Question 59: 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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