# What is Rotational dynamics: Definition and 269 Discussions

Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.
This article assumes that the rotation is also stable, such that no torque is required to keep it going. The kinematics and dynamics of rotation around a fixed axis of a rigid body are mathematically much simpler than those for free rotation of a rigid body; they are entirely analogous to those of linear motion along a single fixed direction, which is not true for free rotation of a rigid body. The expressions for the kinetic energy of the object, and for the forces on the parts of the object, are also simpler for rotation around a fixed axis, than for general rotational motion. For these reasons, rotation around a fixed axis is typically taught in introductory physics courses after students have mastered linear motion; the full generality of rotational motion is not usually taught in introductory physics classes.

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1. ### Why can we use this approximation in rotational work-kinetic energy theorem for small displacements?

If there is a net external torque on the system about ##S##, then this net torque vector must have its direction on the ##z##-axis (since the system is only rotating about this axis at all times). This torque generates angular acceleration $$\tau_{S,z}=I_S\alpha_z\tag{1}$$ where ##I_S## is...
2. ### Angular velocity of a loop after being struck by bullet

First off, I do know how to solve this problem. We use the principle of conservation of angular momentum about the centre of mass of the system which comprises of the loop and the bullet to obtain option B. My doubt is, why do we just not use the principle about the centre of the loop? Where is...
3. ### Rotation and torque about an accelerating point

A solution was provided: We take torques about point B. Note that τ = MgL/2 = Iα so α = (3g)/2L. Everything from here is straightforward. I don't understand why in this step, you can take torque about B without accounting for a fictitious force due to the acceleration of the Rod.Thanks for...

5. ### Maximum angle reached by a cube placed inside a spinning cylinder

I am trying to solve a problem where we have to find the maximum angle before a cube starts sliding when said cube is placed inside a spinning hollow cylinder (the cylinder is placed horizontally). The radius of the cylinder is 0.4 m, the coefficient of static friction between the cube and the...
6. ### B Rolling of non-deforming sphere on a non-deforming rough surface?

According to my current understanding rolling friction rolling friction is the static friction (parallel to the surface on which the object is moving) applied by the frictional surface (rough surface) on the contact point or contact area of the object whose v≠Rw(v is translational velocity and...
7. ### Intuition on the direction of friction in a rotational dynamics problem

The figure illustrates the situation. The radii of the larger and smaller discs are 2R and R, respectively. Their masses are M and 2M, respectively (the largst disc has the smallest mass). Also, m=5/4 M, where m is the mass of the suspended object. The pulley is "massless" (negligible moment...
8. ### What is the "r" in the moment of inertia?

We solved this problem in class as follows: Net torque about the center of the pulley taking counterclockwise rotation to be positive = m1gR - m2gR = I_tot α, where I_tot is the moment of inertia of the full system. My professor said that I_tot = I + m1R^2 + m2R^2, where m1R^2 is the moment...
9. ### Forces when car wheels "lay rubber"

Suppose the car is moving to the right, so if the wheels roll without slipping, they are rolling clockwise. To get the wheel to slip, a counterclockwise torque would need to be applied to cause the wheel to have some angular acceleration. If the wheel was slipping, then the bottom of the wheel...
10. ### Disk with rod attached rotating about the center of the disk

1) Since the rod is uniform, with mass m and length l, it has a linear mass density of ##\lambda=\frac{m}{l}##, so ##I_{rod_O}=\int_{x=r}^{x=r+l}x^2 \lambda dx=\frac{\lambda}{3}[(r+l)^3-r^3]=\frac{\lambda r^3}{3}[(1+\frac{l}{r})^3-1]=\frac{1}{3}mr^2[3+\frac{3l}{r}+\frac{l^2}{r^2}].##...
11. ### A sphere rolling without slipping down a hemisphere

a) From impulse-momentum theorem I have: ##J=mv## so ##v=\frac{J}{m}## and since the ball doesn't slip ##v=\Omega b## so ##\Omega=\frac{J}{mb}## and ##\dot{\theta}=\frac{v}{a+b}=\frac{\Omega b}{a+b}##. b) I considered the angular impulse: ##-J(a+b)=I_0 \Omega_0 \Rightarrow...
12. ### Moment of Inertia of a 4 rod system

This was the question (The line below is probably some translation of upper line in different language) For disc it was ma^2/2 For ring it was ma^2 For square lamina it was 2ma^2/3 For rods It was different Please explain Thank You🙏
13. ### Is the force exerted by a pivot always towards the center of mass?

I thought that the force by the pivot A on the pole AB would be the reaction force to the x-component of the gravitational force on AB. This would mean that the force by the pivot would be parallel to the pole, but in my notes from class the force vector seems to be more along the bisector of...
14. ### What is the final angular displacement of a disk hit by two masses?

1) By conservation of linear momentum: ##m_1 v_1-m_2v_2=(m+m_1+m_2)v_{cm}\Rightarrow v_{cm}=\frac{m_1}{m+m_1+m_2}v_1-\frac{m_2}{m+m_1+m_2}v_2=\frac{3}{8}\frac{m}{s}##; 2) By conservation of angular momentum: ##-Rm_1v_1-Rm_2v_2=I_{total}\omega=(I_{disk}+m_1R^2+m_2R^2)\omega## so...
15. ### Tangential velocity of rotating rod

1) ##LT\sin(\frac{\pi}{2}-\theta)-\frac{L}{2}mg\sin\theta=0\Rightarrow T=\frac{mg}{2}\tan\theta##. ##N_{x}-T=0, N_{y}-mg=0\Rightarrow N=\sqrt{N_x ^2+N_y ^2}=mg\sqrt{(\frac{\tan\theta}{2})^2 +1}## 2) ##E_{k_{fin}}=mg\frac{L}{2}[1+\cos\theta]## 3)...
16. ### Torque calculations: Rotating vertical shaft

I apologize in advance for any errors in my concepts or assumptions. Feel free to correct me wherever I am wrong. Thanks in advance for the help. There is a vertical shaft which will be operated at around 600 rpm (N) which can be achieved in 2 seconds (or even 4 just an assumption). The shaft...

18. ### Question on Moment of Inertia Tensor of a Rotating Rigid Body

Hi. So I was asked the following question whose picture is attached below along with my attempt at the solution. Now my doubt is, since the question refers to the whole system comprising of these thin rigid body 'mini systems', should the Principle moments of Inertia about the respective axes...
19. ### Motion involving Translation & Rotation |Kleppner and Kolenkow

My doubt is with Method 2 of the given example in KK. I'm unable to understand why the torque around A (where we have chosen a coordinate system at A) becomes zero due to the R x F in z direction with a minus sign {Photo Attached} I have tried to reason out that one way to formulate that term...
20. ### Ball rolling on a sphere

My question is this: - Friction exists (for no slipping/pure rolling to occur) - Why is the work done against friction not accounted for in the conservation of energy equation? Thank you!
21. ### Find the velocity and acceleration of a pulley in a mass-spring system

This looks like a classical setup but I can't find a solution. We can calculate the energy of the system by looking at the work done by the gravity and the spring. But how do we divide the energy between the kinetic energy of the pulley and the rotation of the pulley?
22. ### Two rotating masses balanced by a third mass (rotational dynamics)

Here's a diagram of what the system looks like: So far I have figured out what the initial angular velocity is, if the system is balanced (no movement): ## \sum F_m = m*\frac{v^2}{R_0}-\frac{Mg}{2}=0 ## ##m \frac{v^2}{R_0}-\frac{Mg}{2}=0 ## divide both sides by m ##\omega_0 =...
23. ### I Why does wind blow leaves in circles?

Earlier today I realized that, when a strong gust of wind would blow through my area, it would pick up leaves off the ground and typically blow them in circular patterns, and typically the leaves would go in at least several complete circles before coming to rest back down on the ground. Why is...
24. ### Rotational Dynamics - Modeling brake caliper deceleration of a chassis

I am a junior engineer tasked with modeling the dynamics of a small research UAV after landing. The UAV has 3 tires, 1 on the nose landing gear and 2 on the rear landing gear. The rear tires are equipped with disc brake calipers. My coworker has explained that the simplified model (MODEL 1...
25. ### Rotational dynamics and the conservation of energy

I = Icm + mr^2 I = 0.5 mr^2 + mr^2 I= 3/2 mr^2 By COE, mgh = 0.5(3/2 mr^2)(w^2) g(2r) = 3/4(r^2)(w^2) 8g/3 = rw^2 = v^2 / r v = sqrt( 8gr/3) v=0.511m/s ans: v=0.79m/s
26. ### Rotational dynamics: Resolving forces

Resolving the weight of the cylinder c, i get Mgcosθ (-y) and Mgsinθ (-x) mgsinθ - Fs - T = ma ---(1) (where Fs is frictional force and T is tension) τ = I α (where τ is torque and α is angular acceleration) torque is produced by both tension and frictional force (T-Fs) * r = 0.5 m r^2 α...
27. ### Difference between curvilinear and rotational motion

The solution states that there's no rotational motion when ##C## is cut (the motion is curvilinear), so we can take torques with respect to the centre of mass of the plate. But, isn't it rotating? I think of it as a pendulum, which describes a circular motion. What's the difference? Wouldn't the...
28. ### Conservation of angular momentum and its counterpart for linear momentum

Hi, I have just joined the forum. Thank you all for being a part of such places so that people like me can get answers to the questions on their minds! --------------------------- I have been trying to understand how a quadcopter yaws. Referring to the figure below which is bird's eye view of...
29. ### Moment of inertia where mass and torque are at a different positions

The formula for moment of inertia is: I=mr^2 A common derivation for this is: 1. F=ma 2. τ=rma 3. τ=rmrα = r^2 mα This is a rotational version of Newton’s second law, where torque replaces force, moment of inertia replaces mass, and angular acceleration replaces tangential acceleration...
30. ### Solving the First Equation: v1, v2, & ωR

I've marked correct answers above. Have a look at the solutions: How is the first equation justified? Shouldn't v2 and ωR be of opposite signs? What is v1? And how is it equal to v2? My biggest problem is the source of v1 since the ring is not having vertical displacement, then what is v1?
31. ### Stabilizer Leg Linear Actuator Force to Jack up a Truck's rear tyres

Hi, I previously posted about the statically indeterminate truck problem. Thank you to everyone who helped me. However, I now realized that isn't the problem I need to solve. I need to know the force of the linear actuators to lift the rear tyres off the ground. Since the tyres will be...
32. ### What force will be felt by ##B## when a rod is rotated?

We have a rod ##AB## of mass ##m##, a force (perpendicular to AB) is applied at ##A##. I want to know how much force will ##B## going to feel? When ##F_1## is applied at ##A## rod will rotate about its COM (which lies at the Center) and hence the point ##B## will also move (a little downwards...
33. ### Confusion on the concept of point of rotation

--no explanation as conceptual error--
34. ### Mathematically Modeling a Rolling Body with Slipping

Basically, I want to know if my assumptions and workings are correct. This is how I see this situation. First, I'm viewing this body as a series of disconnected points, like I have in this animation I made, modeling purely rolling motion. Modeling the body like that worked in that case, and...
35. ### Finding the angular acceleration of a flywheel

I have solved part a using the conservation of energy, getting a (correct) answer of 47.9 km/h, but I am unable to make headway with part b. Based on the flywheel rotating at 237rev/s when the car is moving at 86.5 km/h, I obtained omega = (237*2pi)v/24=62v. Differentiating both sides should...
36. ### Man rotating in a merry-go-round and grabbing a pendulum

Where: 1) ##A## is the translational acceleration, ##\Omega## the angular velocity and ##\dot \Omega## the angular acceleration (all relative to the inertial frame attached to the ground ##F##). 2) ##r'##, ##v'## and ##a'## are the position, velocity and acceleration vectors, all relative to...
37. ### Rotational dynamics: Rotating rod with two attached balls about a non principal axis

Firstly I deduced that in this situation the moment of inertia I, is not going to be parallel to w. And I calculated it as a matter of the angle, for the rod and the two point particles attached (with a mass 'm'), and the total moment of Inertia ended up being: I=((R²*sin²α)/2)*(M/6 + m) Being...
38. ### Rotational and translational motion

A Uniform rod AB of length 7m is undergoing combined motion such that, at some instant, velocities at top most point A is perpendicular to the rod and magnitude is 11 m/s. The mid point/ centre of mass ,say C, has a velocity of 3 m/s and is also perpendicular to the rod. If both the velocities...
39. ### Rotating Pendulum: Is This Eqn Correct?

is this eqn correct
40. ### Determine the acceleration of the cylinder axis if there is no slip

So I first wrote the moment of inertia of the cylinder, since it says that it is thin-walled, I think that its moment of inertia is ##I=\eta mR^2##. After that I wrote the sum of torques, I think that there are three forces that cause torque, the two forces of friction, the one caused by the...
41. ### Find the acceleration and the work done by a force pulling a spool?

I already solved the first part, but according the my book, the answer isn't quite correct. This is what I did. Finally, I ended up with ##a=\frac{F(r-R\cos\alpha)}{Rm(\gamma+1)}##. But according to my book, the answer is ##a=\frac{F(\cos\alpha-\frac{r}{R})}{m(1+\gamma)}##, what am I doing...
42. ### Rotational Movement of a Disc

I am doing a project, but am struggling to find relationships of proportionality or formulae between my dependent variables (angular velocity, displacement, acceleration of the disc and kinetic energy of the system) and my independent variables (falling masses and then the number of winds) or...
43. ### Rotational Dynamics, 1.68(SS.Krotov) and 1.264(Irodov)

Homework Statement I have attatched both the problems. The figure is essentially the same. Homework Equations 1. Energy conservation 2.mgcosx - N = mv^2/r , where N is zero in limiting case. The Attempt at a Solution I solved the whole question and got the answer given in irodov(0.33...
44. ### Friction between two disks

Homework Statement This problem was originally posted on Physics Problems Q&A: http://physics.qandaexchange.com/?qa=616/friction-between-two-disks Homework Equations Second Newton's law for rotation: $$\tau = I \alpha = RF$$ The Attempt at a Solution I tried to solve this problem as...
45. ### Cylinder lying on conveyor belt

Homework Statement You buy a bottle of water in the store and place it on the conveyor belt with the longitudinal axis perpendicular to the direction of movement of the belt. Initially, both the belt and the bottle are at rest. We can approach the bottle as one cylinder with radius ##R##, mass...
46. ### Rotational dynamics of mono-domain magnetic nanoparticles

Hi, I have a question about the rotation of a single-domain magnetic nanoparticle that is suspended in a ferrofluid immersed in an external field. Specifically, I am trying to work out the path that a normal vector on the surface of the sphere traces out in time. There are 2 ways the...
47. ### Frictional force between two rotating cylinders

Homework Statement .A cylinder P of radius rP is being rotated at a constant angular velocity ωP along positive y-axis with the help of a motor about its axis that is fixed. Another cylinder Q of radius rQ free to rotate about its axis that is also fixed is touched with and pressed on P making...
48. ### Cylinder rolling on a moving board

Homework Statement A rectangular board laying on the ground of mass ##M## is pulled with a force ##F## to the right, while a cylinder of mass ##m##, radius ##R## rolls without slipping on the board. Assume there is no friction between the board and the floor. Which way does the cylinder roll...
49. ### Torque exerted by a flywheel

Homework Statement Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 2.0 m diameter and a mass of 260 kg . Its maximum angular velocity...
50. ### Angular velocity equation

Homework Statement A uniform cylindrical wheel of mass ##m_{1}## and radius ##R_{1}## rotates with angular velocity ##\omega_{1}##. It lies a certain distance (along the same axis) from a static wheel of radius ##R_{2}## and mass ##m_{2}##. The wheels are then pushed against each other with a...