

θ = 30°
150 N
8.0 m
12 s
30 deg
0.45
Governing Equations
Work Equation
W=F⋅d⋅cos⁡(θ)W = F \cdot d \cdot \cos(\theta)
Average Power
P=WtP = \frac{W}{t}
Visual vector projections of force, parallel work, and motion directionFF·cosθdisplacement path (d = 8 m)
Total Work Transferred (W)
1,039.23 J
Avg Power Rate (P)
86.60 W
Parallel force component (F·cosθ)
129.90 N
Perpendicular force (F·sinθ)
75.00 N

Impulse and Momentum - Theory & Concepts - Impulse Momentum

Interactive simulation.

Interactive Physics Simulation

Impulse-Momentum Collision Pulse Simulator

Compare different collision force profiles (Rectangular, Triangular, or realistic Sinusoidal). Scrub through the motion to see how the area under the force-time curve translates to momentum transfer.

Force Pulse Profile (Shape)
900 N
0.18 s
3.0 kg
0.65
Governing Formulas
Impulse-Momentum TheoremJ=∫Fdt=m⋅ΔvJ = \int F dt = m \cdot \Delta v
Impulse (Area under curve)J=12Fpeak⋅ΔtJ = \frac{1}{2} F_{peak} \cdot \Delta t
Force-time pulse and cart responseForceTime
Total Impulse (J)
81.0 N·s
Velocity Increase (Δv)
27.00 m/s

Impulse and Momentum - Theory & Concepts - Collision Momentum

Interactive simulation.

Interactive Physics Simulation

Momentum & Elastic Collision Simulator

Adjust cart masses, incoming velocities, and coefficient of restitution. Scrub through the timeline and watch momentum conservation in action.

Cart A: 2.0kg
Cart B: 3.0kg
Cart A Parameters
2.0 kg
6.0 m/s
Cart B Parameters
3.0 kg
-2.0 m/s
0.25
0.45
Governing Principles
Conservation of Momentum
mAvAi+mBvBi=mAvAf+mBvBfm_A v_{Ai} + m_B v_{Bi} = m_A v_{Af} + m_B v_{Bf}
Restitution Coefficient (e)
e=vBf−vAfvAi−vBie = \frac{v_{Bf} - v_{Af}}{v_{Ai} - v_{Bi}}
Two carts colliding with dynamic velocity vectors and impact physicsimpact planeAB6.0m/s-2.0m/s
Momentum before collision (p_i)
6.00 kg·m/s
Momentum after collision (p_f)
6.00 kg·m/s
Velocity Cart A
6.00 m/s
Velocity Cart B
-2.00 m/s

Rotational Motion - Theory & Concepts - Rotational Kinematics

Interactive simulation.

Interactive Physics Simulation

Rotational Kinematics & Acceleration Vector Simulator

Study the motion of a particle rotating on a disk. Observe how physical tangential and centripetal acceleration vector components scale at radius R.

3.0 rad/s
0.8 rad/s²
1.5 m
4.0 s
Constant Acceleration Formulas
Final Velocity:ω=ω0+αt\omega = \omega_0 + \alpha t
Displacement:θ=ω0t+12αt2\theta = \omega_0 t + \frac{1}{2}\alpha t^2
Linear Vector Magnitudes:ac=ω2Ra_c = \omega^2 R , at=αRa_t = \alpha R

Note that centripetal acceleration increases with the square of speed, while tangential acceleration remains constant under constant $\alpha$.

Angular Velocity (ω)
6.20 rad/s
Angular displacement (θ)
18.40 rad
Centripetal Accel. (ac)
57.66 m/s²
Tangential Accel. (at)
1.20 m/s²

Rotational Motion - Theory & Concepts - Rotational Inertia

Interactive simulation.

Interactive Physics Simulation

Moment Of Rotational Inertia Simulator

Explore how mass and shape distribution affect resistance to rotation. Apply a tangential force to generate torque and angular acceleration.

5 kg
1.2 m
15 N
Inertia Equation for Profile
I=12MR2I = \frac{1}{2} M R^2

A solid cylinder / disk has its mass evenly distributed from the center axle out to the radius R. More mass distributed far from the center yields a larger moment of inertia ($I$).

Moment of Inertia (I)
3.60 kg·m²
Applied Torque (τ = F·R)
18.00 N·m
Angular Accel. (α = τ/I)
5.00 rad/s²
Angular Velocity (ω)
0.00 rad/s

Rotational Motion - Theory & Concepts - Angular Momentum

Interactive simulation.

Interactive Physics Simulation

Angular Momentum Skater Simulator

Pull the rotating masses inward or outward. When friction and external torque are neglected, moment of inertia decreases, causing spin velocity to increase.

1.30 m
0.70 m
2.0 kg·m²
2.5 rad/s
Governing Laws
Moment of Inertia
I=Ibody+2m⋅r2I = I_{body} + 2m \cdot r^2
Conservation of Momentum
L=Ii⋅ωi=If⋅ωfL = I_i \cdot \omega_i = I_f \cdot \omega_f
Rotating skater demonstrating the conservation of angular momentumOrbit expands as spin speed (ω) increases
Angular Momentum (L)
17.68 kg·m²/s
Current Angular Speed (ω)
5.09 rad/s

Equilibrium and Elasticity - Theory & Concepts - Torque Balance

Interactive simulation.

Interactive Physics Simulation

Torque Balance & Rotational Equilibrium Simulator

Place different masses along a balanced beam. Observe CCW and CW torques, pivot reaction forces, and translational/rotational equilibrium states.

✓ Equilibrium (Στ = 0)
Left Block Parameters (CCW)
20 kg
-3.0 m
Right Block Parameters (CW)
30 kg
2.0 m
Live Torque Summation Equation
Στ=τCCW−τCW=588.6 N⋅m−588.6 N⋅m=0.0 N⋅m\Sigma\tau = \tau_{CCW} - \tau_{CW} = 588.6 \text{ N}\cdot\text{m} - 588.6 \text{ N}\cdot\text{m} = 0.0 \text{ N}\cdot\text{m}

For **Rotational Equilibrium**, counter-clockwise (CCW) torque must exactly equal clockwise (CW) torque ($\Sigma \tau = 0$).

CCW Torque (τ1 = m1·g·r1)
588.60 N·m
CW Torque (τ2 = m2·g·r2)
588.60 N·m
Fulcrum Reaction Force (FR)
588.60 N
Net Residual Torque (Στ)
0.00 N·m

Equilibrium and Elasticity - Theory & Concepts - Elasticity Stress Strain

Interactive simulation.

Interactive Physics Simulation

Tensile Specimen & Stress-Strain Curve

Apply axial tensile force on a specimen. Observe Hooke's Law in the linear elastic region, and the horizontal curving path representing plastic flow beyond the yield threshold.

Elastic Region
Material Preset
35 kN
250 mm²
200 GPa
250 MPa
Governing Formulas
Normal Stress
σ=PA\sigma = \frac{P}{A}
Hooke's Law (Elastic Strain)
ϵ=σE(σ≤σy)\epsilon = \frac{\sigma}{E} \quad (\sigma \le \sigma_y)
Stretching specimen and physical stress-strain graph showing elastic and plastic yielding zonesPStrain (ε)Stress (σ)Yield Threshold (σy = 250 MPa)
Normal Stress (σ)
140.0 MPa
Strain (ε)
7.000e-4
Deformation Mode
ELASTIC

Fluid Mechanics - Theory & Concepts - Fluid Pressure

Interactive simulation.

Interactive Physics Simulation

Hydrostatic Pressure Field & Gate Simulator

Submerge a rectangular vertical gate in a fluid field. Watch the hydrostatic pressure prism grow linearly with depth, and observe how the resultant force anchors precisely at the Center of Pressure (y_cp).

Fresh Water
Fluid Preset
5.0 m
1000 kg/m³
2.0 m
1.5 m
Governing Formulas
Resultant Hydrostatic Force
FR=Pc⋅A=(ρghc)⋅(b⋅h)F_R = P_c \cdot A = (\rho g h_c) \cdot (b \cdot h)
Vertical Center of Pressure
ycp=hc+e=hc+h212hcy_{cp} = h_c + e = h_c + \frac{h^2}{12 h_c}
Submerged gate hydrostatic load profile showing center of pressure offsetFree SurfaceFrpressure increases linearly with depth (Pc = ρgh)
Pressure at Centroid (Pc)
49.05 kPa
Resultant Force (Fr)
147.15 kN
Centroid Depth (hc)
5.00 m
Center of Pressure (y_cp)
5.07 m

Fluid Mechanics - Theory & Concepts - Continuity Bernoulli

Interactive simulation.

Interactive Physics Simulation

Continuity & Bernoulli Venturi Simulator

Adjust the upstream and throat cross-sectional areas. Observe how fluid streamlines compress and accelerate inside the narrow constriction, inducing a corresponding pressure drop.

0.12 m²
0.05 m²
0.24 m³/s
0.4 m
Governing Formulas
Continuity Equation
Q=A1⋅v1=A2⋅v2Q = A_1 \cdot v_1 = A_2 \cdot v_2
Bernoulli's Energy Balance
Δh=v22−v122g+hL\Delta h = \frac{v_2^2 - v_1^2}{2g} + h_L
Venturi tube flow field simulator with acceleration and pressure head pipesP₁ headP₂ head
Upstream Velocity (v1)
2.00 m/s
Constriction Velocity (v2)
4.80 m/s
Total Head Drop (Δh)
1.37 m

Oscillations and Waves - Theory & Concepts - Harmonic Motion

Interactive simulation.

Interactive Physics Simulation

Simple Harmonic Motion & Damping

Study the oscillatory behavior of a mass-spring system. Introduce damping to see how the system transitions from standard oscillation to critical and overdamping.

1.0 kg
10 N/m
0.5 m
0.0 kg/s
Damping Ratio (ζ) & State
0.00Undamped
Governing Formulas
Motion Equationmx¨+cx˙+kx=0m \ddot{x} + c \dot{x} + k x = 0
Natural Angular Freqω0=k/m,ζ=c2km\omega_0 = \sqrt{k/m}, \quad \zeta = \frac{c}{2\sqrt{km}}
Natural Freq0.50 Hz
Period (T)1.99 s
Position (x)0.50 m
Velocity (v)0.00 m/s

Oscillations and Waves - Theory & Concepts - Wave Superposition

Interactive simulation.

Interactive Physics Simulation

Wave Superposition & Beats Simulator

Superpose two independent travelling waves. Adjust phase offsets and frequencies to demonstrate constructive/destructive interference and acoustic beats.

Visual Layers Toggle
Wave A Settings (Green)
1.0 m
1.0 Hz
Wave B Settings (Amber)
0.8 m
1.0 Hz
60 deg
Principle of Superposition
ynet(x,t)=yA(x,t)+yB(x,t)y_{net}(x, t) = y_A(x, t) + y_B(x, t)

When two waves cross, the displacement is the sum of individual amplitudes. If frequencies differ slightly, amplitude modulation (**Beats**) occurs.

State: Partial Interference
Combined Peak Envelope
1.56 m
Beat Frequency offset (|fA - fB|)
0.00 Hz

Thermodynamics - Theory & Concepts - Thermodynamics Cycle

Interactive simulation.

Interactive Physics Simulation

Thermodynamics PV Cycle & Heat Engine Simulator

Select standard thermodynamic cycle profiles (Carnot, Otto, Brayton). Adjust compression ratios, gas parameters, and temperature limits to analyze efficiency.

6.5
850 K
300 K
1.40
Cycle Thermal Efficiency Formula
ηCarnot=1−TCTH\eta_{Carnot} = 1 - \frac{T_C}{T_H}

The Carnot cycle represents the absolute maximum theoretical efficiency possible for any heat engine operating between TH and TC. Larger compression/pressure ratios yield higher overall cycle thermal efficiency.

Heat input (Qin)
13,227.8 J
Net Work output (Wnet)
8,559.2 J
Engine Efficiency (η)
64.7 %
Carnot Max Limit (ηmax)
64.7 %

Thermodynamics - Theory & Concepts - Ideal Gas

Interactive simulation.

Interactive Physics Simulation

Ideal Gas Piston Simulator

Study thermodynamics using a movable piston chamber. Experience kinetic theory as molecules buzz and collide with boundaries at speeds proportional to temperature.

Gas Species
1.5 mol
300 K
0.03 m³
Governing Formulas
Ideal Gas State Equation
P⋅V=n⋅R⋅T  ⟹  P=nRTVP \cdot V = n \cdot R \cdot T \implies P = \frac{n R T}{V}
Root-Mean-Square Velocity
vrms=3RTMv_{rms} = \sqrt{\frac{3 R T}{M}}
Thermodynamic cylinder gas chamber showing kinetic gas molecular motionMolecular speed scales as vrms ∝ √T
Pressure (P)
149.65 kPa
Gas Mass Density
1.680 kg/m³

Electricity and Magnetism - Theory & Concepts - Coulombs Law

Interactive simulation.

Interactive Physics Simulation

Coulomb's Law Electrostatic Simulator

Adjust the charges on two point particles and see how the electrostatic force scales with charge magnitudes and separation distance. Opposite charges attract; like charges repel.

Attracts
5 μC
-5 μC
10 cm
Governing Formula
F=ke⋅∣q1⋅q2∣r2F = k_e \cdot \frac{|q_1 \cdot q_2|}{r^2}

Where electrostatic constant $k_e \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$. Force is proportional to charge product and inversely proportional to $r^2$.

Coulomb electrostatic charge interactions and vector forcesr = 10 cmF₁₂F₂₁Charge 1: 5 μC+5Charge 2: -5 μC-5
Force magnitude (F)
22.47 N
Interaction Type
Attractive

Electricity and Magnetism - Theory & Concepts - Circuit Ohms Law

Interactive simulation.

Interactive Physics Simulation

Ohm's Law Circuit Simulator

Switch between series and parallel resistance. Current, power, electron-flow speed, and bulb brightness update immediately.

12 V
6 Ω
4 Ω
Governing Formulas
Ohm's LawI=VRI = \frac{V}{R}
Equivalent Resistance (series)Req=R1+R2R_{eq} = R_1 + R_2
Circuit with battery resistors and bulbR1R2
Equivalent R
10.00 Ω
Current
1.20 A
Power
14.40 W

Optics and Light - Theory & Concepts - Geometric Optics

Interactive simulation.

Interactive Physics Simulation

Snell's Law & Refraction Simulator

Experiment with geometric optics by beaming a light ray from one medium to another. Analyze Snell's Law and Total Internal Reflection.

Standard Refraction
Medium 1 Preset (Top)
Medium 2 Preset (Bottom)
45 deg
1.00
1.52
Governing Formulas
Snell's Lawn1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
Critical Angle (for TIR)θc=arcsin⁡(n2n1)(if n1>n2)\theta_c = \arcsin\left(\frac{n_2}{n_1}\right) \quad (\text{if } n_1 > n_2)
Angle θ₁
45.0°
Angle θ₂
27.7°
Critical Angle (θc)
N/A

Optics and Light - Theory & Concepts - Lens Mirror

Interactive simulation.

Interactive Physics Simulation

Thin Lens & Curved Mirror Ray Simulator

Toggle between curved optical devices and adjust physical metrics. Observe how real incident light rays refract or reflect to form real, virtual, upright, or inverted images.

Real Image
Inverted
30 cm
12 cm
6.0 cm
Governing Formulas
Gaussian Optical Equation
1f=1do+1di  ⟹  di=do⋅fdo−f\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \implies d_i = \frac{d_o \cdot f}{d_o - f}
Linear Magnification
m=−dido=hihom = -\frac{d_i}{d_o} = \frac{h_i}{h_o}
Calibrated optical ray diagram showing thin lens refraction and curved mirror reflectionsconverging LensFF'objimg
Image Distance (di)
20.0 cm
Magnification (m)
-0.67x
Image Quality
REAL