Kinematics and Dynamics of Rigid Bodies

Dynamics deals with bodies in motion. GATE ME tests both particles and rigid bodies in plane motion: translation, rotation about a fixed axis, and general plane motion such as rolling.

Particle kinematics

For constant acceleration a:

• v = u + at
• s = ut + ½at²
• v² = u² + 2as

Projectile launched at speed u and angle θ (g downwards, no air resistance):

• Time of flight T = 2u sin θ / g
• Maximum height H = u² sin² θ / (2g)
• Range R = u² sin 2θ / g (maximum at θ = 45°)

Circular motion at speed v on radius r: centripetal acceleration v²/r = ω²r towards the centre.

Rigid-body kinematics

• Rotation about a fixed axis: v = ωr, tangential acceleration = αr, normal acceleration = ω²r.
• Instantaneous centre (IC) of rotation: at any instant, a body in plane motion rotates about a point of zero velocity. Every point's velocity is perpendicular to the line joining it to the IC, with magnitude ω × distance.
• A wheel rolling without slipping: the contact point is the IC, so v_centre = ωr and the top point moves at 2v.

Newton's laws for a rigid body

• Translation: ΣF = m a_G (a_G = acceleration of the centre of mass).
• Rotation: ΣM_G = I_G α, or about a fixed axis O: ΣM_O = I_O α.

Moments of inertia about the centroidal axis:

Parallel axis theorem: I = I_G + md². Radius of gyration k: I = mk².

Rolling down an incline

A body of radius r and I_G = mk² rolling without slipping down an incline at angle θ has

a = g sin θ / (1 + k²/r²).

A solid cylinder (k²/r² = 1/2) gets a = (2/3) g sin θ; a ring (k²/r² = 1) gets (1/2) g sin θ; a sliding block with no friction gets g sin θ.

Worked example

A uniform rod of length 1.2 m and mass 3 kg is pinned at one end and released from horizontal. Find its angular acceleration at release (g = 9.81 m/s²).

• Moment of weight about the pin: 3 × 9.81 × 0.6 = 17.66 N·m.
• I about the end = mL²/3 = 3 × 1.44/3 = 1.44 kg·m².
• α = 17.66 / 1.44 = 12.26 rad/s² (that is 3g/(2L)).

Practice