跟讀練習: Rotational Motion: Crash Course Physics #11 - 通過影片學習英語口語
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At this point, we’ve covered a lot of the basics when it comes to how things move.
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But we’ve mostly been focusing on only one type of motion: translational motion, which is when something moves through space, but doesn’t rotate.
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But rotational motion is also a thing -- and an important one.
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For example, the spin of a football -- both the soccer and the non-soccer kind -- will affect the way it flies through the air.
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But the physics of rotational motion isn’t all that different from the physics of translational motion.
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It still involves things like position, velocity, and acceleration.
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And many of the equations you use to describe rotational motion will look really familiar to you.
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But there are some important differences.
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Like, instead of positions, there are angles.
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And instead of points on a line, you follow points along an arc.
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And, there are times when rotational motion can explain things that sound impossible, but actually are true.
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Like when a point on a spinning wheel, is actually standing still.
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So, the rules here are the same.
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But rotational motion has what you might call its own “circular logic.” When it comes to translational motion, we tend to talk about position in terms of x and y.
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Where is this object, horizontally? And where is it vertically?
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Those axes make sense, because we’re usually tracking the object’s motion along those directions.
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But for rotational motion, we really want to know the object’s angle -- what we call theta.
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Say we have a big disk with a dot painted on it.
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If we call the top of the disk our starting point, then when the dot is at the top, its angle would be zero.
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And when the dot is at the left, its angle would be 180 degrees -- half of a full circle.
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But! Even though we’ve been measuring angles using degrees until now, there’s another unit that physicists use a lot, and it’ll be the main one we’ll use in this episode and the next.
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That unit is called the radian, and its name comes from the fact that it’s based on the radius of a circle.
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If you think back to basic geometry, you’ll recall that the circumference of a circle is just 2, times pi, times the circle’s radius, right?
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Radians describe angles by essentially telling you how much of that circumference is covered by a given angle.
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So, 360 degrees -- which is a full circle’s worth of angles -- would be 2 pi radians.
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180 degrees -- or half a circle’s worth of angles -- would be pi radians.
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And to convert any number of degrees to radians, you just multiply the degrees times pi and divide by 180.
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So now we know how to describe the angle of something that’s rotating.
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But what about the velocity of its rotation?
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Well, we’ve already learned that plain old translational, or linear, velocity is a measure of an object’s change in position.
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And in the same way, rotational velocity is a measure of an object’s change in angle.
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This is known as its angular velocity, and is represented by the lowercase Greek letter, Omega.
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which I want to point out looks a bit like a W, but itsn't a W.
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And, as you might have guessed by now, angular velocity is the derivative -- or, the rate of change -- of angular displacement with respect to time.
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But we can also describe an object’s rotation in terms of its tangential velocity.
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This is the same type of velocity we used when we talked about the physics of uniform circular motion.
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Remember? With the key spinning around on a string?
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And that vomit-causing carnival ride?
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In those cases, we described how, when an object moves along a circular path, its velocity is perpendicular to the radius of the circle, in the direction of the motion.
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And, when you think about it, any rigid object rotating around a fixed axis is basically a set of points, all moving along circular paths.
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So, at any given moment, each of those points will have a tangential velocity that depends on the path it moves through -- specifically, the radius of that path.
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In fact, its tangential velocity will be equal to its angular velocity, times the radius.
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It’s easy to see why this makes sense, if you picture the spokes of a rotating wheel.
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All of the points along each spoke have to have the same angular velocity, because they all cover the same angular distance in the same amount of time.
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But to get from, say, the right-hand side of the circle to the bottom, the points on the outside of the wheel will pass through a much bigger arc -- covering more space, basically -- than the points on the inside.
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So, the farther that a point on the spoke is from the center of the wheel, the greater its tangential velocity has to be.
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Like circular motion, rotational motion can also be periodic -- when the rotation repeats after a set amount of time, which is represented by a capital T, also called The Period.
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And the equations that describe periodic motion are pretty much the same as the ones we used for a single point moving along a circular path: So the frequency, or number of rotations that happen every second, is equal to one, divided by the period.
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But frequency and angular velocity are really just two different ways of describing the same thing -- they just use different units.
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Frequency is measured in rotations -- or revolutions -- per second, and angular velocity is measured in radians per second.
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And one revolution is equal to the circumference of the circle: 2 pi radians.
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So, in order to convert from frequency to angular velocity, all you need to do is multiply the frequency by 2 pi.
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Now, there’s a special case when it comes to the velocity of rotating objects, and that’s what’s known as rolling without slipping.
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This kind of motion shows up in real life all the time.
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It’s what happens to your car’s tires when you drive down the street, as long as you aren’t skidding -- which, let’s hope you aren’t.
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And it’s what a train’s wheels do as they move along the track.
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But it turns out that the translational velocity at the bottom of the wheel is /super weird/.
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Mainly, because at any given moment, the point at the bottom of the wheel doesn’t have a translational velocity.
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In other words, it doesn’t actually move.
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To figure out why, let’s experiment with a bicycle wheel.
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If you roll the wheel along the floor for one full rotation, that means that the entire circumference of the wheel will touch the floor, one point at a time.
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And the center of the wheel will move forward by a distance that’s equal to the circumference of the wheel -- aka its radius, times 2 pi.
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And the time it took to move that distance was equal to the period of the motion.
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So, the translational velocity of the center was equal to the radius, times the angular velocity.
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Now, what about the top of the wheel?
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It has the same translational velocity as the center of the wheel, plus the tangential velocity that comes from the wheel’s rotation. Because, at the top of the wheel, the tangential velocity is pointing in the direction the wheel is rolling in.
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And in the same way, the bottom of the wheel has the same translational velocity as the center of the wheel, minus the tangential velocity that comes from the wheel’s rotation.
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Because at the bottom of the wheel, the tangential velocity is pointing opposite to the direction the wheel is rolling in.
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Here’s the weird part: We just saw that the translational velocity of the wheel is equal to the radius, times the angular velocity.
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And we know that in general, the magnitude of tangential velocity is also equal to the radius, times the angular velocity.
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So, the top of the wheel will be moving exactly twice as fast as the center of the wheel, relative to the ground.
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Because, to get its total velocity, you add the translational velocity to the tangential velocity.
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But the bottom of the wheel won’t be moving at all … because its total velocity is its translational velocity minus the tangential, since they’re moving in opposite directions.
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As a result, the total velocity at the bottom of the wheel is zero.
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Even though the wheel is clearly moving relative to the ground.
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But if you look at the wheel’s motion at any given instant, you’ll see that whatever point is at the bottom of the wheel can’t be moving relative to the ground.
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If it was moving relative to the ground … that would be what we call slipping.
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Like when a car is skidding on an icy ground: The wheel isn’t turning, but the bottom of the wheel is moving in relation to the ground, because it’s sliding along on top of it.
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But this wheel isn’t slipping -- its bottom has a total velocity of zero, because its velocities cancel out.
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OK: I know I just blew your mind, so while you put your head back together, I want to talk about one more basic quality of rotational motion: angular acceleration.
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Based on what you already know about acceleration, you can already guess that angular acceleration is the derivative of angular velocity.
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It’s represented by the lowercase Greek letter alpha, and it describes how an object’s angular velocity is changing over time.
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And as an object rotates, each point on it can actually accelerate in two different ways.
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Radial acceleration is another term for what we’ve been calling centripetal acceleration up until now.
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It’s the acceleration inward of any point on our rotating object, and it’s equal to the angular velocity, squared, times the radius.
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But there’s also tangential acceleration, which describes whether an individual point on a rotating object is speeding up or slowing down.
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So, like linear velocity, tangential acceleration depends on the distance between the point and the center of the object.
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More specifically, it’s equal to the angular acceleration, times the radius.
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So you see, angular position, velocity, and acceleration relate to each other in much the same ways that linear position, velocity, and acceleration do.
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This allows us to talk about rotational motion with terms and equations that are familiar to us, once we’ve gotten the basics of translational motion under our belts.
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Next time, we’ll see how how the logic of rotational motion applies to another idea: momentum!
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For now, you learned about the qualities of rotational motion, including angular position, angular velocity, periodic motion, and the special case of rolling without slipping.
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We also talked about angular acceleration, as well as constant angular acceleration.
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Crash Course Physics is produced in association with PBS Digital Studios. You can head over to their channel to check out amazing shows like It's Okay to be Smart, Blank on Blank, and Shank's FX.
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This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio with the help of these amazing people and our equally amazing graphics team is Thought Cafe.
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背景與文脈
在這段影片中，講解者深入探討了轉動運動的概念，重點強調了其與平移運動的關係。雖然我們在學習物體的運動時往往聚焦於平移運動，但轉動運動同樣重要，並且在我們的日常生活中無處不在。例如，足球的旋轉方式會影響其在空氣中飛行的軌跡。通過這段影片，觀眾可以理解如何用角度、速度等物理概念來描述運動，這對提升英語聽說能力非常有幫助。
日常交流的五個關鍵短語
- 轉動運動 - 指物體圍繞固定點旋轉的運動形式。
- 角度 - 衡量物體旋轉位置的單位，通常以弧度為主。
- 角速度 - 描述物體旋轉速率的物理量，常用希臘字母Omega表示。
- 切線速度 - 物體在圓形路徑上運動時的速度，依賴於半徑。
- 週期 - 表示旋轉週期的時間，週期越短，頻率越高。
逐步跟讀指導
想要有效地提升英語口語能力，可以藉助這段影片進行英語影子跟讀練習。以下是逐步指導：
- 選擇段落： 從影片中挑選一小段，例如討論角速度的部分，這樣有助於降低難度。
- 聽音辨義： 首先仔細聽講解者的發音，注意重音和語調，嘗試理解每個單詞的含義。
- 重複模仿： 在聽的同時，嘗試跟讀，這是影子跟讀的方法之一，即 shadow speech。開始時可以慢慢跟讀，逐漸加快速度。
- 錄音對比： 將自己的跟讀錄音與原音訊進行對比，找到差距，特別注意語音語調的細微差別。
- 反覆練習： 通過多次練習，提升自己的流暢度和自信心，逐步掌握轉動運動相關的術語和表達。
通過這種英語影子跟讀的方法，你不僅能提升英語口語，還能加深對物理概念的理解，從而在實際生活中更有效地運用這些知識。
影片中的語法
說話人最常用的結構，並附上影片中的原話：
| 結構 | 影片中的用法 |
|---|---|
| 現在完成進行時 have/has been + -ing — 以前開始、現在仍在繼續的動作 | we've been measuring · we've been calling |
| 被動語態 be + 過去分詞 — 強調發生了什麼，而不是誰做的 | is called · is covered · is known |
| 現在完成時 have/has + 過去分詞 — 過去發生但與現在仍有關聯的事 | we've covered · we've already learned · we've gotten |
| 定語從句 who / which + 從句 — 補充說明人或事物 | motion, which is · time, which is |
什麼是跟讀法？
跟讀法 (Shadowing) 是一種有科學依據的語言學習技巧，最初開發用於專業口譯員的培訓，並由多語言者Alexander Arguelles博士普及。這個方法簡單而強大：您在聽英語母語原聲的同時立即大聲重複——就像是一個延遲1-2秒緊跟說話者的影子。與被動聽力或語法練習不同，跟讀法強迫您的大腦和口腔肌肉同時處理並模模擬實的講話模式。研究表明它能顯著提高發音準確性，語調，節奏，連讀，聽力理解和口語流利度——使其成為雅思口語備考和真實英語交流最有效的方法之一。