Posted on Categories:Theoretical mechanics, 物理代写, 理论力学

物理代写|理论力学代写Theoretical Mechanics代考|PHYS61000 Rotation on a Fixed Axis

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物理代写|理论力学代写Theoretical Mechanics代考|Rotation on a Fixed Axis

As shown in Fig. 8.3, if we impose a moment on a door, the door will be closed with respect to a fixed axis. This type of motion is also a basic motion, termed as rotation with respect to a fixed axis. In this rotation process, there always exists a fixed axis on the rigid body. This type of motion takes some special features. If we look through the positive direction of $z$-axis in Fig. 8.3, we can find the orbit of an arbitrary point on the rigid body in rotation, which is actually a circle.

To depict the motion of the rigid body, we introduce the angular displacement $\varphi$, whose sign also obeys the right-hand screw law. The angular velocity can be defined as
$$\omega=\lim _{\Delta t \rightarrow 0} \frac{\Delta \varphi}{\Delta t}=\dot{\varphi} .$$
The angular velocity can also be related with the frequency via
$$\omega=2 \pi f .$$
Introducing the concept of rotation velocity $n$ with the unit of $1 / \mathrm{min}$, one has

$$\omega=\frac{2 \pi n}{60}=\frac{\pi n}{30} .$$
Similarly, the angular acceleration can be further defined as
$$\varepsilon=\lim _{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t}=\dot{\omega}=\ddot{\varphi} .$$
The above relation can be analogous to the displacement, velocity, and acceleration defined in the last chapter.

If the rigid body is in rotation with respect to an axis, any point on the rigid body is in a circular motion. As shown in Fig. 8.4, the radius $r$ is the vertical distance from the axis to the arbitrary point, $s$ is the arc length, and $\varphi$ is the corresponding angular displacement. We then have the following geometric relation:
$$s=r \varphi .$$
Taking derivatives on both sides of the above equation, one has
\begin{aligned} &v=\dot{s}=r \dot{\varphi}=\omega r, \ &a_{\tau}=\dot{v}=\ddot{s}=r \ddot{\varphi}=\varepsilon r . \end{aligned}

物理代写|理论力学代写Theoretical Mechanics代考|Relative Velocity

We consider two points on the planar figure $A$ and $B$, which have the velocity $v_{A}$ and $v_{B}$, respectively, as schematized in Fig. 9.1. We normally name point $A$ as the base point, as it is a reference point. There is a relative velocity $v_{B A}$, with the meaning that $A$ is the base point. As a result, we know that $v_{B A}$ is not equal to $v_{A B}$. According to the velocity superposition, one has
$$v_{B}=v_{A}+v_{B A},$$
where the direction of the relative velocity $\boldsymbol{v}_{B A}$ is perpendicular to the line $A B$, as point $B$ rotates with respect to point $A$ in a circular motion. From the above formula, we can solve the velocity of any point on the planar figure if the base point is given. Therefore, this method of velocity composition is called “Method of base point”.
As a consequence, if we decompose the above velocities in the direction of line $A B$, then one has
\begin{aligned} \left.\boldsymbol{v}{B}\right|{A B} &=\left.\boldsymbol{v}{A}\right|{A B}+\left.\boldsymbol{v}{B A}\right|{A B} \ &=\left.\boldsymbol{v}{A}\right|{A B} . \end{aligned}
This means the projections of the velocities of the two points are equal along their connection line. In fact, this is the second method to study the velocity composition, which is termed as “Method of velocity projection”. As shown in Fig. 9.2, if we have known the angles between the velocities and line $A B$, we then have the relation
$$v_{A} \cos \alpha=v_{B} \cos \beta .$$

物理代写|理论力学代写Theoretical Mechanics代考|Rotation on a Fixed Axis

$$\omega=\lim {\Delta t \rightarrow 0} \frac{\Delta \varphi}{\Delta t}=\dot{\varphi}$$ 角速度也可以通过以下方式与频率相关 $$\omega=2 \pi f$$ 介绍旋转速度的概苡以 $1 / \mathrm{min}$,一个有 $$\omega=\frac{2 \pi n}{60}=\frac{\pi n}{30}$$ 类似地，角加速度可以进一步定义为 $$\varepsilon=\lim {\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t}=\dot{\omega}=\ddot{\varphi} .$$

$$s=r \varphi .$$

$$v=\dot{s}=r \dot{\varphi}=\omega r, \quad a_{\tau}=\dot{v}=\ddot{s}=r \ddot{\varphi}=\varepsilon r .$$

物理代写|理论力学代写Theoretical Mechanics代考|Relative Velocity

$$v_{B}=v_{A}+v_{B A}$$

$$\boldsymbol{v} B|A B=\boldsymbol{v} A| A B+\boldsymbol{v} B A|A B \quad=\boldsymbol{v} A| A B$$

$$v_{A} \cos \alpha=v_{B} \cos \beta .$$

MATLAB代写

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