---
tags: Mechanics, ss, ncu
author: N0-Ball
title: HW2
GA: UA-208228992-1
---

[Toc]

# Question


# Preknowledge

$$
\hat r = cos(\theta) \hat x + sin(\theta) \hat y \\[1em]
\hat \theta = -sin(\theta) \hat x + cos(\theta) \hat y
$$

Since $\hat r \cdot \hat \theta = 0$ and $\hat r = cos(\theta) \hat x + sin(\theta) \hat y$ as definition

$$
\frac{d \hat r}{d \theta} = -sin(\theta) \hat x + cos(\theta) \hat y = \hat \theta
$$

$$
\frac{d \hat \theta}{d \theta} = -cos(\theta) \hat x + -sin(\theta) \hat y = - \hat r
$$

# 1. Find
:::info
$$
\frac{d \hat r}{dt}
$$
:::

$$
\begin{aligned}
\frac{d \hat r}{dt} &= \frac{d \hat r}{d \theta} \times \frac{d \theta}{d t} \\[1em]
&= \hat \theta \cdot \dot \theta
\end{aligned}
$$

## Ans.

$$
\hat \theta \cdot \dot \theta
$$

# 2.
:::info
$$
\frac{d \hat\theta}{dt}
$$
:::

$$
\begin{aligned}
\frac{d \hat \theta}{dt} &= \frac{d \hat \theta}{d \theta} \times \frac{d \theta}{dt} \\[1em]
&= - \hat r \cdot \dot \theta
\end{aligned}
$$

## Ans.

$$
(- \hat r \cdot \dot \theta)
$$

# 3.
:::info
$$
(v_r, v_\theta, v_z)
$$
:::

$$
\begin{aligned}
\vec v &= \frac{\vec r}{dt} = \frac{r \hat r(\theta) + z(t) \hat z}{dt} \\[1em]
&= \dot r \hat r + r \times \frac{d \hat r}{dt} + \dot z \hat z \\[1em]
&= \dot r \hat r + r \times \dot \theta \cdot \hat \theta + \dot z \hat z
\end{aligned}
$$

## Ans.

$$
\begin{aligned}
\vec v &= \dot r \hat r + r \times \dot \theta \cdot \hat \theta + \dot z \hat z\\[1em]
&=(\dot r, r \times \dot \theta, \dot z) = (v_r, v_\theta, v_z)
\end{aligned}
$$

# 4.
:::info
$$
(a_r, a_\theta, a_z)
$$
:::

$$
\begin{aligned}
\vec a &= \frac{d \vec v}{dt} = \frac{d(\dot r \hat r + r \times \dot \theta \cdot \hat \theta + \dot z \hat z)}{dt} \\[1em]
&= \ddot r \hat r + \dot r \dot \theta \hat \theta + \dot r \dot \theta \hat \theta + r \ddot \theta \hat \theta - r \dot \theta \hat r \dot \theta + \ddot z \hat z\\[1em]
&= (\ddot r - r \dot \theta ^ 2) \hat r + (2 \dot r \dot \theta + r \ddot \theta) \hat \theta + \ddot z \hat z
\end{aligned}
$$

## Ans.
$$
\begin{aligned}
\vec a &= (\ddot r - r \dot \theta ^ 2) \hat r + (2 \dot r \dot \theta + r \ddot \theta) \hat \theta + \ddot z \hat z \\[1em]
&= (\ddot r - r \dot \theta ^ 2, 2 \dot r \dot \theta + r \ddot \theta, \ddot z) = (a_r, a_\theta, a_z)
\end{aligned}
$$