Physics Review: Circular Motion, Rotational Motion, and Equilibrium

Chapter 6: Circular Motion, Orbits, and Gravity

Uniform Circular Motion

Uniform circular motion describes the movement of an object along a circular path at constant speed. Although the speed remains unchanged, the direction of motion continuously changes, resulting in acceleration.

• Centripetal Acceleration: The acceleration directed toward the center of the circle, keeping the object in circular motion.

• Formula: $a = \frac{v^2}{r}$

• Newton's Second Law for Circular Motion: $\vec{F}_{\text{net}} = m\vec{a} = \left( \frac{mv^2}{r}, \text{toward center of circle} \right)$

• Period and Frequency: Period ($T$) is the time for one revolution; frequency ($f$) is the number of revolutions per second. $T = \frac{2\pi r}{v}$ $f = \frac{1}{T}$

Universal Gravitation

Newton's law of universal gravitation describes the attractive force between any two masses separated by a distance.

• Formula: $F_{1\text{ on }2} = F_{2\text{ on }1} = \frac{G m_1 m_2}{r^2}$

• Gravitational Constant: $G = 6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$

• Inverse-Square Law: The gravitational force decreases with the square of the distance between objects.

Orbits and Weightlessness

Objects in orbit experience apparent weightlessness because both the object and its surroundings are in free fall under gravity.

• Orbital Speed: $v = \sqrt{\frac{GM}{r}}$

• Orbital Period: $T = 2\pi \sqrt{\frac{r^3}{GM}}$

• Apparent Weight: The normal force felt by an object, which may differ from true weight due to acceleration.

Chapter 7: Rotational Motion

Describing Rotational Motion

Rotational motion involves objects rotating about a fixed axis. Key variables have analogs to linear motion.

• Angular Displacement: $\Delta \theta = \theta_f - \theta_i$ (measured in radians)

• Angular Velocity: $\omega = \frac{\Delta \theta}{\Delta t}$

• Angular Acceleration: $\alpha = \frac{\Delta \omega}{\Delta t}$

• Relationship to Linear Quantities: $v = \omega r$ $a_t = \alpha r$

• Conversion: $1 \text{ rev} = 360^\circ = 2\pi \text{ rad}$

Torque

Torque is the rotational equivalent of force, causing angular acceleration about an axis.

• Definition: $\tau = rF\sin\phi$ where $r$ is the distance from the axis, $F$ is the force, and $\phi$ is the angle between $r$ and $F$.

• Interpretations: 1. $\tau = rF$ (when $F$ is perpendicular to $r$) 2. $\tau = r_{\perp} F$ (using the perpendicular component)

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion, analogous to mass in linear motion.

• General Formula: $I = \sum m_i r_i^2$

• Common Shapes:

  • Hoop: $I_{\text{hoop}} = MR^2$

  • Disk: $I_{\text{disk}} = \frac{1}{2}MR^2$

  • Sphere: $I_{\text{sphere}} = \frac{2}{5}MR^2$

  • Rod (center): $I_{\text{rod}} = \frac{1}{12}ML^2$

Newton's Second Law for Rotation

Analogous to linear motion, the net torque on an object produces angular acceleration.

• Formula: $\tau_{\text{net}} = I\alpha$

• Comparison: Linear: $F_{\text{net}} = ma$; Rotational: $\tau_{\text{net}} = I\alpha$

Center of Gravity

The center of gravity is the point where the total weight of an object can be considered to act.

• Coordinates: $x_{cg} = \frac{\sum m_i x_i}{\sum m_i}$ $y_{cg} = \frac{\sum m_i y_i}{\sum m_i}$

Rotational Kinematics

Equations for rotational motion under constant angular acceleration mirror those for linear motion.

• $\omega_f = \omega_i + \alpha \Delta t$

• $\theta_f = \theta_i + \omega_i \Delta t + \frac{1}{2} \alpha (\Delta t)^2$

• $\omega_f^2 = \omega_i^2 + 2\alpha \Delta \theta$

Rolling Motion

For objects rolling without slipping, the linear velocity at the center is related to angular velocity.

• Formula: $v = \omega R$

• The velocity at the top of the object is twice that of the center.

Chapter 8: Equilibrium and Elasticity

Static Equilibrium

An object is in static equilibrium if it has no net force and no net torque acting on it.

• Conditions: $\sum F_x = 0$ $\sum F_y = 0$ $\sum \tau = 0$

• The pivot point for torque calculations can be chosen for convenience.

Springs and Hooke's Law

When a spring is stretched or compressed, it exerts a restoring force proportional to the displacement from its equilibrium length.

• Hooke's Law: $F_{\text{spring}} = -k \Delta x$

• Spring Constant ($k$): Measures the stiffness of the spring; larger $k$ means a stiffer spring.

Additional info:

• Students should be able to use scientific notation, significant figures, and keep units consistent in all calculations.

• Vector components and the use of trigonometric functions (sin, cos, tan) are essential for solving problems.

• Connections between translational and rotational motion are important for understanding physical systems.

• Problem-solving strategies for rotational dynamics mirror those for linear dynamics, including drawing diagrams, identifying forces, and checking units.

Linear Motion	Rotational Motion
Position ($x$)	Angle ($\theta$)
Velocity ($v$)	Angular velocity ($\omega$)
Acceleration ($a$)	Angular acceleration ($\alpha$)
Mass ($m$)	Moment of inertia ($I$)
Force ($F$)	Torque ($\tau$)