【Physics】Circular Motion and Universal Gravitation: Mastering the Rules of the Universe!
Hello! In this chapter, we will learn about "Circular Motion" and "Universal Gravitation." While they might seem difficult at first glance, everything from a car turning a corner to the Moon orbiting the Earth follows the exact same rules. It might feel overwhelming at first because of the many formulas, but once you learn the basic patterns, you'll be able to solve them like a puzzle. Let's take it one step at a time and have fun!
What you will learn in this chapter:
1. New terminology to describe circular motion (such as angular velocity)
2. The forces that maintain circular motion (centripetal force) and centrifugal force
3. The force that acts on every object in the universe (universal gravitation)
4. The rules governing the movement of planets (Kepler's Laws)
1. Basics of Uniform Circular Motion: Moving in a Circle
First, let’s master "Uniform Circular Motion," where an object moves in a circle at a constant speed. The key here is using "angles" to describe the movement.
① Angular Velocity \(\omega\) (Omega)
Normal velocity \(v\) represents "distance traveled per second," whereas angular velocity \(\omega\) represents the "angle (in radians) rotated per second."
\( \omega = \frac{\theta}{t} \) [rad/s]
② Relationship between speed \(v\) and angular velocity \(\omega\)
When moving in a circle of radius \(r\), the speed \(v\) along the circumference is given by the following formula:
\( v = r\omega \)
(Think about it: the larger the radius, the faster the outer edge must move to maintain the same rotational speed!)
③ Period \(T\) and Frequency \(f\)
・Period \(T\): The time taken for one full rotation.
\( T = \frac{2\pi r}{v} = \frac{2\pi}{\omega} \) [s]
・Frequency \(f\): The number of rotations per second.
\( f = \frac{1}{T} \) [Hz]
【Pro Tip!】
In exams, calculations using \(v = r\omega\) and \(T = \frac{2\pi}{\omega}\) appear frequently. Make sure to learn them as a set!
2. Acceleration and Force in Circular Motion
You might wonder: "How can there be 'acceleration' if the speed is uniform?" Actually, a change in direction is also a form of acceleration.
① Centripetal Acceleration \(a\)
An object in circular motion is constantly accelerating toward the center of the circle.
\( a = r\omega^2 = \frac{v^2}{r} \)
② Centripetal Force \(F\)
Since there is acceleration, there must be a force acting there. We call this force centripetal force.
\( F = ma = mr\omega^2 = m\frac{v^2}{r} \)
【Common Mistake】
There is no special force specifically called "centripetal force." Instead, forces like tension, friction, or gravity act as the "centripetal force" to maintain circular motion. When drawing diagrams, first list the forces from objects actually in contact and any remote forces.
③ Centrifugal Force (Inertial Force)
From the perspective of someone moving along with the object, it feels as though a force is pushing it outward. This is centrifugal force.
Its magnitude is the same as the centripetal force (\(m\frac{v^2}{r}\)), but its direction is opposite to the center (outward).
(Example: The feeling of your body being pushed outward when a car takes a sharp turn!)
【Summary: Steps to solve circular motion problems】
1. Set the direction toward the center of the circle as positive and list the acting forces.
2. Apply the equation of motion \(ma = F\).
3. Substitute \(a\) with either \(r\omega^2\) or \(\frac{v^2}{r}\), whichever is more convenient.
3. Universal Gravitation: Attraction on a Cosmic Scale
Newton realized that the same "attractive force" is responsible for both an apple falling and the Moon orbiting the Earth.
① Law of Universal Gravitation
The force \(F\) acting between objects of mass \(M\) and \(m\) is inversely proportional to the square of the distance \(r\) between them.
\( F = G \frac{Mm}{r^2} \) (where \(G\) is the gravitational constant)
Fun Fact:
In reality, there is a gravitational pull between you and the person sitting next to you. However, because the value of \(G\) is extremely small, this force is so weak that it cannot be felt unless a massive object like the Earth is involved.
② Gravity and Universal Gravitation
Gravity \(mg\) at the Earth's surface (radius \(R\)) is actually the universal gravitational pull from the Earth (ignoring the effects of rotation).
From \( mg = G \frac{Mm}{R^2} \), we can derive the relation \( g = \frac{GM}{R^2} \).
* The substitution \(GM = gR^2\) is a frequently used technique in calculation problems!
③ Gravitational Potential Energy
Using an infinitely distant point (infinity) as the reference (0), potential energy \(U\) becomes negative.
\( U = -G \frac{Mm}{r} \)
(Why negative?: Since the object is being "pulled" from infinity, it is in a state of lower energy compared to the reference point.)
4. Kepler's Laws: Rules of Planetary Motion
Kepler discovered three rules regarding how planets move.
First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus.
Second Law (Law of Equal Areas): Planets move faster when closer to the Sun and slower when further away. (\( \frac{1}{2}rv\sin\phi = \text{constant} \))
Third Law (Harmonic Law): The square of the orbital period \(T\) is proportional to the cube of the semi-major axis \(a\).
\( \frac{T^2}{a^3} = k \) (constant)
【Pro Tip!】
The third law is extremely powerful when comparing multiple planets orbiting the same central body (e.g., the Sun)!
5. Cosmic Velocities: How to Escape Earth?
Finally, let’s summarize the "cosmic velocities" that often appear on tests.
First Cosmic Velocity
The speed required to maintain circular motion just above the Earth's surface.
(Determined from the equation: Centripetal force = Gravity)
\( v_1 \approx 7.9 \text{ km/s} \)
Second Cosmic Velocity (Escape Velocity)
The minimum speed required to break free from Earth's gravity and travel to infinity.
(Determined from the Law of Conservation of Energy: Kinetic Energy + Potential Energy \(\geqq 0\))
\( v_2 \approx 11.2 \text{ km/s} \)
★ Final Advice ★
The trick to solving circular motion problems is to always be aware of "which way is toward the center?" The formulas might look complex at first, but your hands will learn them through practice.
Doesn't it make physics feel a bit more relatable to know that stars in space and playground rides are all described by the same \(m\frac{v^2}{r}\)?
Don't rush—take your time to digest the meaning of each formula as you go. I'm rooting for you!