Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For the derivations, diagrams and worked examples, use the full study guide.
Recap
- Rotational equilibrium: the net torque on the system is zero. Then ω is constant. The constant can be zero (a resting shelf) or not (a fan at steady speed).
- Newton’s first law, rotational form: ω stays constant only if Στ = 0. Corollary: unbalanced torques mean ω is changing.
- Translational equilibrium (ΣF = 0) is a separate condition. A couple gives ΣF = 0 with Στ ≠ 0; a tumbling object in free fall has Στ = 0 about its center of mass with ΣF ≠ 0.
- Draw an extended force diagram: the object’s outline, each force starting where it acts, the axis marked.
- One plane of rotation only: state “counterclockwise positive” and give every torque a sign.
Key relationships
| Relationship | Meaning | When to use |
|---|---|---|
| τ = rF sin θ = r⊥F | size of one torque | every force; r⊥ is the lever arm |
| Στ = 0 | rotational equilibrium | ω constant |
| ΣF_x = 0, ΣF_y = 0 | translational equilibrium | v_cm constant |
| τ_grav = −g∫x dm = −Mg x_cm | weight acts at the center of mass | extended or non-uniform objects |
| x_cm = (1/M)∫x λ(x) dx, M = ∫λ dx | center of mass of a rod | λ(x) given |
| τ_Q = τ_P − d × ΣF | torque about a new point | if ΣF = 0, the torque point does not matter |
Assumptions behind the numbers
- Rigid objects; all forces and rotation in one plane.
- g = 9.8 m/s². Light (massless) strings and cables unless a mass is given.
- “About to tip” means the support on the far side has just reached zero force.
Mistakes to avoid
- Weight at the middle of a non-uniform object. Integrate for x_cm first.
- Distance instead of lever arm. Use the perpendicular distance to the line of action.
- Ignoring a pivot force in ΣF. It has no torque about the pivot but still balances forces.
- “Spinning means not in equilibrium.” Constant ω is equilibrium.
- “ΣF = 0 means no rotation.” Look for a couple.
- Choosing a torque point that keeps every unknown. Take torques where an unknown force acts.
- Forcing a unique answer when there are more unknowns than equations (statically indeterminate).
Quick self-check
- A uniform plank (10 kg, 2.0 m) rests on supports at its left end and 1.6 m from it. Find both support forces. (Torques about the left end: N(1.6) = 98(1.0), so N = 61 N; the left support gives 98 − 61 = 37 N.)
- A 0.90 m rod has λ = kx, where x is measured from its thin end. Where is its balance point? (x_cm = ∫kx² dx ÷ ∫kx dx = 2L/3 = 0.60 m from the thin end.)
- Two opposite 5.0 N forces act on a free board along parallel lines 0.30 m apart. Find ΣF and the net torque. (ΣF = 0; Στ = 5.0 × 0.30 = 1.5 N·m about every point, so ω changes.)
Next: practice questions.