You will be able to
- Convert angles between degrees and radians and relate arc length to angle and radius
- Derive the rotational kinematic equations by analogy with linear motion
- Compute the kinetic energy and moment of inertia of a system of point masses
- Calculate torque from force, lever arm and angle, and apply the rotational form of Newton's second law
- Compute moments of inertia of a ring, disk and rod by integration
- Apply the parallel axis theorem to shift a moment of inertia to a new axis
Good to know first
- Newton's second law and linear kinematics in one dimension
- Work and kinetic energy, including the work-energy theorem
- Basic integration with polynomials and trigonometric functions
- Vector components and the sine and cosine of an angle
Syllabus
9 lessons
- 1
Rigid bodies: translation plus rotation Free
A rigid body keeps its shape, so describing its motion needs more than a position: it can translate and rotate at once. The chapter introduces the idea of splitting a complicated motion into a translation followed by a rotation, and starts by studying pure rotation about a fixed point.
2 exercises4 questions - 2
Describing orientation with angle and radians Beyond Pass
A rigid body pinned at one point can only rotate, and one angle theta tells you its full orientation, just as one coordinate x tells you where a body is in one dimension. Measuring that angle in radians makes arc length simple: Delta s = R Delta theta.
3 formulas3 exercises3 questions - 3
Angular velocity and acceleration, the rotational twins of velocity and acceleration Beyond Pass
Angular velocity ω and angular acceleration α describe how fast a rigid body turns and how fast that turning changes, just as velocity and acceleration do for straight-line motion. A point on a spinning body has a tangential speed v = ωr and two accelerations: a centripetal one that is always there and a tangential one that exists only when the spin speeds up or slows down.
3 exercises4 questions - 4
Kinetic energy and moment of inertia of point masses Beyond Pass
A rigid body spinning at angular velocity ω has kinetic energy written as ½Iω², where I is the moment of inertia. The chapter shows how I comes from adding up m r² over all the masses, and how it plays the role that mass played for straight-line motion.
4 formulas3 exercises3 questions - 5
Work and torque: the rotational work-energy relation Beyond Pass
A force that turns a rigid body does work equal to torque times angle, and that work changes the body's rotational kinetic energy. Equating the two gives the rotational law tau = I alpha, with torque defined as tau = F r sin(theta).
3 exercises3 questions - 6
Torque Direction and a Pendulum Worked Example Beyond Pass
You learn how to find the torque needed to hold a pendulum at an angle, and how to compute the work done by that torque by integrating over the angle. A cross-check with potential energy shows the answer is right.
2 exercises3 questions - 7
Moment of inertia of a ring and a disk by integration Beyond Pass
This lesson shows how to find the moment of inertia of a continuous body by splitting it into tiny pieces and adding their contributions. It works through a ring, where every piece is the same distance from the axis, and then a disk, built from concentric thin rings.
2 exercises3 questions - 8
Moment of inertia of a uniform disc and a uniform rod Beyond Pass
We find the moment of inertia of a uniform disc (MR²/2) by slicing it into thin rings and integrating, then do the same for a thin rod about its end (mL²/3). We also learn a sanity check: the answer must be a fraction of MR² or mL², never a bigger number.
3 exercises3 questions - 9
The parallel axis theorem: shifting the axis of rotation Beyond Pass
The moment of inertia of a rod depends on where the axis of rotation is placed, and it is smallest through the center of mass. The parallel axis theorem gives the moment of inertia about any parallel axis from the value at the center of mass plus M d².
2 formulas3 exercises3 questions
Free preview · Lesson 1 in full
Rigid bodies: translation plus rotation
A rigid body keeps its shape, so describing its motion needs more than a position: it can translate and rotate at once. The chapter introduces the idea of splitting a complicated motion into a translation followed by a rotation, and starts by studying pure rotation about a fixed point.
Draw and write next to the video.
The idea
A rigid body is an object that does not bend or change shape when forces act on it. Real objects bend a little, but a coin or a meter stick is a good approximation. The technical definition: pick any two points on the body, and the distance between them stays the same during the motion. A living snake is not rigid, because it bends and moves its body; a dead snake would be closer to rigid.
For a point mass, all you need to know is where it is and how fast it moves. A point mass can only translate, meaning move from one place to another, on a straight line or a curve. It has no orientation, so turning it around changes nothing you could notice. A real object with size and shape is different. If you throw a football, it travels along a parabolic path (translation) and it also spins and tumbles (rotation). Knowing where its centre is does not tell you which way the football points.
The key claim of the chapter is not obvious: any motion of a rigid body from one instant to the next can be achieved by first translating the body and then rotating it slightly to reach the final orientation. So translation and rotation are the two building blocks of rigid-body motion.
To make the problem easier, the chapter splits it into two parts. First, study a body that only rotates and does not translate. Imagine holding one point of the body fixed. At any instant, the motion is a rotation about an axis passing through that fixed point. Points on the axis do not move, and the point you are holding must lie on that axis. Once this is understood well, you add back translation.
The chapter also warns that one dimension is too simple. In one dimension a rigid body cannot rotate at all, so you need at least two or three dimensions to see rotation. In three dimensions the motion of a rigid body is fairly complicated, which is why it helps to start with the simplest case and make it more complex step by step.
Key points
- A rigid body keeps its shape: the distance between any two of its points never changes. 0:09
- A point mass only translates; an extended body with shape can also rotate, so it needs orientation as well as position. 1:07
- Any motion of a rigid body between two instants can be done as a translation followed by a rotation. 1:59
- To study rotation, hold one point fixed: the body then rotates about an axis through that point. 2:58
- In one dimension a rigid body cannot rotate, so rotation needs at least two dimensions. 4:17
Words to know
- Rigid body
- An object whose shape does not change under forces: the distance between any two of its points stays fixed.
- Translation
- A motion in which every point of the body moves in the same way, so the body shifts from one place to another without turning.
- Rotation
- A motion in which the body turns about an axis; points on the axis stay still.
- Point mass
- An idealised object with mass but no size or shape, so it has no orientation and can only translate.
- Axis of rotation
- The line through a rotating body whose points do not move during the rotation.
Practice
In your own words, explain why a point mass is described by only its position and velocity, but a football needs more information to describe its motion.
Solution
- A point mass has no size, shape, or orientation, so turning it changes nothing you could observe.
- A football has size and shape, so it has an orientation (which way it points) in addition to where it is.
- Its motion can include rotation as well as translation, so you also need to know how it is oriented and how it turns.
A point mass has no orientation, so position and velocity are enough. A football also has a shape and an orientation, and it can rotate while its centre translates, so you need extra information about how it is turned.
A rigid rod is held at one end, so that end does not move. In a rotation about the held end, a point on the rod is 2 m from the held end. How far is the point from the held end at any moment during the rotation? Give the answer in metres.
Hint
The held point is fixed, and the distance from it to a point on a rigid body does not change.
Solution
- The rod is rigid, so the distance between any two of its points stays the same.
- The held end does not move, and the point is 2 m from it.
- So the distance from the held end to that point stays 2 m throughout the rotation.
2 m
On your scratchpad
Draw a simple rigid object, such as a rectangle or a stick. Mark one point on it as the fixed point, draw a line through that point as the axis, and draw an arrow showing the object turning around that axis.
Quiz
What is the technical definition of a rigid body given in the video?
- Its mass is evenly spread out.
- The distance between any two of its points does not change during the motion.
- It never rotates.
- It always moves in a straight line.
Show the answer
B. The distance between any two of its points does not change during the motion. · The video defines a rigid body by saying that if you pick a couple of points, the distance between them does not change during the motion.
0:28 watch the momentWhy is a point mass not enough to describe a football thrown in the air?
- A football has too much mass.
- A football has size and shape, so it also has an orientation and can rotate as well as translate.
- A football does not follow a parabolic path.
- A point mass cannot move in a curve.
Show the answer
B. A football has size and shape, so it also has an orientation and can rotate as well as translate. · The football's position alone does not say which way it points, because it can also rotate; a point mass has no orientation.
1:07 watch the momentAccording to the video, how can any motion of a rigid body from one instant to the next be achieved?
- By a rotation followed by a translation only.
- By only a translation.
- By a translation followed by a slight rotation.
- By a rotation about the centre of mass only, with no translation.
Show the answer
C. By a translation followed by a slight rotation. · The video says the motion can be achieved by first translating the body and then doing a slight rotation to reach the configuration you want.
1:59 watch the momentWhy can a rigid body not rotate in one dimension?
- Because one-dimensional bodies are too heavy.
- Because there is no room to turn: rotation needs at least two dimensions.
- Because rigid bodies never rotate.
- Because forces cannot act in one dimension.
Show the answer
B. Because there is no room to turn: rotation needs at least two dimensions. · The video states that if you take a rigid body in one dimension it cannot rotate at all.
4:17 watch the moment
Lessons 2–9
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Made by Beyond from the video “9. Rotations, Part I: Dynamics of Rigid Bodies” by YaleCourses. The video plays in YouTube's own player. Watch it on YouTube · All lessons