Chapter 5.6, Problem 1E

For an object in simple harmonic motion with amplitude a and period 2π/ω, find an equation that models the displacement y at time t if

1. (a) y = 0 at time t = 0: y = __________.
2. (b) y = a at time t = 0: y = __________.

To determine

To fill: The equation for the displacement of the object in simple harmonic motion.

Answer to Problem 1E

The equation for the displacement of the object in simple harmonic motion is $y=a\sin \omega t$.

Explanation of Solution

Given:

The displacement y is 0 at time $t=0$.

Definition used:

The equation for the simple harmonic motion which describes the displacement y of an object at time t is,

$y=a\sin \omega t$ or $y=a\cos \omega t$.

If both the equation are satisfied, then the object is said to be in simple harmonic motion.

Where,

Amplitude is equal to $\left|a\right|$.

Period is equal to $\frac{2\pi }{\omega }$.

Frequency is equal to $\frac{\omega }{2\pi }$.

Calculation:

From the definition of the simple harmonic motion, displacement can be written as,

$y=a\sin \omega t$ (1)

$y=a\cos \omega t$ (2)

The displacement y is 0 at $t=0$.

Substitute the value 0 for t, in equation (1),

$\begin{array}{c}y=a\sin \left(\omega \cdot 0\right)\\ =a\sin 0\left[\therefore \sin 0=0\right]\\ =0\end{array}$

Thus, the displacement is 0 at time $t=0$ for the equation (1).

Now, substitute the value 0 for t in equation (2),

$\begin{array}{c}y=a\cos \left(\omega \cdot 0\right)\\ =a\cos 0\left[\therefore \cos 0=1\right]\\ =a\end{array}$

Thus, the value of displacement is a for the time $t=0$ for the sine curve.

Hence the equation for the displacement of the models of simple harmonic motion at time $t=0$ is $y=a\sin \omega t$.

To determine

To fill: The equation for the displacement of the object in simple harmonic motion.

Answer to Problem 1E

The equation for the displacement of the object in simple harmonic motion is $y=a\cos \omega t$.

Explanation of Solution

Given:

The displacement y is a at time $t=0$.

Definition used:

The equation for the simple harmonic motion which describes the displacement y of an object at time t is,

$y=a\sin \omega t$ or $y=a\cos \omega t$.

If both the equation are satisfied, then the object is said to be in simple harmonic motion.

Where,

Amplitude is equal to $\left|a\right|$.

Period is equal to $\frac{2\pi }{\omega }$.

Frequency is equal to $\frac{\omega }{2\pi }$.

Calculation:

From the definition of the simple harmonic motion, displacement can be written as,

$y=a\sin \omega t$ (1)

$y=a\cos \omega t$ (2)

The displacement y is a at $t=0$.

Substitute the value 0 for t, in equation (1),

$\begin{array}{c}y=a\sin \left(\omega \cdot 0\right)\\ =a\sin 0\left[\therefore \sin 0=0\right]\\ =0\end{array}$

Thus, the displacement is 0 at time $t=0$ for the equation (1).

Now, substitute the value 0 for t in equation (2),

$\begin{array}{c}y=a\cos \left(\omega \cdot 0\right)\\ =a\cos 0\left[\therefore \cos 0=1\right]\\ =a\end{array}$

Thus, the value of displacement is a for the time $t=0$ for the cosine curve.

Hence the equation for the displacement a of the object in simple harmonic motion at time $t=0$ is $y=a\cos \omega t$.