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In the I ist-I below, four different paths of a particle are g

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In the I ist-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 List-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 List-I with those quantities in List-II, which are conserved for that pathi

Uploaded: 2023-05-19T12:31:58.000Z
Duration: 1 min

Views: 5,995 students

Updated on: May 19, 2023

Book:

Mechanics Volume 2 (DC Pandey)

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Theodore Discussed

In the I ist-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 List-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 List-I with those quantities in List-II, which are conserved for that pathi

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Text solutionVerified

(a) When force

F = 0 \Rightarrow

potential energy

U =

constant

F \neq = 0 \Rightarrow

force is conservative

\Rightarrow

Total energy

E =

constant
List-I
(P)

r (t) = αα \hat{i} + β \hat{j}

\frac{d r}{d t} = v = α \hat{i} + β \hat{j} = constant \Rightarrow p = constant

∣ v ∣ = α^{2} + β^{2} = constant

\Rightarrow K = constant

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

E = U + K = constant

L = m (r \times v) = 0

L = constant

P \to 1, 2, 3, 4, 5

(Q)

\frac{d r}{d t} \Rightarrow p \neq = constant ∣ v ∣ \Rightarrow K a = \Rightarrow But K \neq = constant \Rightarrow U \neq = constant r (t) = α cos ω t \hat{i} + β sin ω t \hat{j} = v = α ω sin ω t (- \hat{i}) + β ω cos ω t \hat{j} \neq = constant = ω (α sin ω t)^{2} + (β cos ω t)^{2} \neq = constant \neq = constant \frac{d v}{d t} = - ω^{2} r \neq = 0 E = constant = K + U L = m (r \times v) = mω α β (\hat{k}) = constant

Q \to 2, 5

(R)

r (t) = α (cos ω t \hat{i} + sin ω \hat{j})

\frac{d r}{d t} = v = α ω [sin ω t (- \hat{i}) + cos ω t \hat{j}]] \neq =

constant

\Rightarrow p \neq =

constant

∣ v ∣ = α ω =

constant

\Rightarrow K =

constant

a = \frac{d v}{d t} = - ω^{2} r \neq = 0 \Rightarrow E =

constant,

U =

constant

L = m (r \times v) = mω α^{2} \hat{k} =

constant

R \to 2, 3, 4, 5

(S)

r (t) = α t \hat{i} + \frac{β}{2} t^{2} \hat{j}

\frac{d r}{d t} = v = α \hat{i} + β \hat{j} \neq =

constant

\Rightarrow p \neq =

constant
(S)

r (t) = α t \hat{i} + \frac{β}{2} t^{2} \hat{j}

\frac{d r}{d t} = v = α \hat{i} + β \hat{j} \neq = constant \Rightarrow p \neq = constant

∣ v ∣ = α^{2} + (βt)^{2} \neq = constant \Rightarrow K \neq = constant

a = \frac{d v}{d t} = β \hat{j} \neq = 0 \Rightarrow E = constant = K + U

But K \neq = constant

∴ U \neq = constant

L = m (r \times v) = \frac{1}{2} α β t^{2} \hat{k} \neq = constant

S \to 5

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Question Text	In the I ist-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 List-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 List-I with those quantities in List-II, which are conserved for that pathi
Updated On	May 19, 2023
Topic	Work, Energy and Power
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 2
Upvotes	302
Avg. Video Duration	11 min

Text solutionVerified

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