---
# System prepended metadata

title: HW3
tags: [Mechanics, ' ss', ' ncu']

---

---
tags: Mechanics, ss, ncu
author: N0-Ball
title: HW3
GA: UA-208228992-1
---

[Toc]

# Question

![](https://i.imgur.com/bKJPVav.jpg)


# Preknowledge

$$
\begin{aligned}
\hat r &= cos\theta\ \hat z + sin \theta\ (cos \phi \hat x + sin \phi \hat y) \\[1em]
\hat \theta &= -sin\theta\ \hat z + cos\theta\ (cos\phi\ \hat x + sin\phi\ \hat y) \\[1em]
\hat \phi &= -sin\phi\ \hat x + cos\phi\ \hat y
\end{aligned}
$$

**Therefore**

$$
\begin{aligned}
\frac{\partial \hat r}{\partial \theta} &= -sin\theta\ \hat z + cos\theta\ (cos\phi\ \hat x + sin\phi\ \hat y) &= \hat \theta \\[1em]
\frac{\partial \hat r}{\partial \phi} &= sin\theta\ (-sin\phi \hat x + cos\phi\ \hat y) &= sin\theta\ \hat \phi \\[1em]
\frac{\partial \hat \theta}{\partial \theta} &= -cos\theta\ \hat z + -sin\theta\ (cos\phi\ \hat x + sin\phi\ \hat y) &= - \hat r\\[1em]
\frac{\partial \hat \theta}{\partial \phi} &= cos\theta\ (-sin\phi\ \hat x + cos\phi\ \hat y) &= cos\theta\ \hat \phi \\[1em]
\frac{d \hat \phi}{d \phi} &= -cos\phi\ \hat x - sin\phi\ \hat y \\[1em]
&= -sin\theta\ [cos\theta\ \hat z + sin\theta\ (cos\phi\ \hat x + sin\phi\ \hat y)] - cos\theta\ [-sin\theta\ \hat z + cos\theta\ (cos\phi\ \hat x + sin\phi\ \hat y)] &= -sin\theta\  \hat r - cos\theta\ \hat \theta
\end{aligned}
$$

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

$$
\begin{aligned}
\frac{d \hat r}{dt} &= \frac{\partial \hat r}{\partial \theta} \cdot \frac{d \theta}{dt} + \frac{\partial \hat r }{\partial \phi} \cdot \frac{d \phi}{dt} \\[1em]
&= \underline{\dot \theta \hat \theta + sin\theta\ \dot \phi \hat \phi}
\end{aligned}
$$

## Ans.

$$
\dot \theta \hat \theta + sin\theta\ \dot \phi \hat \phi
$$

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

$$
\begin{aligned}
\frac{d \hat \theta}{dt} &= \frac{\partial \hat \theta}{\partial \theta} \cdot \frac{d \theta}{dt} + \frac{\partial \hat \theta}{\partial \phi} \cdot \frac{d \phi}{dt} \\[1em]
&= \underline{-\dot \theta \hat r + cos\theta\ \dot \phi \hat \phi}
\end{aligned}
$$

## Ans.

$$
-\dot \theta \hat r + cos\theta\ \dot \phi \hat \phi
$$

# 3.
:::info
Find $\frac{d \hat \phi}{dt}$
:::

$$
\begin{aligned}
\frac{d \hat \phi}{dt} &= \frac{d \hat \phi}{d \phi} \cdot \frac{d \phi}{dt} \\[1em]
&= (-sin\theta\ \hat r - cos\theta\ \hat \theta) \cdot \dot \phi \\[1em]
&= \underline{-sin\theta\ \dot\phi \hat r - cos\theta\ \dot\phi \hat\theta}
\end{aligned}
$$

## Ans.

$$
-sin\theta\ \dot\phi \hat r - cos\theta\ \dot\phi \hat\theta 
$$

# 4.
:::info
Find $(v_r, v_\theta, v_\phi)$
:::

$\text{By Definition, }\vec r = r(t)\hat r[\theta(t), \phi(t)]$

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

## Ans.

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

# 5.
:::info
Find $(a_r, a_\theta, a_\phi)$
:::

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

## Ans.

$$
\begin{aligned}
\vec a &= (\ddot r - r\dot\theta^2 - r sin^2\theta\ \dot\phi^2) \hat r + (2\dot r \dot \theta + r \ddot \theta - r sin\theta\ cos\theta\ \dot\phi^2) \hat \theta + (2sin\theta\ \dot r \dot \phi + 2r cos\theta\ \dot \theta \dot \phi + rsin\theta\ \ddot \phi) \hat \phi \\[1em]
&= (\ddot r - r\dot\theta^2 - r sin^2\theta\ \dot\phi^2, 2\dot r \dot \theta + r \ddot \theta - r sin\theta\ cos\theta\ \dot\phi^2, 2sin\theta\ \dot r \dot \phi + 2r cos\theta\ \dot \theta \dot \phi + rsin\theta\ \ddot \phi) = (a_r, a_\theta, a_\phi)
\end{aligned}
$$