Question 82: Laws of Motion

Question

A particle moves in $x$-$y$ plane under the influence of a force $\vec{F}$ such that its linear momentum is $\overrightarrow{\mathrm{p}}(\mathrm{t})=\hat{i} \cos (\mathrm{kt})-\hat{j} \sin (\mathrm{kt})$. If $\mathrm{k}$ is constant, the angle between $\overrightarrow{\mathrm{F}}$ and $\overrightarrow{\mathrm{p}}$ will be :

Accepted Answer

To find the angle between $\vec{F}$ and $\overrightarrow{\mathrm{p}}$, we first need to understand the relationship between force and momentum. The force $\vec{F}$ acting on a particle is related to the rate of change of its linear momentum $\overrightarrow{\mathrm{p}}$ with respect to time, as described by Newton's second law of motion:
$\vec{F} = \frac{d\overrightarrow{\mathrm{p}}}{dt}$
Given the expression for the momentum $\overrightarrow{\mathrm{p}}(t) = \hat{i} \cos (kt) - \hat{j} \sin (kt)$, we can find $\vec{F}$ by differentiating $\overrightarrow{\mathrm{p}}$ with respect to $t$:
$\frac{d\overrightarrow{\mathrm{p}}}{dt} = -k\hat{i} \sin (kt) - k\hat{j} \cos (kt)$
So, $\vec{F} = -k\hat{i} \sin (kt) - k\hat{j} \cos (kt)$.
Now, to find the angle between $\vec{F}$ and $\overrightarrow{\mathrm{p}}$, we use the dot product formula:
$\vec{F} \cdot \overrightarrow{\mathrm{p}} = |\vec{F}| |\overrightarrow{\mathrm{p}}| \cos(	heta)$,
where $	heta$ is the angle between $\vec{F}$ and $\overrightarrow{\mathrm{p}}$. However, in this case, it's more insightful to see if $\vec{F}$ and $\overrightarrow{\mathrm{p}}$ are orthogonal (at a $\frac{\pi}{2}$ angle to each other), because the dot product of two perpendicular vectors is zero.
The dot product of $\vec{F}$ and $\overrightarrow{\mathrm{p}}$ is:
$(-k\hat{i} \sin (kt) - k\hat{j} \cos (kt)) \cdot (\hat{i} \cos (kt) - \hat{j} \sin (kt)) =$
$- k \sin (kt) \cos (kt) + k \cos (kt) \sin (kt) = 0$
The result is zero, indicating that the angle between $\vec{F}$ and $\overrightarrow{\mathrm{p}}$ is indeed $\frac{\pi}{2}$.
Therefore, the correct option is:
Option A $\frac{\pi}{2}$.

Answer

$\frac{\pi}{2}$

Answer

$\frac{\pi}{3}$

Answer

$\frac{\pi}{4}$

Answer

$\frac{\pi}{6}$

Laws of Motion

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