⚙ Meccanica

Cinematica, dinamica, lavoro ed energia, sistemi di punti, urto, momenti di inerzia e gravitazione.

📋 Formulario

Complete Theory

Worked Examples

Example 1Elastic 1D collision — equal mass billiard balls
Example 2Perfectly inelastic collision — vehicle crash

Exercises with Solutions

Exercise 1Elastic 2D collisionHard
📋 Problem to solve
Two balls of equal mass m1=m2=0.3 kgm_1 = m_2 = 0.3\,\mathrm{kg} collide elastically. Ball 1 moves initially with speed v1=6 m/sv_1 = 6\,\mathrm{m/s} east, while ball 2 is at rest. After the collision, ball 1 is deflected by θ1=30°\theta_1 = 30° from its original direction. Determine: (a) the speed v1′v_1' of ball 1 after the collision, (b) the speed v2′v_2' and direction θ2\theta_2 of ball 2 after the collision.
📌 Given data
m_1 = m_2 = 0.3\,kg (equal masses)v_1 = 6\,m/s (initial speed of ball 1)v_2 = 0 (ball 2 at rest)\theta_1 = 30° (deflection angle of ball 1)
Exercise 2Rolling diskVery Hard
📋 Problem to solve
A uniform solid disk of mass M=2 kgM = 2\,\mathrm{kg} and radius R=0.15 mR = 0.15\,\mathrm{m} rolls without slipping down an incline of θ=25°\theta = 25° and length L=3 mL = 3\,\mathrm{m}, starting from rest. Determine: (a) the acceleration of the centre of mass, (b) the CM speed at the bottom of the incline, (c) the static friction force required for rolling.
📌 Given data
M = 2\,kg (disk mass)R = 0.15\,m (disk radius)\theta = 25° (incline angle)L = 3\,m (incline length)
📚

Recommended Books

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

As an Amazon Associate I earn from qualifying purchases.

🔥

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
h_0 = 50\,\mathrm{m}m = 0.025\,\mathrm{kg}v_0 = 400\,\mathrm{m/s}M = 4.0\,\mathrm{kg}L = 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.
📌 Problem data
k = 6000\,\mathrm{N/m}x_0 = 0.25\,\mathrm{m}\theta=30°,\;L=5\,\mathrm{m},\;\mu_d=0.06M_{disk}=3.0\,\mathrm{kg},\;R=0.15\,\mathrm{m}H_{top}=L\sin\theta=2.5\,\mathrm{m}m_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