⚙ 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 1Mass-spring oscillator: period, energy, and maximum velocity
Example 2Simple pendulum — Earth vs Moon

Exercises with Solutions

Exercise 1SHM with initial conditionsHard
📋 Problem to solve
A mass-spring oscillator (m=0.3 kgm = 0.3\,\mathrm{kg}, k=120 N/mk = 120\,\mathrm{N/m}) is set in motion with x0=0x_0 = 0 and v0=2.5 m/sv_0 = 2.5\,\mathrm{m/s}. (a) Determine the amplitude AA and initial phase ϕ\phi of the motion. (b) Calculate the position xx at time t=0.1 st = 0.1\,\mathrm{s}. (c) Find the first time (t>0t > 0) when the mass passes through x=−A/2x = -A/2.
📌 Given data
m = 0.3\,\mathrm{kg}k = 120\,\mathrm{N/m}x0=0x_0 = 0 (starts at equilibrium)v0=2.5 m/sv_0 = 2.5\,\mathrm{m/s} (initial velocity)
Exercise 2Damping — underdamped regimeHard
📋 Problem to solve
A damped mass-spring oscillator has m=0.5 kgm = 0.5\,\mathrm{kg}, k=50 N/mk = 50\,\mathrm{N/m}, damping constant b=1.0 N⋅s/mb = 1.0\,\mathrm{N\cdot s/m}. (a) Determine the damping regime by comparing γ\gamma and ω0\omega_0. (b) Calculate the damped angular frequency ω1\omega_1. (c) Find the time t1/2t_{1/2} required for the amplitude to drop to half its initial value. (d) Compute the quality factor QQ and interpret it.
📌 Given data
m = 0.5\,\mathrm{kg}k = 50\,\mathrm{N/m}b = 1.0\,\mathrm{N\cdot s/m}
Exercise 3Physical pendulum — rotating diskVery Hard
📋 Problem to solve
A solid homogeneous disk of mass M=2 kgM = 2\,\mathrm{kg} and radius R=0.3 mR = 0.3\,\mathrm{m} is suspended from a horizontal pivot at a point on its rim. The disk oscillates as a physical pendulum. (a) Calculate the moment of inertia of the disk about the pivot. (b) Determine the period TT of small oscillations. (c) Find the length LeqL_{eq} of a simple pendulum that would have the same period.
📌 Given data
M = 2\,\mathrm{kg}R = 0.3\,\mathrm{m}d=R=0.3 md = R = 0.3\,\mathrm{m} (pivot-to-CM distance)
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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