Chapter 6: Q4P (page 150)

Question: The coordinates of an object moving in theplane vary with time according to the equations $x = - 5.00 \sin ω t$ and $y = 4.00 - 5.00 \cos ω t$ , where $ω$ is a constant, x and y are in meters, and t is in seconds.
(a) Determine the components of velocity of the object at t = 0 .
(b) Determine the components of acceleration of the object at t = 0.
(c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t > 0
(d) Describe the path of the object in an xyplot.

Short Answer

Expert verified

(a)The x and y components of velocity are $v_{x} = - 5 ω$ and $v_{y} = 0$ .

(b)The components of the acceleration are $a_{x} = 0 m / s^{2}$ and $(a_{y}) = 5 ω^{2} {m/s}^{2}$ .

(c)The expressions for the vectors is-

$\vec{r} = (- 5.00 m) \sin (ωt) \hat{i} + [(4.00 m) - (5.00 m) \cos (ωt)] \hat{j}$

localid="1663668893261" $\vec{v} = (5.00 m) ω (- \cos ω t \hat{i} + \sin ω t \hat{j})$

$\vec{a} = (5.00 m) ω^{2} (\sin ω t \hat{i} + \cos ω t \hat{j})$

(d)The path of the object in the XY plot is a circle of radius $r = 5.00 m$

Step by step solution

Definition of velocity and equations that used for solution.

Velocity is the rate of change of displacement, whereas acceleration is the rate of change of velocity, with respect to time.

Position vector of the given object is

$\vec{r} (t) = x \hat{i} + y \hat{j}$

Velocity vector of the given object is

$\vec{v} (t) = \frac{d}{dt} [\vec{r} (t)]$

Acceleration of the given object is

$\vec{a} (t) = \frac{d}{dt} [\vec{v} (t)]$

Determine the components of the velocity.

(a)

Velocity of the given object along x axis is calculated as

$v_{x} = \frac{d x}{d t} = \frac{d}{dt} (- 5 \sin ω t) = - 5 ω \cos ω t m / s$
Therefore, the velocity of the given object at the time $t = 0 sis$ is

${(v_{x})}_{t = 0} = - 5 ω \cos 0 ° = - 5 ω m / s$

Velocity of the given object along y axis is calculated as

$v_{y} = \frac{d y}{d t} = \frac{d}{d t} (4.00 - 5.0 \cos ω t) = 0 - \frac{d}{d t} (5.0 \cos ω t) = 5.0 ω \sin ω t m / s$

At the time , the velocity of the given object is
${(V_{y})}_{t = o} = 5.0 ω \sin 0 ° = 0 m / s$

Thus, the components of velocity of $v_{x} = - 5 ω$ and $v_{y} = 0$

Determine the components of the acceleration.

(b)

Acceleration of the given object along x axis is

At the time component of the acceleration of the given object is
${(a_{x})}_{t - 0} = 5 ω^{2} \sin 0 ° = 0 m / s^{2}$

Acceleration of the given object along y axis S

$a_{y} = \frac{d v_{y}}{d t} = \frac{d}{d t} (5.0 ω \sin ω t) = 5.0 ω \frac{d}{d t} (\sin ω t) = 5.0 ω^{2} \cos ω t m / s^{2}$

At the time component of the acceleration of the given object is
${(a_{y})}_{t - 0} = 5 ω^{2} \cos 0 ° = 5 ω^{2} m / s^{2}$

Thus, the components of the acceleration is $a_{x} = 0 m / s^{2}$ and $(a_{y}) = 5 ω^{2} {m/s}^{2}$ .

Expressions for the velocity, position and the acceleration vectors.

(c)

Position vector of the object as a function of time is
$\vec{r} = (- 5.00 m) \sin (ω t) \hat{i} + [(4.00 m) - (5.00 m) \cos (ω t)] \hat{j}$

Velocity vector,

$\vec{v} = \frac{d r}{d t} = \frac{d}{d t} ((4.00 m) \hat{j} + (5.00 m) (- \sin ω t \hat{i} - \cos ω t \hat{j})) = (5.00 m) ω (- \cos ω t \hat{i} + \sin ω t \hat{j})$
Acceleration Vector of the object as a function of time is
$\vec{a} = \frac{d \vec{v}}{d t} \vec{a} = (5.00 m) ω^{2} (\sin ω t \hat{i} + \cos ω t \hat{j})$

Thus, the expressions for the vectors is,
$\vec{r} = (- 5.00 m) \sin (ω t) \hat{i} + [(4.00 m) - (5.00 m) \cos (ω t)] \hat{j}$
$\vec{v} = (5.00 m) ω (- \cos ω t \hat{i} + \sin ω t \hat{j})$
$\vec{a} = (5.00 m) ω^{2} (\sin ω t \hat{i} + \cos ω t \hat{j})$