⚙ 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 1Escape velocity from Earth and Moon
Example 2Geostationary orbit altitude

Exercises with Solutions

Exercise 1Elliptical orbitHard
📋 Problem to solve
A planet orbits the Sun on an elliptical orbit with semi-major axis a=2.5 AUa = 2.5\,\mathrm{AU} (Astronomical Units, 1 AU = 1.496×1011 m1.496\times10^{11}\,\mathrm{m}) and eccentricity e=0.4e = 0.4. Given: a=3.74×1011 ma = 3.74\times10^{11}\,\mathrm{m}, GM⊙=1.327×1020 m3/s2GM_\odot = 1.327\times10^{20}\,\mathrm{m^3/s^2}. Determine: (a) the orbital period TT in years, (b) the speed at perihelion vpv_p and aphelion vav_a, knowing that rp=a(1−e)r_p = a(1-e) and ra=a(1+e)r_a = a(1+e).
📌 Given data
a = 3.74\times10^{11}\,m (semi-major axis)e = 0.4 (eccentricity)GM_\odot = 1.327\times10^{20}\,m^3/s^2 (solar constant)
Exercise 2Three collinear massesHard
📋 Problem to solve
Three point masses are arranged along the x-axis: m1=5×1010 kgm_1 = 5\times10^{10}\,\mathrm{kg} at x=0x=0, m2=2×1010 kgm_2 = 2\times10^{10}\,\mathrm{kg} at x=4 mx=4\,\mathrm{m}, m3=3×1010 kgm_3 = 3\times10^{10}\,\mathrm{kg} at x=10 mx=10\,\mathrm{m}. Determine the net gravitational force (magnitude and direction) acting on m2m_2.
📌 Given data
m_1 = 5\times10^{10}\,kg (at x=0)m_2 = 2\times10^{10}\,kg (at x=4\,m)m_3 = 3\times10^{10}\,kg (at x=10\,m)
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Recommended Books

Introductory
Physics for Scientists and Engineers
Serway, Jewett
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Advanced
Classical Mechanics
Goldstein, Poole, Safko
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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