PHSX 220 Physics I (w/ Calculus) · Unit 2: Applying Newton, statics, fluids, energy · on Exam 2
Ch 6.3–6.4 Centripetal force, banked curves, and drag
Study notes for Centripetal force, banked curves, and drag 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
- "Centripetal force" is not a new force. It is the name for the net force component toward the center, $\sum F_{\text{radial}} = mv^2/r$, supplied by whatever real forces are present: tension, friction, normal force, gravity.
- Flat curve: friction supplies it, so $v_{\max} = \sqrt{\mu_s g r}$. Banked curve with no friction: $\tan\theta = v^2/(rg)$.
- Vertical circles: at the bottom $T - mg = mv^2/r$ (tension largest); at the top $T + mg = mv^2/r$, so the minimum speed to keep a taut string or stay on a loop is $v = \sqrt{gr}$.
- Drag grows with speed ($F_D = \tfrac12 C\rho A v^2$ for air). Terminal speed is reached when drag equals weight: net force zero, constant velocity, $v_t = \sqrt{2mg/(C\rho A)}$.
Key formulas
Radial second law
$$\sum F_{\text{radial}} = \frac{mv^2}{r}$$
Curves
$$v_{\max} = \sqrt{\mu_s g r}\ \text{(flat)},\qquad \tan\theta = \frac{v^2}{rg}\ \text{(banked, no friction)}$$
Vertical circle
$$T_{\text{bottom}} = m\!\left(g + \frac{v^2}{r}\right),\quad T_{\text{top}} = m\!\left(\frac{v^2}{r} - g\right),\quad v_{\min,\text{top}} = \sqrt{gr}$$
Terminal speed
$$v_t = \sqrt{\frac{2mg}{C\rho A}}$$
Worked example
A 1200 kg car rounds a flat curve of radius 50 m at 15 m/s. Find the required friction force and the minimum $\mu_s$.
Show the solution
$F = mv^2/r = 1200(225)/50 = 5400$ N toward the center. $\mu_s \ge F/(mg) = 5400/11760 = 0.46$.
Common mistakes
- Adding a "centrifugal force" pointing outward on the free-body diagram.
- Forgetting gravity in a vertical circle.
- Taking the top-of-loop minimum speed as zero.
Exam tipOn the radial axis, take "toward the center" as positive and set the sum equal to mv²/r. Everything else is bookkeeping.
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