Problem set 2

1. What would be the approximate length of the day if the earth spun so fast that bodies floated on the equator? Take the radius of the earth $R=6\times 10^6 m$ and $g=9.8 m/s^2$
1. 12 hours
2. 6 hours
3. 3 hours
4. 1.5 hour
Body remains on earth's surface because
$gravitational~ pull ~(body~weight)\geq centrifugal~ force$
Body will start floating when
$centrifugal~force > gravitational~pull~i.e. body~weight$
$$m\omega^2r>mg$$ or $$\omega>\sqrt{\frac{g}{r}}$$ But $$\omega=\frac{2\pi}{T}$$ $$T(in~ seconds)>2\pi\sqrt{\frac{r}{g}}$$
$$T(in~ hours)>2\pi\sqrt{\frac{r}{g}}\frac{1}{3600}=\frac{2\times3.14\times1000\times\sqrt{6}}{3600\times\sqrt{9.8}}=1.5~ hours$$
2. The real matrix $A=\begin{pmatrix}a&-f&-g\\ f&a&h\\g&-h&a\end{pmatrix}$ is skew symmetric when
   1. $a=0$
   2. $f=0$
   3. $g=h$
   4. $f=g$ $
   Matrix $A$ is skew symmetric when $A_{ij}=-A_{ji}$. Hence, $a=0$
3. If $A$ and $B$ are matrices such that $AB=B$ and $BA=A$ then $A^2+B^2$ equals
1. 2AB
2. 2BA
3. A+B
4. AB
$$\begin{align*} A^2+B^2&=(BA)^2+(AB)^2\\ &=BABA+ABAB \end{align*}$$ $$\begin{align*} A^2+B^2&=B\underline{AB}A+A\underline{BA}B\\ &=B\underline{BA}+A\underline{AB}\\ &=BA+AB=A+B \end{align*}$$
4. The average value of function $f(x)=4x^3$ in the interval 1 to 3 is
1. 15
2. 20
3. 30
4. 40
$$\begin{align*} <f(x)>&=\frac{\int_1^3f(x)dx}{\int_1^3dx}\\ &=\frac{\int_1^34^3dx}{\int_1^3dx}\\ &=\frac{[x^4]_1^3}{[x]_1^3}\\ &=\frac{81-1}{3-1}=40 \end{align*}$$