⚙ Mechanics

Kinematics, dynamics, work and energy, systems of particles, collisions, moments of inertia and gravitation. Every topic with step-by-step theory, interactive diagrams and solved exercises.

📋 Formulario

Complete Theory

Worked Examples

Example 1Race down the ramp — sphere vs cylinder
Example 2Bar against a wall — static equilibrium
Example 3Thrown ball that slips: translating and rotating with kinetic friction

Exercises with Solutions

Exercise 1Moment of inertiaMedium
📋 Problem to solve
A uniform disk of mass m=2m = 2 kg and radius R=0.3R = 0.3 m rotates about its central axis at constant angular velocity ω=10\omega = 10 rad/s. Compute: (a) the moment of inertia of the disk, (b) the rotational kinetic energy stored, (c) the magnitude of the angular momentum. Interpret the results physically.
📌 Given data
m = 2 kgR = 0.3 mω = 10 rad/sUniform disk → I = ½mR²
Exercise 2Rolling motionMedium
📋 Problem to solve
A uniform solid cylinder of mass m=3m = 3 kg and radius R=0.1R = 0.1 m rolls without slipping on a horizontal plane. The center of mass moves at constant speed vCM=2v_{CM} = 2 m/s. Compute the total kinetic energy of the cylinder and the fraction of energy stored in rotation. What would change if it were a solid sphere?
📌 Given data
m = 3 kgR = 0.1 mv_CM = 2 m/sSolid cylinder → I = ½mR²
Exercise 3Static equilibriumHard
📋 Problem to solve
A ladder of mass m=10m = 10 kg and length L=4L = 4 m leans against a smooth (frictionless) vertical wall at an angle θ=60°\theta = 60° with the horizontal ground, which is rough (static friction present). A worker of mass M=70M = 70 kg climbs to a position 3/43/4 of the way up the ladder measured from the bottom. Compute the reaction forces from the ground and the wall. Verify that static friction is sufficient (μs=0.5\mu_s = 0.5).
📌 Given data
m = 10 kg (ladder mass)L = 4 m (ladder length)M = 70 kg (worker mass)θ = 60° (angle with ground)Smooth wall → F_W only horizontalµ_s = 0.5 (static friction coefficient at ground)
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Recommended Books

Introductory
Physics for Scientists and Engineers
Serway, Jewett
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Advanced
Classical Mechanics
Goldstein, Poole, Safko
Buy on Amazon →

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Integrative Problems

Problems combining all chapters — exam level
Problem 1Tower, Ballistic Pendulum, and Keplerian OrbitEXTREME
A cannon is placed on top of a tower h0=50 mh_0 = 50\,\mathrm{m} tall and fires a projectile of m=0.025 kgm = 0.025\,\mathrm{kg} horizontally at v0=400 m/sv_0 = 400\,\mathrm{m/s}.

The projectile strikes and embeds in a wooden block M=4.0 kgM = 4.0\,\mathrm{kg} hanging from a rope of length L=2.0 mL = 2.0\,\mathrm{m} (ballistic pendulum), at ground level.

The Earth-Moon system is then used as a reference for Kepler's third law.
📌 Problem data
h0=50 mh_0 = 50\,\mathrm{m}m=0.025 kgm = 0.025\,\mathrm{kg}v0=400 m/sv_0 = 400\,\mathrm{m/s}M=4.0 kgM = 4.0\,\mathrm{kg}L=2.0 mL = 2.0\,\mathrm{m}
(a)Uniformly Accelerated Motion(b)Inelastic Collision(c)Potential Energy + Pendulum(d)Moment of Inertia — Rigid Body(e)Gravitation — Kepler's Third Law
Problem 2Spring, Rolling Disk, Inclined Plane Collision, and ConservationEXTREME
A spring (k=6000 N/mk = 6000\,\mathrm{N/m}, compressed x0=0.25 mx_0 = 0.25\,\mathrm{m}) launches a solid disk (M=3.0 kgM = 3.0\,\mathrm{kg}, R=0.15 mR = 0.15\,\mathrm{m}) up an inclined plane (θ=30°\theta=30°, L=5 mL=5\,\mathrm{m}, μd=0.06\mu_d=0.06) that rolls without slipping.

At the top the disk is launched horizontally and strikes a pendulum (mp=2.0 kgm_p=2.0\,\mathrm{kg}, l=1.5 ml=1.5\,\mathrm{m}) — perfectly inelastic collision. θ=30°springdisk (rolls)H=2.5 mhoriz. launchpendulumθ_max✱ What is asked (solved below, a→e): (a) the disk's speed at the top of the plane; (b) the range and impact speed of the horizontal launch; (c) the speed after the inelastic collision with the pendulum and the energy lost; (d) the pendulum's maximum angle, the maximum tension, and whether it completes the loop; (e) the full energy balance (from spring to maximum angle).
📌 Problem data
k=6000 N/mk = 6000\,\mathrm{N/m}x0=0.25 mx_0 = 0.25\,\mathrm{m}θ=30°,  L=5 m,  μd=0.06\theta=30°,\;L=5\,\mathrm{m},\;\mu_d=0.06Mdisk=3.0 kg,  R=0.15 mM_{disk}=3.0\,\mathrm{kg},\;R=0.15\,\mathrm{m}Htop=Lsin⁡θ=2.5 mH_{top}=L\sin\theta=2.5\,\mathrm{m}mp=2.0 kg,  l=1.5 mm_p=2.0\,\mathrm{kg},\;l=1.5\,\mathrm{m}
(a)Energy + Rigid Body (rolling)(b)Kinematics — Projectile(c)Inelastic Collision + CM(d)Pendulum Dynamics + Forces(e)Conservation Laws — Complete Energy Balance