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 - 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- p is the linear momentum 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 Column I with those quantitites in Column II- which are conserved for that path- Column I Column II- A rt - - t - - - t- 1- p- B rt - - cos - t - - - sin - t - 2- L- C rt - -cos - - - sin - t- 3- K- D rt - - t- - -2t2- 4- U- 5- E- -

A → 1,2,3,4,5; B → 2,5; C → 2,3,4,5; D → 5 When force is conservative and nbsp; nbsp;F =0 ⇒ potential energy U = constant When force is conservative F ≠ 0 ⇒ ...

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Byju's Answer

Standard XII

Physics

Conservative and Non Conservative Forces

In the Column...

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: $\to p$ is the linear momentum $\to L$ 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 Column I with those quantitites in Column II, which are conserved for that path.
$\begin{matrix} Column~I & Column~II (A) \to r (t) = α t^i + β t^j & 1. \to p (B) \to r (t) = α cos ω t^i + β sin ω t^j & 2. \to L (C) \to r (t) = α (cos ω^i + sin ω t^j) & 3. K (D) \to r (t) = α t^i + \frac{β}{2} t^{2}^j & 4. U 5. E \end{matrix}$

A \to 1, 2, 3, 4, 5; B \to 2, 5; C \to 2, 3, 4, 5; D \to 5

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A \to 1, 2, 3, 4, 5; B \to 3, 5; C \to 1, 2, 3, 4; D \to 4

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A \to 1, 2, 3, 4, 5; B \to 1, 5; C \to 2, 3, 4, 5; D \to 1

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A \to 1, 2, 3, 4, 5; B \to 4; C \to 1, 3, 4, 5; D \to 3

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Solution

The correct option is A $A \to 1, 2, 3, 4, 5; B \to 2, 5; C \to 2, 3, 4, 5; D \to 5$
$A \to 1, 2, 3, 4, 5; B \to 2, 5; C \to 2, 3, 4, 5; D \to 5$
When force is conservative and $F = 0$ $\Rightarrow$ potential energy $U =$ constant
When force is conservative $F \neq 0 \Rightarrow$ $\Rightarrow$ Total energy $E =$ constant
$(A) \to r (t) = α t^i + β t^j$
$\frac{d \to r}{d t} = \to v = α^i + β^j = constant \Rightarrow \to p = constant$

$| \to v | = \sqrt{α^{2} + β^{2}} = constant \Rightarrow K = constant$

$\frac{d \to v}{d t} = \to a = 0 \Rightarrow F = 0 \Rightarrow U = constant$

$E = U + K = constant$
$\to L = m (\to r \times \to v) = 0$
$\to L = constant$
$A \to 1, 2, 3, 4, 5$

$(B) \to r (t) = α cos ω t^i + β sin ω t^j$
$\frac{d \to r}{d t} = \to v = α ω s i n ω t (-^i) + β ω c o s ω^j \neq constant \Rightarrow \to p \neq constant$

$| \to v | = ω \sqrt{(α s i n ω t)^{2} + (β cos ω t)^{2}} \neq constant \Rightarrow K \neq constant$

$\to a = \frac{d \to v}{d t} = - ω^{2} \to r \neq 0$
$E = constant = K + U$
But $K$ $\neq$ constant
$\Rightarrow U \neq constant$
$\to L = m (\to r \times \to v) = m ω α β (\to k) = constant$
$B \to 2, 5$

$(C) \to r (t) = α (c o s ω t^i + s i n ω t^j$
$\frac{d \to r}{d t} = \to v = α ω [s i n ω t (-^i) + c o s ω t^j \neq constant \Rightarrow \to p \neq constant$

$| \to v | = α ω = constant \Rightarrow K = constant$
$\to a = \frac{d \to v}{d t} = - ω^{2} \to r \neq 0 \Rightarrow E = constant, U = constant$

$\to L = m (\to r \times \to v) = m ω α^{2}^k = constant)$
$C \to 2, 3, 4, 5$

$(D) \to r (t) = α t^i + \frac{β}{2} t^{2}^j$
$\frac{d \to r}{d t} = \to v = α^i + β t^j \neq constant \Rightarrow \to p \neq constant$
$\to a = \frac{d \to v}{d t} = β^j \neq 0 \Rightarrow E = constant = K + U$
But $K \neq constant$
So, U $\neq constant$
$\to L = m (\to r \times \to v) = \frac{1}{2} α β t^{2}^k \neq constant$
$D \to 5$

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Similar questions

Q. 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:

\to p

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is the angular momentum about the origin,

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is the kinetic energy,

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Match each path in Column I with those quantitites in Column II, which are conserved for that path.

\begin{matrix} Column~I & Column~II (A) \to r (t) = α t^i + β t^j & 1. \to p (B) \to r (t) = α cos ω t^i + β sin ω t^j & 2. \to L (C) \to r (t) = α (cos ω^i + sin ω t^j) & 3. K (D) \to r (t) = α t^i + \frac{β}{2} t^{2}^j & 4. U 5. E \end{matrix}

Q. In the Column I below, four different paths of a particle are given as functions of time.
In these functions,

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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.
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Match each path in Column I with those quantitites in Column II, which are conserved for that path.

\begin{matrix} Column~I & Column~II (A) \to r (t) = α t^i + β t^j & 1. \to p (B) \to r (t) = α cos ω t^i + β sin ω t^j & 2. \to L (C) \to r (t) = α (cos ω^i + sin ω t^j) & 3. K (D) \to r (t) = α t^i + \frac{β}{2} t^{2}^j & 4. U 5. E \end{matrix}

Q. In the List-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 B

. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned;

\to p

is the linear momentum

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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 the path.

List - I	List - II
P. $\to r (t) = α t^i + β t^j$	1. $\to p$
Q. $\to r (t) = α c o s (ω t)^i + β s i n (ω t)^j$	2. $\to L$
R. $\to r (t) = α (c o s (ω t)^i + s i n (ω t)^j)$	3. K
S. $\to r (t) = α t^i + \frac{β}{2} t^{2}^j$	4. U
	5. E

Q. In the List-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 B

. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned;

\to p

is the linear momentum

\to L

List - I	List - II
P. $\to r (t) = α t^i + β t^j$	1. $\to p$
Q. $\to r (t) = α c o s (ω t)^i + β s i n (ω t)^j$	2. $\to L$
R. $\to r (t) = α (c o s (ω t)^i + s i n (ω t)^j)$	3. K
S. $\to r (t) = α t^i + \frac{β}{2} t^{2}^j$	4. U 5. E

Q. In the List-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 B

. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned;

\to p

is the linear momentum

\to L

List - I	List - II
P. $\to r (t) = α t^i + β t^j$	1. $\to p$
Q. $\to r (t) = α c o s (ω t)^i + β s i n (ω t)^j$	2. $\to L$
R. $\to r (t) = α (c o s (ω t)^i + s i n (ω t)^j)$	3. K
S. $\to r (t) = α t^i + \frac{β}{2} t^{2}^j$	4. U
	5. E

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