IAT IISER 2023

1

PYP

IAT IISER 2023

Two fixed point particles, each of charge $+q$, are separated by a distance $2 L$. Another point charge $+q$ of mass $m$ is oscillating about its equilibrium position as indicated in the figure below. The time period of oscillation is given by $T=$ $2 \pi^{3 / 2} \alpha \sqrt{m} / q$. Given that $\epsilon_0$ is the

permittivity of free space, which of the following options is the dimensionally correct expression for $\alpha$ in SI units?

A

$\epsilon_0^{3 / 2} L$

B

$\epsilon_0^{1 / 2} L$

C

$\epsilon_0^{3 / 2} L^{1 / 2}$

D

$\epsilon_0^{1 / 2} L^{3 / 2}$

2

PYP

IAT IISER 2023

The velocity $v(t)$ of a particle moving in one dimension is given by:

$$ v(t)= \begin{cases}\alpha t, & 0 \leq t \leq T / 3 \\ \alpha T / 3, & T / 3 \leq t \leq 2 T / 3 \\ \alpha(T-t), & 2 T / 3 \leq t \leq T\end{cases} $$

where $\alpha(\neq 0)$ is a constant. What is the displacement of the particle from time $t=0$ to $T$ ?

A

$\frac{2 \alpha T^2}{9}$

B

$\frac{4 \alpha T^2}{9}$

C

$\frac{8 \alpha T^2}{9}$

D

$\frac{7 \alpha T^2}{9}$

3

PYP

IAT IISER 2023

A ball is thrown vertically upwards with an initial speed $u$ from a height $h$ above the ground. The ball eventually hits the ground with a speed $v$. The acceleration due to gravity is $g$ and air resistance is negligible. What is the average speed of the ball over its entire trajectory?

A

$\frac{g h}{2(u+v)}$

B

$\frac{u+v}{2}$

C

$\frac{u^2+v^2}{2(u+v)}$

D

$\frac{u^2+g h}{2(u+v)}$

4

PYP

IAT IISER 2023

A quantity has been measured to have a value of 0.00230 in some units. How many significant figures does the measured value have?

A

2

B

3

C

4

D

5

5

PYP

IAT IISER 2023

Two spherical bodies A and B each of mass $M$ and radius $R$ are located such that their centers are apart by a distance of $4 R$ as shown in the figure. An object $C$ of mass $m$ is thrown from the surface of A directly towards the center of B with a speed $v_0=2 v_{\text {min }}$, where $v_m$ in is the minimum speed needed for $C$ to reach the surface of $B$. Given that $G$ is the universal gravitational constant, how does the speed $v(x)$ of C change as a function of its distance from the center of A?

A

$ v(x)=\frac{2 \sqrt{2 G M R}}{x^{\frac{1}{2}}(4 R-x)^{\frac{1}{2}}}$

B

$v(x)=\frac{2 R \sqrt{2 G M R}}{x(4 R-x)}$

C

$v(x)=\frac{\sqrt{2 G M R}}{x^{\frac{1}{2}}(4 R-x)^{\frac{1}{2}}}$

D

$v(x)=\frac{6 R^2 \sqrt{2 G M R}}{x^{\frac{3}{2}}(4 R-x)^{\frac{3}{2}}}$