PHSX 220 Physics I (w/ Calculus) · Unit 2: Applying Newton, statics, fluids, energy · on Exam 2
Ch 12.1–12.2 Static equilibrium and torque
Study notes for Static equilibrium and torque in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Torque measures a force's ability to rotate an object about a pivot: $\tau = rF\sin\theta = F\,d_\perp$ (lever arm $d_\perp$ is the perpendicular distance from the pivot to the line of the force). Counterclockwise is usually taken positive.
- Static equilibrium needs both $\sum\vec F = 0$ and $\sum\tau = 0$ about any point. Choose the pivot where an unknown force acts so that it drops out of the torque equation.
- Weight acts at the center of gravity. A uniform beam's weight acts at its midpoint.
- Classic set-ups: seesaw/balance ($m_1d_1 = m_2d_2$), beam on two supports, ladder against a wall (friction at the floor, normal at the wall), hanging sign with a strut.
Key formulas
Torque
$$\tau = rF\sin\theta = F\,d_\perp,\qquad \vec\tau = \vec r\times\vec F$$
Equilibrium
$$\sum F_x = 0,\quad \sum F_y = 0,\quad \sum\tau = 0\ \text{(any pivot)}$$
Balance
$$m_1 d_1 = m_2 d_2$$
Worked example
A 6.0 m uniform plank (20 kg) rests on supports at its ends. A 60 kg person stands 2.0 m from the left end. Find the support forces.
Show the solution
Torques about the left end: $F_R(6) = 20(9.8)(3) + 60(9.8)(2) = 588 + 1176 = 1764 \Rightarrow F_R = 294$ N. Forces: $F_L = (80)(9.8) - 294 = 490$ N.
Common mistakes
- Using the full distance instead of the perpendicular lever arm for an angled force.
- Forgetting the beam’s own weight at its center.
- Assigning the same sign to clockwise and counterclockwise torques.
Exam tipPick the pivot at the point with the most unknown forces. One clean torque equation usually solves the whole problem.
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