All Questions
Tagged with kinematics classical-mechanics
34 questions with no upvoted or accepted answers
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Why is the ratio of components of kinetic energy equal to the ratio of kinetic energy to total energy for a projectile whose range is maximized?
The launch angle $\theta$ that maximizes the range of a projectile in a uniform gravitational field is
\begin{align}
\theta = \arctan\left(\frac{v_o}{\sqrt{v_o^2 + 2gh}}\right), \tag{1}
\end{align}
...
- 41
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Kinematics of a rolling disk on a static disk (variation of the Euler disk)
I'm puzzled by the following problem. Consider a simple tilted disk $\mathcal{D}$ of radius 1 (in any unit) rolling without sliding on top of a static horizontal disk $\mathcal{S}$. The normal $\...
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Why does the photon in Compton Scattering have a minimum frequency/maximum wavelenght?
Using conservation of four-momentum one finds that, with respect to the angle of deviation of the photon from its original direction $\theta$, the wavelength and frequency of the emitted photon are:
$
...
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419
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Is there is an entropy cost of moving an object?
Is there an entropy cost associated with moving an object from one point to another, even if all forces involved are conservative? Or, is there some condition on what kind of move has an entropy cost?
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Equation of Motion of Rigid Body Represented by Twist and Derivative of Twist
This question is an extension of question Understanding terms Twist and Wrench.
Assuming there is a rigid body with body twist denoted as $\mathcal{V}_{b}=\left(\boldsymbol \omega_{b}, \boldsymbol v_{...
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Proving that the relative angular velocity of any particle with respect to any other particle is the same in a rigid body
Claim: The angular velocity of any point mass of a rigid body relative to any other point mass is the same, i.e., $\vec{\omega_{i,j}} = \vec{\omega}\;\,\forall{i}\,\forall{j}$, where $\vec{\omega}$ is ...
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4
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Uniform Circular Motion w/ Tension and Friction
So I had a problem today which I couldn't make any sense of. I don't have access to it at the moment but this is a pretty accurate approximation.
Basically, a mass (m) is swinging horizontally on ...
- 21
1
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1
answer
46
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Linearly Interpolating in a Non-Inertial Frame
I am working on an engineering task where we have a flying object which knows its height above a curved surface as well as how close it is to other flying objects. If I want to interpolate between two ...
- 171
1
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1
answer
59
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Statistical Analysis of motion under central force
My question is very simple. If we were to plot the motion of a particle inside circular walls governed by a central attractive force with perfectly elastic collisions, would the statistics of the ...
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Difference between eigenvalues of the potential energy Hessian vs. "generalized" eigenvalues with respect to a kinetic energy "metric"
Simple version
Consider if we have a Lagrangian defined by
$$L(q,\dot{q}) = \frac{1}{2} g_{ij}(q) \dot{q}^i \dot{q}^j - U(q) \tag{1a}$$
where the potential energy $U(q)$ has a single minimum at $q=0$ (...
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1
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Finding The Trajectory of particle Kept on a rotating surface
Suppose we "Keep" a particle on a circular, rotating turn-table, rotating with constant angular velocity $\dot{\theta}$, (i.e, the at $t=0$,when the particle makes contact with the table, the ...
- 1,325
1
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1
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Determining acceleration in order to meet a space-time-velocity reservation
An Example
A train on a single line track is told to be at position $x_{res}$, at time $t_{res}$, and to be moving at exactly $v_{res}$ when it hits that point. Where $_{res}$ denotes the reservation....
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1
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How should I think about a block-and-tackle configuration weight-lifting machine?
At my gym, some of the weight machines provide a double mechanical advantage using a gun tackle pulley configuration.
For a long time when I was using such machines, I convinced myself that if I were ...
- 339
1
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Move body in 1-dimension to desired state by applying force
Consider a simple point mass of $M$ in vacuum in 1-dimension with initial position $x_0$, velocity $v_0$ and acceleration $a_0$. Our goal is to move this body to position $x_1$ with velocity $v_1$ and ...
- 505
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Computing angular acceleration of a composition of coordinate frame transformations
Given two coordinate frames $A$ and $B$, we can define the transformation between them by:
$[\vec r_{AB}]_A$, the vector from the origin of $A$ to the origin of $B$ (resolved in frame $A$), and
$q_{...
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1
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Why is the tangent of the rear wheel path of a bicycle parallel to the frame?
In "bicycle problems" (made famous for example by the book "Which Way Did the Bicycle Go?") the relevant point is the following: If $r(t)$ and $f(t)$ are the points of contact of ...
- 213
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21
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Position and displacement vector in Arc coordinate system
In Arc coordinate system the position of the particle is given by the length of the path(which is pre-determined and may also be curved) that it has travelled so how can we write it's position vector ...
- 51
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27
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How to mathematically model a lumped-mass cable model?
I am trying to simulate a 2D cable as a lumped-mass model. I think that it's like a chain of pendulums and I would preferably like to be damped. Just to make sure, this is the cable model that I'm ...
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1
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129
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Absolute Angular Velocity - How to use?
From my dynamics course, we were introduced to the so-called absolute angular velocity of a rigid body. Below is a short diagram:
The following equation for the velocity of point P on a rigid body is ...
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1
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97
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Angle, and maximum area of projectile motion accounting air resistance
Recently I was wondering about what will happen to the particle when subjected various elevation angle of projectile motion if we account air resistance. I want to know what the angle of elevation ...
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Necessary and sufficient conditions for periodic motion
Let us fix a reference frame $S$ with origin in $O$ in the euclidean space $\Bbb R^3$, then let us also define a periodic motion in the following manner:
A motion is periodic if and only if the time-...
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Minimum seperation of moving objects doubt
Let there be $2$ objects $P_1$(initial velocity $u$ $ms^{-1}$ & acceleration $a$ $ms^{-2}$) & $P_2$ (initial velocity $U$ $ms^{-1}$ & acceleration $A$ $ms^{-2}$) initially separated by ...
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85
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Cartesian coordinate velocity and generalized coordinate velocity
use $x_k$ to denote the kth component of cartesian coordinate, and $q_k$ to denote the generalized coordinate.
Taking the derivate of $x_k(q_1,q_2,q_3,t)$ w.r.t. time, we have
$$\frac{d x_k(q_1,q_2,...
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45
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If we know the location of object at different times, what more can we find out from it?
I'm brushing up on my physics, and I've been struggling with this problem in my head. I hope you can help.
Let's assume that we have some object/particle that only moves along a single axis ($x$). We ...
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How can I isolate the drag constant (C) for vertical motion with quadratic drag from the equation for peak height?
Some background on my problem: I have calculated the peak height of a rocket (with quadratic drag). As this will be used for a PID software application, I need to have the equation be expressing peak ...
- 101
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1
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91
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The velocity of an object leaving the inclined plane
I saw the question below in our physics textbook. It asks
A block of mass m is released and slides down on the frictionless inclined surface ( the inclined plane is of concrete and doesn't ...
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Quick question: can you describe the direction of motion relative to a particle?
In Gregory R.D.'s Classical Mechanics, he has written "Relative to a single particle, the only thing that can be specified is distance from that particle".
My question is, isn't it possible to ...
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2
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What is the direction of linear motion after this impact?
I've been thinking about a situation where I am not sure what the resulting motion would be.
This is what I am thinking,
1) P is the point of application of force. If there are two force components ...
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1
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78
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circular(ish) motion?
how would one describe as a function of time the position vector of an object with initial position $\vec{x_0}$ and velocity $\vec{ v_0}$ experiencing a central force of constant magnitude around the ...
user225234
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Particle Disintegration Range of Angles (Landau& Lifshitz)
In the particle disintegration problem in the book by Landau and Lifshitz (pg 44 question 3), it is asked to find the range for $\theta=\theta_1+\theta_2$, where $\theta_1$ and $\theta_2$ are the ...
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Why do systems with a fixed gear-ratio still use gears?
From my understanding, there are two uses of a gearing system: to change the speed of output rotation (trading it with torque), and to change the axis of rotation. Now, in a car, for example, it is ...
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Optimal "Blow up" Configuration
Suppose you have three balls glued together. Two are red and one is blue. The system of balls is blown up by an explosion of pure energy (that conserves the center of mass frame) exactly at the center,...
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Confusion about valid set of virtual displacement
The answer of the problem below makes me confused about my thinking about virtual displacement...
Consider the system in the figure below, similar to mechanisms used for aerial filming of sporting ...
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How to find the direction of acceleration if an object is changing its direction of velocity but not magnitude then how we can find the direction
I am new at this topic so please do mind if my question doest make sense to you.I am trying to find out that what will be the direction of acceleration if object changes Direction of velocity but not ...
