Chapter 8.5, Problem 19AYU

(a)

To determine

The motion of the object, where the displacement $d$ in meters at time $t$ seconds is given by $d(t)=3+7\cos \left(3\pi t\right)$

Answer to Problem 19AYU

Solution:

The motion of the object modeled by the displacement function $d(t)=3+7\cos \left(3\pi t\right)$ is simple harmonic.

Explanation of Solution

Given information:

The displacement $d$ of the object in meters and time $t$ in seconds is given by the function $d(t)=3+7\cos \left(3\pi t\right)$.

Explanation:

In simple harmonic motion, an object that moves on a coordinate axis, so that the displacement $d$ from the rest position at time $t$ is given by either $d(t)=a\cos \left(\omega t\right)+b$ or $d(t)=a\sin \left(\omega t\right)+b$, where $a$ and $\omega >0$ are constants, moves with simple harmonic motion has amplitude $\left|a\right|$ and period $T=\frac{2\pi }{\omega }$ and $b$ is a vertical shift.

By comparing $d(t)=3+7\cos \left(3\pi t\right)$ with an equation of the simple harmonic motion, the motion of an object is simple harmonic.

Therefore, motion of the object is simple harmonic.

(b)

To determine

To calculate: The maximum displacement of the object from its rest position, where the displacement $d$ in meters at time $t$ seconds is given by $d(t)=3+7\cos \left(3\pi t\right)$

Answer to Problem 19AYU

Solution:

The maximum displacement of the object from its rest position is $7$ meters.

Explanation of Solution

Given information:

The displacement $d$ of the object in meters and time $t$ in seconds is given by the function $d(t)=3+7\cos \left(3\pi t\right)$

Formula used:

In simple harmonic motion, the displacement $d(t)=a\cos \left(\omega t\right)+b$ with amplitude $\left|a\right|$ and period $T=\frac{2\pi }{\omega }$ and $b$ is a vertical shift.

Calculation:

The maximum displacement of the object from its rest position is the amplitude.

Compare $d(t)=3+7\cos \left(3\pi t\right)$ with $d(t)=a\cos \left(\omega t\right)+b$ gives $a=7\Rightarrow \left|a\right|=7$

The maximum displacement of the object from its rest position is $7$ meters since $a$ represents amplitude.

Therefore, the maximum displacement of the object from its rest position is $7$ meters.

(c)

To determine

To calculate: The time required for one oscillation, where the displacement $d$ in meters at time $t$ seconds is given by $d(t)=3+7\cos \left(3\pi t\right)$.

Answer to Problem 19AYU

Solution:

The time required for one oscillation is $\frac{2}{3}$ seconds.

Explanation of Solution

Given information:

The displacement $d$ of the object in meters and time $t$ in seconds is given by the function $d(t)=3+7\cos \left(3\pi t\right)$.

Formula used:

In simple harmonic motion, the displacement $d(t)=a\cos \left(\omega t\right)+b$ with amplitude $\left|a\right|$ and period $T=\frac{2\pi }{\omega }$ and $b$ is the vertical shift.

Calculation:

The time required for one oscillation is period.

Compare $d(t)=3+7\cos \left(3\pi t\right)$ with $d(t)=a\cos \left(\omega t\right)+b$ gives $\omega =3\pi$

Plug $\omega =3\pi$ in the formula for period $T=\frac{2\pi }{\omega }$

$\Rightarrow T=\frac{2\pi }{3\pi }=\frac{2}{3}$

Thus, period is $\frac{2}{3}$ seconds.

Therefore, the time required for one oscillation is $\frac{2}{3}$ seconds.

(d)

To determine

To calculate: Frequency of an object, where the displacement $d$ in meters at time $t$ seconds is given by $d(t)=3+7\cos \left(3\pi t\right)$.

Answer to Problem 19AYU

Solution:

The frequency of an object is $1.5$ oscillation per second.

Explanation of Solution

Given information:

The displacement $d$ of the object in meters and time $t$ in seconds is given by the function $d(t)=3+7\cos \left(3\pi t\right)$.

Formula used:

In simple harmonic motion, the displacement $d(t)=a\cos \left(\omega t\right)+b$ with amplitude $\left|a\right|$ and period $T=\frac{2\pi }{\omega }$ and the frequency is reciprocal of period that is $f=\frac{1}{T}$

Calculation:

From part (c), $T=\frac{2}{3}$ seconds

The frequency is defined as $f=\frac{1}{T}$ so $f=\frac{3}{2}=1.5$

Thus, there is $1.5$ oscillation per second.

Therefore, the frequency of an object is $1.5$ oscillation per second.