Chapter 17 — Planar Kinematics of Rigid Bodies

Rigid Body in Planar Motion — §17.1

Is the point fixed to the rigid body you are analyzing?

Details

A rigid body keeps the distance between any two of its points constant. Planar motion: every point moves parallel to one fixed plane, and ω = ωk is perpendicular to that plane.

Name the body first. A piston moves relative to the ground, but its pin is fixed to the connecting rod.

YES — the point is fixed to the body

How does the body move?

Details

Clues: attached, pinned, welded, a point “of the body”, the center of a rolling wheel.

NO — the point moves relative to the body

It slides along the body, or it is seen from a moving frame. Frame xyz fixed to the body, origin B:

v_A = v_B + v_{A rel} + ω × r_A/B

a_A = a_B + a_{A rel} + 2ω × v_{A rel} + α × r_A/B − ω²r_A/B

Need ω, α of that body? Get them from the YES side first.

Details

v_{A rel}, a_{A rel}: the motion of A seen by an observer riding on the body (along the slot or bar).

2ω × v_{A rel} is the Coriolis acceleration: it appears when A slides on a body that rotates.

If A does not move on the body, v_{A rel} = a_{A rel} = 0 and these become the YES-side equations.

Translation — §17.1

The body moves without rotating.

ω = 0, α = 0

v_A = v_B, a_A = a_B

Details

Every point has the same velocity and acceleration: describe one point (Chapter 13).

The points may move on curved paths while the body stays level (a swing’s seat bar).

Rotation About a Fixed Axis — §17.2

The body turns about an axis fixed in the ground.

ω = dθ/dt, α = dω/dt

v = rω, a_t = rα, a_n = rω²

v = ω × r, a = α × r − ω²r

Details

This is §17.3/§17.5 with B on the axis (v_B = a_B = 0).

Constant α only: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt², ω² = ω₀² + 2α(θ − θ₀).

Meshing gears:

r_Aω_A = r_Bω_B, r_Aα_A = r_Bα_B

Same speed and tangential acceleration at the contact point; the normal accelerations differ.

General Planar Motion

The body translates and rotates (connecting rod, rolling wheel, sliding ladder).

Velocities first — then accelerations.

Sliding Contacts — §17.6

A slides along a moving or rotating body.

Pin in a slot, collar on a rotating bar. Attach xyz to the body A slides on.

Details

Typical unknowns: ω of the slotted bar, v_{A rel} along the slot. Match i and j components.

A piston in a fixed guide is not §17.6: its pin is fixed to the connecting rod (YES side).

Moving Reference Frames — §17.7

The general version of §17.6: motion seen from any moving frame.

An observer in a vehicle, a ship, a plane, or on a turntable. Same equations; B is the frame’s origin.

Details

Frame on a vehicle moving on a curve of radius ρ: ω = v_B/ρ, α = (a_t)_B/ρ.

Using ΣF = ma in a frame that accelerates or rotates (Earth, ship, plane): see §17.7. An Earth-fixed frame can usually be treated as inertial.

Relative Velocity — §17.3

You know the velocity of another point of the same body.

v_A = v_B + ω × r_A/B

Details

A moves on a circle about B, so v_A/B = r_A/Bω, ⊥ to r_A/B.

Write ω = ωk (counterclockwise +). A negative answer means clockwise.

Sliders and walls fix the direction of a velocity: use that component equation first.

Instantaneous Center — §17.4

You can find a point with zero velocity right now.

v_P = ω r_P/C, ⊥ to r_P/C

Velocities only — never accelerations.

Details

C is where the perpendiculars to two known velocity directions meet.

Rolling without slipping: the contact point is C; v_G = Rω, top point 2Rω.

If the perpendiculars are parallel, C is at infinity: the body translates (ω = 0).

Relative Acceleration — §17.5

Asked for accelerations? (Only velocities asked: stop above.)

a_A = a_B + α × r_A/B − ω²r_A/B

Needs every ω from the velocity step.

Details

α × r_A/B is ⊥ to r_A/B; −ω²r_A/B points from A toward B.

Rolling without slipping: a_G = Rα; the contact point has a_C = Rω² toward the center — not zero.

Linkages: write the acceleration of the shared pin from each bar and match i and j components.