Rigid body mechanics
Rigid Body Mechanics
Until now, we’ve treated objects as point particles where all mass concentrates at a single point. This simplification works well for translational motion, but real objects have size and shape. They can rotate, spin, roll, and pivot, not just slide.
Rigid body: An idealized extended object in which the distance between any two points remains constant regardless of external forces. While no real object is perfectly rigid (all materials deform slightly under stress), many objects approximate rigid body behavior well enough for practical analysis: wheels, doors, rods, gears, planets.
Rigid body mechanics extends Newton’s laws to rotational motion.
Rotational Kinematics: Angular Quantities
Angular displacement (θ): The angle through which a point, line, or body has rotated about a specified axis.
SI unit: radian (rad), where 2π rad = 360° = 1 complete revolution
Angular velocity (ω): The rate of change of angular displacement.
ω = dθ/dt (instantaneous) | ω_avg = Δθ/Δt (average)
SI unit: rad/s (radians per second)
Angular acceleration (α): The rate of change of angular velocity.
α = dω/dt (instantaneous) | α_avg = Δω/Δt (average)
SI unit: rad/s² (radians per second squared)
Relationship Between Linear and Angular Quantities
s = rθ (arc length)
v = rω (tangential speed)
a_t = rα (tangential acceleration)
a_c = v²/r = rω² (centripetal acceleration)
Example: A wheel of radius 0.3 m rotates at 5 rad/s. Find the linear speed of a point on its rim.
v = rω = (0.3)(5) = 1.5 m/s
Rotational Kinematics Equations
When angular acceleration is constant, rotational kinematics equations directly parallel linear equations:
| Linear | Rotational |
|---|---|
| v = u + at | ω = ω₀ + αt |
| s = ut + ½at² | θ = ω₀t + ½αt² |
| v² = u² + 2as | ω² = ω₀² + 2αθ |
| s = ½(u + v)t | θ = ½(ω₀ + ω)t |
Torque
τ = r × F (vector product)
Magnitude: τ = rF sin θ
r = lever arm (perpendicular distance to axis)
θ = angle between r and F
Torque is the rotational equivalent of force. It’s the tendency of a force to cause rotation about an axis.
SI unit: N·m (newton-meter)
Sign convention:
- Counterclockwise torque: positive (+)
- Clockwise torque: negative (-)
Example: You push a door with 20 N perpendicular to its surface at a point 0.8 m from the hinges. Calculate the torque.
τ = rF sin 90° = (0.8)(20)(1) = 16 N·m
Moment of Inertia
For a point mass: I = mr²
For extended objects: I = Σmᵢrᵢ² or I = ∫r²dm
Moment of inertia (rotational inertia) is the rotational analog of mass. It measures an object’s resistance to angular acceleration about a particular axis.
SI unit: kg·m²
The moment of inertia depends on:
- Mass (more mass → larger I)
- Distribution of mass relative to axis (mass farther from axis → much larger I, because of r²)
- Choice of rotation axis
Common Moments of Inertia
| Object | Axis | Moment of Inertia |
|---|---|---|
| Thin rod (length L) | Through center, perpendicular | I = (1/12)ML² |
| Thin rod (length L) | Through end, perpendicular | I = (1/3)ML² |
| Solid sphere (radius R) | Through center | I = (2/5)MR² |
| Hollow sphere (radius R) | Through center | I = (2/3)MR² |
| Solid cylinder (radius R) | Through central axis | I = (1/2)MR² |
| Thin hoop (radius R) | Through center, perpendicular | I = MR² |
Parallel Axis Theorem: I = I_cm + Md²
I_cm = moment about center of mass, d = distance between parallel axes
Newton's Second Law for Rotation
τ_net = Iα
This is the rotational analog of F_net = ma.
Example: A wheel with I = 0.5 kg·m² experiences a net torque of 2 N·m. Find its angular acceleration.
α = τ_net/I = 2/0.5 = 4 rad/s²
Rotational Kinetic Energy
KE_rot = ½Iω²
For an object undergoing both translation and rotation (like a rolling wheel):
KE_total = ½mv²_cm + ½I_cm ω²
Example: A solid cylinder (M = 2 kg, R = 0.1 m) rolls without slipping at v = 3 m/s. Calculate total kinetic energy.
I = ½MR² = ½(2)(0.1)² = 0.01 kg·m²
v = Rω → ω = 3/0.1 = 30 rad/s
KE_trans = ½(2)(3)² = 9 J
KE_rot = ½(0.01)(30)² = 4.5 J
KE_total = 13.5 J
Work Done by Torque
W = τθ (for constant torque)
P = τω (power delivered by torque)
θ must be in radians.
Angular Momentum
L = Iω (for rigid body rotating about fixed axis)
L = r × p = mvr sin θ (for a point mass)
Angular momentum is the rotational analog of linear momentum.
SI unit: kg·m²/s or J·s
Conservation of Angular Momentum
If τ_ext = 0, then L = constant
ΣL_initial = ΣL_final
Example: Figure skater spin
Skater spins with arms extended (large I, modest ω). Pulling arms in decreases I. To conserve L = Iω, ω must increase — the skater spins faster.
If I_f = ½I_i, then ω_f = 2ω_i (angular velocity doubles).
Equilibrium of Rigid Bodies
For a rigid body in complete equilibrium:
ΣF = 0
Στ = 0 (about any axis)
Example: A uniform beam (length 5 m, mass 100 kg) is supported at its center. A 50 kg person stands 1.5 m from the left end. Where should a 60 kg person stand to balance?
Taking torques about the support point (center):
τ_cw = (50)(9.81)(1.0) = 490.5 N·m
τ_ccw = (60)(9.81)(x) = 490.5 → x = 0.833 m to the right of center
Rolling Motion
Rolling without slipping: The contact point between the rolling object and surface is instantaneously at rest.
v_cm = Rω
For an object rolling down an incline from rest through vertical height h:
mgh = ½mv²_cm + ½I_cm ω²
mgh = ½mv²_cm(1 + I_cm/(MR²))
Objects with smaller I/(MR²) accelerate faster down inclines.
- Solid sphere (I/(MR²) = 2/5) → fastest
- Hollow sphere (I/(MR²) = 2/3)
- Hollow cylinder (I/(MR²) = 1) → slowest