Chapter 4: Q12Q (page 83)

In Fig. 4-29, particle P is in uniform circular motion, centered on the origin of an xy coordinate system. (a) At what values of $θ$ is the vertical component $r_{y}$ of the position vector greatest in magnitude? (b) At what values of $θ$ is the vertical component $v_{y}$ of the particle’s velocity greatest in magnitude? (c) At what values of $θ$ is the vertical component $a_{y}$
of the particle’s acceleration greatest in magnitude?

Short Answer

Expert verified

$θ = 90 ° or 270 °$

$θ = 0 ° or 180 °$
$θ = 90 ° or 270 °$

Step by step solution

Given information

Position vector $\vec{r}$ is directed at an angle $θ$ from center.

To understand the concept

This problem is based on concept on uniform circular motion. It is the motion of a body moving at a constant speed on a circular path.

Formulae:

Position vector can be written as:

$\vec{r} = r_{x} \hat{i} + r_{y} \hat{j}$

Where,

$r_{x} = r \cos (θ)$

$r_{y} = r \sin (θ)$

(a) To find the values of at which the vertical component of the position vector greatest in magnitude?

Vertical component of position vector as:

$r_{y} = r \sin (θ)$

And it would be greatest when $θ = 90 °$ or $270 °$ , because sine function has max value at $90 °$ and $70 °$

(b) To find the values of at which the vertical component of the particle’s velocity greatest in magnitude?

Vertical component of velocity:

$v_{y} = \frac{d (r_{y})}{d t} = r \cos (θ)$

And it would be greatest when $θ = 0 ° or 180 °$ because cos function has max value at $θ = 0 ° and 180 °$ .

(c) To find the values of at which the vertical component of the particle’s acceleration greatest in magnitude?

Vertical component of particle’s acceleration:

$a_{y} = \frac{d (v_{y})}{d t} = - r \sin (θ)$

Sine function have max value at $90 ° and 270 °$ . Hence, acceleration would be max at $θ = 90 ° or 270 °$ .

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Short Answer

Step by step solution

Given information

To understand the concept

(a) To find the values of at which the vertical component of the position vector greatest in magnitude?

(b) To find the values of at which the vertical component of the particle’s velocity greatest in magnitude?

(c) To find the values of at which the vertical component of the particle’s acceleration greatest in magnitude?

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