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Confusion about the Spinning Top

I think your equation $\Omega\wedge P_c=r_c\wedge Mg$ is not correct. The l.h.s. should be the derivative of the total angular momentum w.r.t. to point of contact with the floor, not w.r.t. the center ...
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Inertial accelerations like the Coriolis effect are well known. Are there also 'inertial jerks' and what are some examples?

An astronaut sits in a rocket on the launch pad. The rocket is launched. From the astronaut's point of view, the rocket is stationary. A time dependent fictitious force sprang into being and pushed ...
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Inertial accelerations like the Coriolis effect are well known. Are there also 'inertial jerks' and what are some examples?

The Coriolis effect arises when you use co-ordinates fixed in an accelerating frame of reference (in this case, the surface of a rotating planet). If you use co-ordinates fixed in a frame of reference ...
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Physical intuition about the inertia tensor

I have also encountered the same problem until recently I understood the meaning behind the indices. Although there are a lot of definitions of a tensor, we are left to decide which one is palatable ...
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Faster Small Tires vs Slower Large Tires

Long distance trucks use large tires in part because of rolling resistance, as @AgniusVasiliauskas pointed out. But also a large tire has a more gentle curvature. This means a larger contact patch. ...
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Faster Small Tires vs Slower Large Tires

As per rolling resistance law, ratio of engine work done opposing rolling resistance force negative work ,in case big and small tires is : $$ \frac {W_{_R}}{W_r} = \sqrt {\frac {r}{R}} ~~~~~~~~~~~~~~~~...
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How to solve for the trajectory of the center of mass?

If $\mathbf J(x(t),y(t))$ is the external force acting on square between times $t_0, t$, then the total impulse is $\int_{t_0}^{t}\mathbf J(x(u),y(u)))du$. So we get $\int_{t_0}^{t}\mathbf J(x(u),y(u))...
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How to solve for the trajectory of the center of mass?

The motion (acceleration) of the center of mass (CM) is $\vec a_{CM} = {\vec F_{ext} \over M}$ where $\vec F_{ext}$ is the total applied external force and M is the mass of the square. The rotation ...
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1 vote

The utility of HorsePower vs Torque

Neglecting air resistance and other forms of drag the general expression for acceleration as a function of engine power (regardless of gearing etc) is $$ a = \frac{P}{m\, v} \tag{1}$$ Note: All ...
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The utility of HorsePower vs Torque

Which vehicle would experience greater acceleration (G) The acceleration is given by the force at the road. The force at the road is in turn specified by taking the power and dividing by the speed. ...
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1 vote

What is the $xy$ equation of a point particle of a rigid body experiencing rolling motion where $v_{cm}=R \omega$?

The curve traced by a point belonging to a disc in pure rolling is a cycloid. In Cartesian coordinates it reads $$x=R\cos^{-1}(1-y/R)-\sqrt{y(2R-y)}.$$
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Representing vectors in non-inertial rotating frame of reference

Consider two frames of reference, O and O*. O is the origin of a fixed point in an inertial frame. O* is the origin of a fixed point in a noninertial frame. In general O* can be accelerating with ...
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Finding the time period under variable acceleration

You know that the tangential acceleration $Rd\omega/dt=\omega ^2 R$. So now you can integrate twice and find $\theta(t)$.
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High school Momentum/Torque problem

A torque $T$ is given as: $$T = F x_{\perp}$$ where $F$ is the force and $x_{\perp}$ is the perpendicular distance from the acting force to the pivot of it's axis. The force of the wind will start to ...
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When to apply $I_c \underline{\omega} = \underline{M_c}$?

Cylinder rolling in the inclined plane. starting with the free body diagram , you obtain two equations translation $$m\,\ddot s=-F\tag 1$$ rotation $$I_{\text{CM}}\,\ddot\varphi=F\,r+\tau\tag 2$$ now ...
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High school Momentum/Torque problem

The problem should also state "assume the assume the point where the base remains on the floor does not move". Consider the torque to turn the set over, about the fixed point of the base on ...
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Acceleration calculation for an object with is rotating about it's COM and it's COM is revolving about another object

the position vector to the object A given in inertial system is: $$\begin{bmatrix} x \\ y \\ \end{bmatrix}= \left[ \begin {array}{c} r\cos \left( \omega\,t \right) +l\cos \left( \varphi(t) \...
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When to apply $I_c \underline{\omega} = \underline{M_c}$?

The expression $ \underline{M}_c = \mathrm{I}_c \,\underline{\omega}$ is never correct. I think you forgot the time derivative of rotational velocity here. Also, the are Coriolis torques that relate ...
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Acceleration calculation for an object with is rotating about it's COM and it's COM is revolving about another object

In the frame of the satellite, the acceleration of A is just acceleration from rotation around the satellite. In the frame planet, it's the sum of the accelerations of A wrt the satellite plus the ...
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1 vote

Spinning top fixed point

When calculating rotation problems, with torques and suchlike, it is necessary to choose an origin for the coordinate system. This choice is ultimately arbitrary since the equation of motions are ...
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11 votes

Why do we talk about inertia tensor?

A rank 2 tensor is something that relates two vectors. In this case, the MMOI tensor relates the rotational velocity vector to the angular momentum vector. Given a solid whose internal particles are ...
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9 votes

Why do we talk about inertia tensor?

The moment of inertia you mentioned is only for a single, given axis of rotation. It's used to compute the angular momentum related to his axis: $$L=I\omega$$ If you want to generalize the result in ...
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What does inextensible string mean and how can that be applied in real life

It sounds like you have the right understanding of inextensible. All real strings (ropes, cables, chains etc.) will stretch to a certain degree when under tension so describing a string inextensible ...
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6 votes

Can the value of friction force ever exceed value of applied force?

First, there is an important qualification to "the value of friction force can never be greater than the applied force". This refers to static friction. Indeed, you pose a scenario where &...
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2 votes

Can the value of friction force ever exceed value of applied force?

To answer the question in the title, the friction force can be larger than the normal force that is producing it. There is no restriction. Coefficients of friction larger than 1 are not common in ...
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20 votes

Can the value of friction force ever exceed value of applied force?

No. You've discovered the Class 2 Lever, which places the load between the input force and the fulcrum. You have correctly calculated that the friction force is 4 times the input force. The reason for ...
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Lagrangian Mechanics - Is the Given Answer Incorrect?

$\dot{\phi}$ is not a constant of the motion, so you can't treat it as constant when taking the derivative of $V_\text{eff}$ with respect to $\theta$. If you leave $V_\text{eff}$ in its original form,...
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What specifically about the torque vector is perpendicular? Is the torque vector like this only so that it works smoothly with linear algebra?

It's neither calculus nor linear algebra; rather, it's geometry, or perhaps representation theory, or Clifford algebra... each of which can lead to a long an in-depth answer. That will not be ...
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Can the direction of angular momentum be changed without external torque?

In the absence of external torques with respect to a particular origin, the magnitude and direction of the angular momentum of a system are constant (with respect to that origin). So the most ...
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3 votes

Rotating a lever — small end has more force but less speed?

There are two errors here. First, $ F_{2} = F_{1}\frac{R}{r} $ is about the balancing of two torques in static equilibrium. That is, the usually case where this holds is where $F_1 R$ is one torque, ...
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Rotating a lever — small end has more force but less speed?

What accelerates object A is not the force $F_1$ that you applied to it, it's the net force that it experiences. $F_1$ is one component of the net force, but there is also a reaction force from the ...
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How to know the direction of static friction with angular momentum?

In this case, the static friction is responsible for generating the circulation in the case of rolling without slipping. The reverse occurs when friction is a resistance to rolling. This is similar to ...
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Work Done on a rotating thin rod by hinge Forces

There is no friction so the only force on the rod at the hinge is the hinge reaction force. Is there a displacement of the hinge reaction force along the line of action of the hinge reaction force? As ...
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Resisting Precession of a Gyroscope

I will first provide a link, and then I will go to your question specifically. For discussion of the underlying mechanics I refer to my 2012 answer: gyroscopic precession. (That discussion, ...
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Resisting Precession of a Gyroscope

Gravity provides a torque that would pull the gyro over if it is not spinning. When the gyro is spinning, in the non-inertial frame precessing with the gyro, fictitious forces are present. Consider ...
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Work done by static friction on a car

The answer by @stafusa is correct. The increase in kinetic energy (KE) of the car is due to the decrease in internal energy of the car (from combustion, batteries, etc.) This can be understood using ...
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1 vote

Traction Torque Effect On A Rotating Wheel

Here is a much simplified schematic diagram of the rear end of the transmission chain as a car is accelerating towards you. For the transmission shaft the net torque on it, $\tau_{\rm TE}-\tau_{\rm ...
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1 vote

Traction Torque Effect On A Rotating Wheel

For any mechanical system, every piece of it will obey Newton's laws. For a system like the one you've shown, each colored part will obey Newton's laws during operation (this analysis is usually done ...
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2 votes
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Traction Torque Effect On A Rotating Wheel

engineers usually say that the wheels driving torque is always equal to the traction force times the radius of the wheel. That's a simplification where the mass of the wheel is insignificant compared ...
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Traction Torque Effect On A Rotating Wheel

If the torque due to traction equals that of the motor the angular acceleration is indeed zero. Obviously when a vehicle accelerates the driving force of the motor overcomes that of the friction. It ...
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Traction Torque Effect On A Rotating Wheel

This wheel would never accelerate as two opposing torques of equal magnitude are acting on the wheels such they should cancel each other creating ZERO acceleration. No. You need to learn what free ...
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What is the general no slip condition for rotations?

The general no-slip condition for a body is expressed as a dot product between the translational velocity of the body at the point of contact $\boldsymbol{v}_A$, and the contact normal vector $\...
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1 vote

What is the general no slip condition for rotations?

Landau & Lifshitz explain an equivalent problem. Some observations: The Instataneous Axis of Rotation (IAR) is a straight line passing for the two poins of the disks, in contact with the flat ...
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When does a free body moving on a smooth circular path make a complete revolution?

To do a loop over the top the velocity at the top ($v_{top}$) has to be such that the centripetal acceleration is at least $g$ (otherwise it will fall down). This is: $$a= v^2 / r$$ So: $$g = {v_{top}}...
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When does a free body moving on a smooth circular path make a complete revolution?

From what I think you mean from Like a ball in a closed circular tube the radial or normal force from the tube, $N$ in the diagram above, can only be positive. If $N$ is negative, the ball will fall ...
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Seemingly correct and useful expression for kinetic energy of 3D rigid body not found anywhere

Perhaps the expression is not common, I would be surprised if it had not been mentioned in some reference, because it is obvious to the eye. If one intends to use the principle of conservation of ...
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Seemingly correct and useful expression for kinetic energy of 3D rigid body not found anywhere

Considering the sum of particles on a rigid body the mass moment of inertia tensor about some arbitrary point A is $$ \mathrm{I}_A = \int \left( \vec{r}\cdot\vec{r} - \vec{r} \odot \vec{r} \right) \,{\...
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Seemingly correct and useful expression for kinetic energy of 3D rigid body not found anywhere

The expression you gave is very common. You mostly get if when going through the proof of König's theorem for a continous system, which involves going to the center-of-mass frame (hence getting the ...
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6 votes

Seemingly correct and useful expression for kinetic energy of 3D rigid body not found anywhere

The integral $$ I= \int\mathrm dm\ r_\perp^2 $$ is the moment of inertia about the given axis. You may be partway along the path to computing the moment of inertia tensor, which is useful describing ...
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Serious confusion regarding central concept of torque

One thing that might give you better intuition is that the definition of an angle is the ratio of 2 lengths: $$\theta = \frac{s}{r}$$ With $s$ being the arc length traveled along a circle, and $r$ the ...
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