The position vector - r of a particle of mass m is given by the following equation- r t - t 3 - - t 2 - - where - -10-3 ms -2 - - -5 ms -2 and m-0-1 kg- At t-1 s- which of the following statements is-are true about the particle-

The position vector is nbsp; nbsp; nbsp;r⃗ (t) nbsp;= α t^3 î + β t^2 ĵ ∴ Velocity of particle nbsp; nbsp; nbsp;v⃗(t) nbsp;= d r⃗ (t)dt ...

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Byju's Answer

Standard XII

Physics

Characteristics of Bohr's Atomic Model

The position ...

Question

The position vector $\to r$ of a particle of mass $m$ is given by the following equation
$\to r (t) = {α t}^{3}^i + β t^{2}^j$ , where $α = 10 / 3 {m s}^{- 2}, β = 5 {m s}^{- 2}$ and $m = 0.1 k g$ . At $t = 1$ s, which of the following statements is/are true about the particle?

The velocity

\to v

is given by

\to v = (10^i + 10^j) {m s}^{- 1}

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The angular momentum

\to L

with respect to the origin is given by

\to L = (- 5 / 3)^k N m s

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The force

\to F

is given by

\to F = (^i + 2^j) N

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The torque

\to τ

with respect to the origin is given by

\to τ = - (20 / 3)^k N m

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Solution

The correct option is D The torque $\to τ$ with respect to the origin is given by $\to τ = - (20 / 3)^k N m$
The position vector is $\to r (t) = α t^{3}^i + β t^{2}^j$
$∴$ Velocity of particle $\to v (t) = \frac{d \to r (t)}{d t} = 3 α t^{2}^i + 2 β t^j$
$⟹$ $\to v (t = 1) = 3 \times \frac{10}{3} \times 1^{2}^i + 2 (5) (1)^j = 10^i + 10^j$
Thus option A is correct.

Position vector at $t = 1$ s $\to r (t = 1) = \frac{10}{3} \times 1^{3}^i + 5 (1^{2})^j = \frac{10}{3}^i + 5^j$
Angular momentum $\to L (t = 1) = m [\to r (t = 1) \times \to v (t = 1)]$
$∴$ $\to L (t = 1) = (0.1) [(\frac{10}{3}^i + 5^j) \times (10^i + 10^j)] = (0.1) [\frac{100}{3}^k - 50^k]$
$⟹ \to L (t = 1) = - \frac{5}{3}^k$ N ms
Thus option B is correct.

The acceleration of the particle $\to a (t) = \frac{d \to v (t)}{d t} = 6 α t^i + 2 β^j$
$∴$ $\to a (t = 1) = 6 \times \frac{10}{3} \times 1^i + 2 (5)^j = 20^i + 10^j$
Force on the particle $\to F (t = 1) = m \to a (t = 1) = 0.1 (20^i + 10^j) = 2^i +^j$ N
Thus option C is incorrect.

Torque w.r.t origin $\to τ (t = 1) = \to r \times \to F$
$∴$ $\to τ (t = 1) = [(10 / 3)^i + 5^j] \times [2^i +^j]$
$⟹$ $\to τ (t = 1) = \frac{10}{3}^k - 10^k = \frac{- 20}{3}^k N m$
Thus option D is correct.

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Similar questions

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The position vector $\to r$ of a particle of mass $m$ is given by the following equation
$\to r (t) = {α t}^{3}^i + β t^{2}^j$ , where $α = 10 / 3 {m s}^{- 2}, β = 5 {m s}^{- 2}$ and $m = 0.1 k g$ . At $t = 1$ s, which of the following statements is/are true about the particle?