In the Column-I below, four different paths of a particle are given as functions of time. In these functions, and are positive constants of appropriate dimensions and . [2018]

Question

In the Column-I below, four different paths of a particle are given as functions of time. In these functions, $α$ and $β$ are positive constants of appropriate dimensions and $α \neq β$ . In each case, the force acting on the particle is either zero or conservative. In Column-II, five physical quantities of the particle are mentioned $\vec{p}$ is the linear momentum, $\vec{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path. [2018]

	Column-I		Column-II
P.	$\vec{r} (t) = α t \hat{i} + β t \hat{j}$	1.	$\vec{p}$
Q.	$\vec{r} (t) = α \cos ω t \hat{i} + β \sin ω t \hat{j}$	2.	$\vec{L}$
R.	$\vec{r} (t) = α (\cos ω t \hat{i} + \sin ω t \hat{j})$	3.	$K$
S.	$\vec{r} (t) = α t \hat{i} + \frac{β}{2} t^{2} \hat{j}$	4.	$U$
		5.	$E$

	Column-I		Column-II
P.	$\vec{r} (t) = α t \hat{i} + β t \hat{j}$	1.	$\vec{p}$
Q.	$\vec{r} (t) = α \cos ω t \hat{i} + β \sin ω t \hat{j}$	2.	$\vec{L}$
R.	$\vec{r} (t) = α (\cos ω t \hat{i} + \sin ω t \hat{j})$	3.	$K$
S.	$\vec{r} (t) = α t \hat{i} + \frac{β}{2} t^{2} \hat{j}$	4.	$U$
		5.	$E$

Smart Achievers · Accepted Answer

Solution

1 P → 1, 2, 3, 4, 5; Q → 2, 5; R → 2, 3, 4, 5; S → 5

2 P → 1, 2, 3, 4, 5; Q → 3, 5; R → 2, 3, 4, 5; S → 2, 5

3 P → 2, 3, 4; Q → 5; R → 1, 2, 4; S → 2, 5

4 P → 1, 2, 3, 5; Q → 2, 5; R → 2, 3, 4, 5; S → 2, 5

Ans.
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