Let and represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector . The direction of force is [2025]

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Q.
Let $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector $\vec{r} = a (\hat{i} \cos ω t + \hat{j} \sin ω t)$ . The direction of force is [2025]

1 Opposite to the direction of $\vec{r}$

2 Opposite to the direction of $\vec{L}$

3 Opposite to the direction of $\vec{P}$

4 Opposite to the direction of $\vec{L} \times \vec{P}$

Ans.
(1)

$\vec{r} = a \cos ω t \hat{i} + a \sin ω t \hat{j}$

$\vec{v} = \frac{d \vec{r}}{d t} = - a ω \sin ω t \hat{i} + a ω \cos ω t \hat{j}$

$\vec{a} = \frac{d \vec{v}}{d t} = - a ω^{2} \cos ω t \hat{i} - a ω^{2} \sin ω t \hat{j}$

$\vec{a} = - ω^{2} (a \cos ω t \hat{i} + a \sin ω t \hat{j}) = - ω^{2} \vec{r}$

$\vec{a} is antiparallel to \vec{r}$

Hence, $\vec{F}$ is opposite to the direction of $\vec{r}$

Solution

Q.
Let $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector $\vec{r} = a (\hat{i} \cos ω t + \hat{j} \sin ω t)$ . The direction of force is [2025]

1 Opposite to the direction of $\vec{r}$

2 Opposite to the direction of $\vec{L}$

3 Opposite to the direction of $\vec{P}$

4 Opposite to the direction of $\vec{L} \times \vec{P}$

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Solution

Q. Let L→ and P→ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector r→=a(i^ cos ωt+j^ sin ωt). The direction of force is [2025]

1 Opposite to the direction of r→

2 Opposite to the direction of L→

3 Opposite to the direction of P→

4 Opposite to the direction of L→×P→

Ans. (1) r→=a cos ωti^+a sin ωtj^ v→=dr→dt=–aω sin ωti^+aω cos ωtj^ a→=dv→dt=–aω2cos ωti^–aω2sin ωtj^ a→=–ω2(a cos ωti^+a sin ωtj^)=–ω2r→ a→ is antiparallel to r→ Hence, F→ is opposite to the direction of r→

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Q.
Let $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm' having position vector $\vec{r} = a (\hat{i} \cos ω t + \hat{j} \sin ω t)$ . The direction of force is [2025]

1 Opposite to the direction of $\vec{r}$

2 Opposite to the direction of $\vec{L}$

3 Opposite to the direction of $\vec{P}$

4 Opposite to the direction of $\vec{L} \times \vec{P}$