⚙ 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 1Block on inclined plane with friction
Example 2Atwood machine
Example 3Compressed spring — launch speed

Exercises with Solutions

Exercise 1Inclined plane — equilibrium?Hard
📋 Problem to solve
A block of mass m=8 kgm = 8\,\mathrm{kg} rests on an incline at θ=35°\theta = 35°. The coefficient of static friction is μs=0.45\mu_s = 0.45, kinetic friction μk=0.35\mu_k = 0.35. Determine: (a) whether the block remains at rest or slides, (b) if it slides, the acceleration, (c) the distance travelled after 2 seconds starting from rest.
📌 Given data
m = 8\,kg (block mass)\theta = 35° (incline angle)\mu_s = 0.45 (static friction)\mu_k = 0.35 (kinetic friction)
Exercise 2Bullet embedding in pendulumVery Hard
📋 Problem to solve
A bullet of mass mp=0.02 kgm_p = 0.02\,\mathrm{kg} (20 g) travels at v0=300 m/sv_0 = 300\,\mathrm{m/s} and embeds itself in a wooden block of mass M=2 kgM = 2\,\mathrm{kg} suspended by a string of length L=0.8 mL = 0.8\,\mathrm{m} (ballistic pendulum). Determine: (a) the speed of the block+bullet immediately after the collision, (b) the maximum height reached by the pendulum, (c) the tension in the string right after the collision (at the lowest point).
📌 Given data
m_p = 0.02\,kg (bullet mass)v_0 = 300\,m/s (bullet speed)M = 2\,kg (block mass)L = 0.8\,m (string length)
Exercise 3Road curveHard
📋 Problem to solve
A car of mass m=1200 kgm = 1200\,\mathrm{kg} travels around a circular curve of radius r=80 mr = 80\,\mathrm{m} at speed v=20 m/sv = 20\,\mathrm{m/s} (72 km/h). The coefficient of static friction between tyres and asphalt is μs=0.5\mu_s = 0.5. (a) Is friction sufficient to keep the car on the curve? If not, what is the maximum safe speed? (b) If the road is banked at an angle φ=15°\varphi = 15°, what is the ideal speed (no friction needed)?
📌 Given data
m = 1200\,kg (car mass)r = 80\,m (curve radius)v = 20\,m/s (speed)\mu_s = 0.5 (static friction)
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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