Connecting Linear and Rotational Motion
One spin, many speeds.
Describe the linear motion of a point on a rotating rigid system that corresponds to the rotational motion of that point, and vice versa.
Everything on this page serves that one sentence.
Think about two riders on a merry-go-round, one sitting out at the rail and one sitting near the middle. They go around together, once every twelve seconds, both of them. Do you think they are moving at the same speed? If not, which one do you think is moving faster?
Both of those statements are true at the same moment, and this topic is about what keeps them together. What the two riders share is the angle. What they do not share is the distance.
1. Arc length is the distance a point actually travels
A point on a rotating system travels along an arc. How long that arc is depends on only two things: how far the system turned, and how far the point sits from the axis. The distance the point covers along its arc is called the arc length.
\(\Delta s = r\,\Delta\theta\)meters
This is the main reason we use radians. In radians, the relationship is exactly this simple, with no conversion factor hidden inside. In degrees, you would have to more complicated trig calculations.
Every point on the system shares the same angle, and no two of them share the same radius, so no two of them cover the same arc length in the same amount of time. When you move a point twice as far from the axis, it travels twice as far through the same turn, and nothing else about the motion changes.
One disk with two points that you can drag to any distance from the axis. Turn the disk by dragging it, and both arcs are traced and measured as you go. The panel shows each radius, the shared angle, and each arc length, and it divides one arc by the other so you can compare that ratio with the ratio of the radii.
Test thisPut one point at \(r = 0.20\) m and the other at \(r = 0.60\) m. Turn the disk through about one radian.
Report what happenedWhat is the ratio of the two arc lengths, and what is the ratio of the two radii? Explain in one sentence why those two ratios have to agree.
Now without the simulationA point sits \(0.40\) m from the axis of a wheel that turns through \(2.5\) rad. The distance the point travels is
Topic questions
1 Why does the angle have to be in radians when you calculate an arc length from the radius and the angle?
2 Two points on one rotating disk sit at different distances from the axis. Through one turn of the disk, the two points have
2. Linear speed is the radius times the angular velocity
When you divide both sides of the arc length equation by the time the turn took, the left side becomes a speed.
\(v = r\omega\)meters per second
The speed in that equation applies to one point, along its own arc, at one specific instant. It points along the direction that the point is traveling, which is to say along the tangent to the circle, and not toward the axis.
Two points on one wheel share one \(\omega\) and have two different values of \(v\), and the difference between them is just the radius, or the distance from the rotational axis. This is why a bicycle with larger wheels covers more ground for each turn of the pedals, even though both bicycles are turning their wheels at the same rate.
A disk turning at an angular velocity you determine, with velocity arrows drawn at several distances from the axis. Drag the slider and every arrow changes together. Drag a point in or out and watch how only that arrow changes. The arrows are drawn to one scale, so you can compare their lengths directly.
Test thisSet the angular velocity to \(3.0\) rad/s. Drag one point to \(r = 0.10\) m and another to \(r = 0.30\) m.
Report what happenedRead both speeds from the panel. By what factor do they differ, and by what factor do the radii differ? Then explain what happens to both speeds when you double the angular velocity.
Now without the simulationA wheel of radius \(0.25\) m turns at \(8.0\) rad/s. The speed of a point on its rim is
Topic questions
3 Point X sits twice as far from the axis as point Y on the same rotating disk. Comparing their linear speeds,
4 The linear velocity of a point on a rotating rigid system points
3. Tangential acceleration is the radius times the angular acceleration
If the angular velocity is changing, then the speed of every point on the system is changing too, and we have an similar to the one from Section 2.
\(a_T = r\alpha\)meters per second squared
The subscript is very important because it is only the part of the acceleration that lies along the direction of travel, the part that changes how fast the point is moving. A point going around a circle also has a centripetal acceleration pointing at the axis, and that part changes the direction of travel instead. They are two perpendicular parts of one acceleration, and this equation gives us one of them.
So a point on a steadily spinning wheel does have an acceleration even though its speed never changes, and its tangential acceleration is zero the whole time. When you give the wheel an angular acceleration, the tangential part appears next to the centripetal part.
A disk you can give an angular acceleration to, with one point you can move in and out. The tangential arrow and the centripetal arrow are drawn separately and then added. It starts at rest, so the tangential arrow is the only one there at first. Set the angular acceleration to zero once the disk is spinning and watch which arrow is left.
Test thisStart the disk from rest with an angular acceleration of \(2.0\) rad/s\(^2\) and a point at \(r = 0.30\) m. Let it run, then set the angular acceleration to zero without stopping the disk.
Report what happenedWhich arrow disappeared and which one remained? Explain what each of the two arrows does to the motion of the point.
Now without the simulationA wheel of radius \(0.50\) m has an angular acceleration of \(4.0\) rad/s\(^2\). The tangential acceleration of a point on its rim is
Topic questions
5 A disk turns at a constant angular velocity. A point on its rim has
6 Two points on one disk sit at different radii while the disk speeds up. Which quantity is the same for both points?
4. Each linear quantity carries one factor of r
Try to see what the three relationships next to each other have in common.
\(\Delta s = r\,\Delta\theta \qquad v = r\omega \qquad a_T = r\alpha\)one factor of the radius, every time
Each linear quantity is its angular counterpart multiplied by the distance from the axis. That one factor of \(r\) connects the two descriptions of the same motion, and once you see it, there is nothing further to memorize.
Worth remembering. Anything shared by the whole system is easier to state in angles. Anything about one particular point is easier to state in meters. A question that mixes the two is asking you to convert, and converting is one multiplication in either direction.
The common mistake here is not getting the multiplication wrong. It is forgetting that a conversion is needed at all, and comparing an angular quantity with a linear one as though they were the same kind of thing. A wheel turning at 10 rad/s and a cart moving at 10 m/s are not doing anything similar, and the matching numbers basically mean nothing helpful.
Two turntables side by side, turning at the same angular velocity, with one marker on each at a radius you choose. Run the race. The lap counter shows that they finish together every time, while the distance counters never agree. Switch the readout between angles and meters to see the same motion described both ways.
Test thisSet one marker near the axis and the other near the rim, then run one full lap.
Report what happenedWhat was the same for the two markers at the end of the lap, and what was different? Give the two numbers that were the same and the two that were not.
Now without the simulationA wheel of radius \(0.20\) m rolls without slipping. Its angular velocity is \(15\) rad/s. The speed of its center is
Topic questions
7 A student writes that a disk turning at 6.0 rad/s has a rim speed of 6.0 m/s. The error is that
8 A question asks how long a merry-go-round takes to complete one turn. Which description makes the calculation simplest?
That is as far as Unit 5 goes on the site for now. Topic 5.3 brings in torque, which is what makes an angular acceleration happen in the first place.