The position vector of a particle of mass m-6kg is given as r-3t2-6t-4t3- m- Findi the force - F-ma- - acting on the particle-ii the torque - -r-F- - with respect to the origin- acting on the particle-iii the momentum - p-mv- - of the particle-iv the angular momentum - L-r-p- - of the particle with respect to the origin-

The correct option is A (i) (36î 144tĵ)N (ii) ( 288 t^3+864t^2)k̂ (iii) (36t 36)î 72 t^2ĵ (iv) ( 72 t^4+288 t^3)k̂ v⃗=dr⃗/dt=(6t 6)î+( ...

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Byju's Answer

Standard XII

Physics

Multiplication with Vectors

The position ...

Question

The position vector of a particle of mass $m = 6 k g$ is given as $\to r = [(3 t^{2} - 6 t)^i + (- 4 t^{3})^j] m$ . Find
(i) the force $(\begin{matrix} \to F = m \to a \end{matrix})$ acting on the particle.
(ii) the torque $(\begin{matrix} \to τ = \to r \times \to F \end{matrix})$ with respect to the origin, acting on the particle.
(iii) the momentum $(\begin{matrix} \to p = m \to v \end{matrix})$ of the particle.
(iv) the angular momentum $(\begin{matrix} \to L = \to r \times \to p \end{matrix})$ of the particle with respect to the origin.

(i)

(36^i - 144 t^j) N

(ii)

(- 288 t^{3} + 864 t^{2})^k

(iii)

(36 t - 36)^i - 72 t^{2}^j

(iv)

(- 72 t^{4} + 288 t^{3})^k

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(i)

(63^i - 144 t^j) N

(ii)

(- 288 t^{3} + 864 t^{2})^k

(iii)

(36 t - 36)^i - 72 t^{2}^j

(iv)

(- 72 t^{4} + 288 t^{3})^k

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(i)

(36^i - 144 t^j) N

(ii)

(- 28 t^{3} + 84 t^{2})^k

(iii)

(36 t - 36)^i - 72 t^{2}^j

(iv)

(- 72 t^{4} + 288 t^{3})^k

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(i)

(36^i - 144 t^j) N

(ii)

(- 288 t^{3} + 864 t^{2})^k

(iii)

(3 t - 6)^i - 72 t^{2}^j

(iv)

(- 72 t^{4} + 288 t^{3})^k

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Solution

The correct option is A (i) $(36^i - 144 t^j) N$ (ii) $(- 288 t^{3} + 864 t^{2})^k$ (iii) $(36 t - 36)^i - 72 t^{2}^j$ (iv) $(- 72 t^{4} + 288 t^{3})^k$
$\to v = \frac{d \to r}{d t} = (6 t - 6)^i + (- 12 t^{2})^j m / s$
$\to a = \frac{d \to v}{d t} = (6^i - 24 t^j) m / s^{2}$
(i) $\to F = m \to a = 6 (6^i - 24 t^j) = (36^i - 144 t^j) N$
(ii) $\to τ = \to r \times \to F = [\begin{matrix} (3 t^{2} - 6 t)^i + (- 4 t^{3})^j \end{matrix}] \times [\begin{matrix} 36^i - 144 t^j \end{matrix}]$
$= [(- 144 \times 3 t^{2}) + (144 \times 6 T^{2}) + 144 t^{3}]^k$
$= (- 288 t^{3} + 864 t^{2})^k$
(iii) $\to p = m \to v = 6 [(6 t - 6)^i + (- 12 t^{2})^j]$
$= [36 (t - 1)^i - 72 t^{2} t^{2}^j]$
(iv) $\to L = \to r \times \to p = [(3 t^{2} - 6 t)^i + (- 4 t^{3})^j] \times [36 (t - 1)^i - 72 t^{2}^j]$
$= [- 72 t^{4} + 288 t^{3}]^k$

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