Problem set 88

1. The moment of inertia of a thin disc of radius $R$ about an axis passing through its center and perpendicular to the plane of disc is:
1. $MR^2$
2. $\frac{2}{3}MR^2$
3. $\frac{3}{2}MR^2$
4. $\frac{1}{2}MR^2$

$\frac{1}{2}MR^2$

Hence, answer is (D)

2. A coin is tossed four times what is the probability of getting two heads and two tails?
1. $\frac{3}{8}$
2. $\frac{1}{2}$
3. $\frac{5}{8}$
4. $\frac{3}{4}$

When a coin is tossed once, there are two out comes possible. Hence, the coin is tossed four times, the total number of outcomes are $2^4=16$. Out of which we will two heads exactly $\frac{4!}{2!(4-2)!}=6$ times. Hence, probability of getting two heads and two tails is $\frac{6}{16}=\frac{3}{8}$

Hence, answer is (A)

3. Consider three vectors $\vec a=\hat i+\hat j+\hat k$, $\vec b=\hat i-\hat j+\hat k$ and $\vec c=\hat i-\hat j-\hat k$. Which of the following statement is true?
1. $\vec a$, $\vec b$, $\vec c$ are linearly independent
2. $\vec a$, $\vec b$ are linearly independent
3. $\vec b$ and $\vec c$ are right angle to each other
4. $\vec a$ and $\vec c$ are parallel

The vectors in a subset $S = \{ \vec v_1 ,\vec v_2 ,\cdots,\vec v_n \}$ of a vector space $V$ are said to be linearly dependent, if there exist a finite number of distinct vectors $\vec {v}_{1},\vec {v}_{2},\dots ,\vec {v}_{k}$ in $S$ and scalars $a_{1},a_{2},\dots ,a_{k}$, not all zero, such that $$a_{1}\vec {v}_{1}+a_{2}\vec {v}_{2}+\cdots +a_{k}\vec {v}_{k}=\vec {0}$$ where zero denotes the zero vector. Let $$a_1\vec a+a_2\vec b+a_3\vec c=0$$ $$\begin{bmatrix}1&1&1\\1&-1&-1\\1&1&-1\end{bmatrix}\begin{bmatrix}a_1\\a_2\\a_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$ If $\vec a$, $\vec b$ and $\vec c$ are linearly independent, then the only solution to this system of equations is the trivial solution, $a_1=a_2=a_3=0$. For homogeneous systems this happens precisely when the determinant is non-zero. If determinant is zero then vectors are linearly dependent. $$\begin{vmatrix}1&1&1\\1&-1&-1\\1&1&-1\end{vmatrix}=4$$ Hence, $\vec a$, $\vec b$ and $\vec c$ are linearly independent.

Hence, answer is (A)

4. Which of the following defines a conservative force field?
1. $\vec\nabla\cdot\vec F=0$
2. $\vec\nabla\times\vec F=0$
3. $\oint\vec F\cdot\vec dr=0$
4. $\frac{d\vec F}{dt}=0$

A force field $\vec F$, is called a conservative force or conservative vector field if it satisfies following three equivalent conditions: $$\vec\nabla\times\vec F=0$$ $$\oint\vec F\cdot\vec dr=0$$ $$\vec F=-\vec\nabla\Phi$$ where $\Phi$ is scalar potential.

Hence, answer is (B and C)

5. $\nabla\left(\frac{1}{|\vec r|}\right)$ is given by
1. $\frac{1}{r}\hat r$
2. $\frac{1}{r^3}(\hat i+\hat j+\hat k)$
3. $-\frac{\vec r}{r^3}$
4. $r(\hat i-\hat j-\hat k)$

$$\begin{align*} \nabla\left(\frac{1}{|\vec r|}\right)&={\scriptstyle \left(\hat i\frac{\partial}{\partial x}+\hat j\frac{\partial}{\partial y}+\hat k\frac{\partial}{\partial z}\right)\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)}\\ &={\scriptstyle -\left(\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}\right)\left(x\hat i+y\hat j+z\hat k\right)}\\ &=-\frac{\vec r}{r^3}\\ \end{align*}$$

Hence, answer is (C)