The time dependence of the position of a particle of mass 2 kg is given by r t -2 t-3 t 2- m- Its angular momentum with respect to origin at time t -2 s is given by - N k- kg - m 2 - s- the value of N is

We know that angular momentum, L=r×p .....(1) r(t)=2t î 3t^2 ĵ At t=2 s, nbsp;r=4î 12ĵ m nbsp; Velocity of particle, v=drdt=2î 6tĵ At t=2 s, nbsp;v=2î 12ĵ ...

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Standard XII

Physics

Angular Analogue of Linear Momentum

The time depe...

Question

The time dependence of the position of a particle of mass $2 k g$ is given by $\to r (t) = 2 t^i - 3 t^{2}^j m$ . Its angular momentum with respect to origin at time $t = 2 s$ is given by $- N^k {kg.m}^{2} /s$ , the value of $N$ is

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Solution

We know that angular momentum, $\to L = \to r \times \to p . . . . . (1)$
$\to r (t) = 2 t^i - 3 t^{2}^j$
At $t = 2 s$ , $\to r = 4^i - 12^j m$
Velocity of particle, $\to v = \frac{\to d r}{\to d t} = 2^i - 6 t^j$
At $t = 2 s$ , $\to v = 2^i - 12^j m/s$
Now linear momentum,
$\to p = m \to v = 4^i - 24^j$ .........(2) $[m = 2 k g]$
From Eq. $(1)$ and $(2)$ ,
$\to L = \to r \times \to p$
$\to L = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 4 & - 12 & 0 4 & - 24 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$\to L =^i (0 - 0) -^j (0 - 0) +^k (- 96 - (- 48))$
$\Rightarrow \to L = - 48^k {kg.m}^{2} /s$
Since, $\to L = - N^k {kg.m}^{2} /s$
Hence on comparing we get,
$\Rightarrow N = 48$

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Angular Analogue of Linear Momentum

Standard XII Physics

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The time dependence of the position of a particle of mass 2 kg is given by →r(t)=2t^i−3t2^j m. Its angular momentum with respect to origin at time t=2 s is given by −N^k kg.m2/s, the value of N is

The time dependence of the position of a particle of mass $2 k g$ is given by $\to r (t) = 2 t^i - 3 t^{2}^j m$ . Its angular momentum with respect to origin at time $t = 2 s$ is given by $- N^k {kg.m}^{2} /s$ , the value of $N$ is