Mechanics: Rotational Motion, Gravitation, and Oscillations

TL;DR

This topic explores how things spin, how gravity pulls them together, and how they swing back and forth. You'll learn about rotational equivalents of linear motion, the universal force of gravity, and the patterns of repetitive motion. Understanding these concepts helps explain everything from planets orbiting stars to pendulums ticking.

1. The Mental Model

Imagine you're trying to push a heavy door open (rotational motion), or how an apple falls from a tree (gravitation), or even a child on a swing (oscillations). These aren't just isolated events; they're all governed by a few fundamental physics principles. We're translating what you know about straight-line pushes and pulls into spinning, falling, and swaying.

2. The Core Material

Rotational Motion

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Just like linear motion has displacement, velocity, and acceleration, rotational motion has angular displacement ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$). The key is that objects rotate around an axis.

• Torque ($\tau$): This is the rotational equivalent of force. It's what causes an object to start rotating or change its rotational speed. Think of it as the "turning power" of a force. Torque depends on the force applied and the distance from the pivot point (lever arm).
  • $\tau = r \times F \times \sin(\theta)$, where $r$ is the lever arm, $F$ is the force, and $\theta$ is the angle between them.
• Moment of Inertia ($I$): This is the rotational equivalent of mass. It's a measure of an object's resistance to changes in its rotational motion. The more mass an object has and the further that mass is from the axis of rotation, the larger its moment of inertia.
  • For a point mass $m$ at distance $r$ from the axis, $I = mr^2$. For extended objects, it involves integration or specific formulas for common shapes (e.g., solid cylinder $I = \frac{1}{2}MR^2$).
• Rotational Kinematics: These are just like linear kinematic equations, but with angular variables:
  • $\omega_f = \omega_i + \alpha t$
  • $\theta_f = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$
  • $\omega_f^2 = \omega_i^2 + 2\alpha \Delta\theta$
• Rotational Kinetic Energy ($KE_{rot}$): Objects that are spinning have energy because of their motion.
  • $KE_{rot} = \frac{1}{2}I\omega^2$

Gravitation

Newton's Law of Universal Gravitation describes the attractive force between any two objects with mass.

• Gravitational Force ($F_g$): Every particle attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
  • $F_g = G \frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant ($6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$), $m_1$ and $m_2$ are the masses, and $r$ is the distance between their centers.
• Gravitational Potential Energy ($U_g$): This is the energy stored in an object due to its position in a gravitational field. When an object moves away from a large mass, its gravitational potential energy increases.
  • $U_g = -G \frac{m_1 m_2}{r}$ (note the negative sign, meaning potential energy increases as objects move further apart, approaching zero at infinite separation).

Oscillations (Simple Harmonic Motion - SHM)

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Oscillations are repetitive motions where an object moves back and forth around an equilibrium position. Simple Harmonic Motion (SHM) is a special type of oscillation caused by a restoring force that is proportional to the displacement from equilibrium.

• Hooke's Law: The classic example is a spring. The force exerted by a spring is $F = -kx$, where $k$ is the spring constant (stiffness) and $x$ is the displacement from equilibrium. The negative sign means the force always opposes the displacement.
• Period ($T$) and Frequency ($f$):
  • Period: The time it takes for one complete oscillation (e.g., one full swing of a pendulum).
  • Frequency: The number of oscillations per unit time ($f = 1/T$).
• Mass-Spring System: For a mass $m$ attached to a spring with constant $k$, the period of oscillation is:
  • $T = 2\pi\sqrt{\frac{m}{k}}$
• Simple Pendulum: For a simple pendulum of length $L$ with small angles of swing, the period is:
  • $T = 2\pi\sqrt{\frac{L}{g}}$, where $g$ is the acceleration due to gravity.

graph TD
    A["Initial Concept"] --> B["Linear Motion Basics (回顾)"]

    B --> C["Rotational Motion"]
    C --> C1["Torque (τ)"]
    C --> C2["Moment of Inertia (I)"]
    C --> C3["Angular Kinematics"]
    C --> C4["Rotational KE"]

    B --> D["Gravitation"]
    D --> D1["Newton's Law (Fg)"]
    D --> D2["Gravitational Potential Energy (Ug)"]

    B --> E["Oscillations (SHM)"]
    E --> E1["Restoring Force (e.g., Hooke's Law)"]
    E --> E2["Period & Frequency"]
    E --> E3["Mass-Spring System (T = 2π√(m/k))"]
    E --> E4["Simple Pendulum (T = 2π√(L/g))"]

    C1 & C2 & C3 & C4 & D1 & D2 & E1 & E2 & E3 & E4 --> F["Applications & Problem Solving"]

3. Worked Example

Let's consider a simple pendulum problem.

Problem: A simple pendulum has a length of 1.5 meters. What is its period of oscillation on Earth, where $g = 9.8 \text{ m/s}^2$?

Solution:
We use the formula for the period of a simple pendulum:
$T = 2\pi\sqrt{\frac{L}{g}}$

Given:
$L = 1.5 \text{ m}$
$g = 9.8 \text{ m/s}^2$

Substitute the values into the formula:
$T = 2\pi\sqrt{\frac{1.5 \text{ m}}{9.8 \text{ m/s}^2}}$
$T = 2\pi\sqrt{0.15306 \text{ s}^2}$
$T = 2\pi \times 0.3912 \text{ s}$
$T \approx 2.458 \text{ s}$

So, the period of oscillation for this pendulum is approximately 2.46 seconds.

4. Key Takeaways

• Rotational motion is analogous to linear motion but involves angular quantities like torque and moment of inertia.
• Torque is the rotational equivalent of force, causing angular acceleration.
• Moment of inertia describes an object's resistance to rotational changes, depending on mass distribution.
• Gravitational force is always attractive and depends on the product of masses and the inverse square of the distance between them.
• Gravitational potential energy increases as objects move farther apart.
• Oscillations are repetitive motions, and Simple Harmonic Motion is driven by a restoring force proportional to displacement.
• The period of a mass-spring system depends on mass and spring constant, while a pendulum's period depends on its length and gravity.

Common Mistakes to Avoid:
- Confusing linear and angular quantities (e.g., using force instead of torque for rotation).
- Forgetting the negative sign in gravitational potential energy, which indicates attraction.
- Applying the simple pendulum formula for large oscillation angles where it's no longer accurate.
- Mixing up radius and lever arm in torque calculations if they're not the same.

5. Now Try It

You have a mass of 0.5 kg attached to a spring. When you pull the mass 0.1 meters from its equilibrium position and release it, it oscillates with a period of 0.75 seconds. Calculate the spring constant ($k$) of the spring. What success looks like: You'll provide the calculated spring constant in N/m.