title: "Uniform Circular Motion" course: "honors-physics" unitSlug: "unit-2-circular-motion" topicSlug: "uniform-circular-motion" tags:

• "centripetal acceleration"
• "uniform circular motion"
• "giancoli" chapter: 5 giancoliRef: "7th Ed. Chapter 5, Section 1-3"

Uniform Circular Motion

An object moving in a circle of radius $r$ at a constant speed $v$ is undergoing uniform circular motion.

Even though the speed is constant, the velocity vector $\vec{v}$ continuously changes direction. Therefore, the object experiences a centripetal acceleration $\vec{a}_c$ directed toward the center.

$\vec{a}_c = \frac{v^2}{r} \hat{r}$ Centripetal acceleration vector equals velocity squared divided by radius, in the radial direction.

Concept Focus

In this topic, students should be able to:

• Explain why constant speed can still produce acceleration.
• Identify the direction of centripetal acceleration.
• Use $a_c = \frac{v^2}{r}$ to solve circular motion problems.

Sample Problem (Giancoli Ch. 5)

A $1{,}200\text{ kg}$ car rounds a flat curve of radius $r = 65\text{ m}$ on a wet road with coefficient of static friction $\mu_s = 0.25$.

1. Radial force requirement: $F_f = m \frac{v^2}{r}$
2. Maximum friction: $F_f = \mu_s F_N = \mu_s mg$
3. Equate and solve:

$v = \sqrt{\mu_s g r}$ Speed equals the square root of static friction coefficient times gravity times radius.

$v = \sqrt{(0.25)(9.8\text{ m/s}^2)(65\text{ m})} \approx 12.6\text{ m/s}$ Substituting values gives a safe speed of approximately 12.6 meters per second.

Reflection Prompt

If the road condition changes and $\mu_s$ decreases, what must happen to the safe speed and why?