Question

a body of mass 100gm is suspended from the end of a light eleastic spring and set up into oscillatory motion.the equation of its displacement is x=2cos[2pie+pie/4].
find the phase constant and the period of oscillatory motion.what is the force acting on the body?

Answer 1

Answer on Question #58438, Physics / Mechanics | Relativity

a body of mass $100\mathrm{gm}$ is suspended from the end of a light eleastic spring and set up into oscillatory motion. The equation of its displacement is $x = 2\cos [2pie + pie / 4]$. Find the phase constant and the period of oscillatory motion. What is the force acting on the body?

Find: $\varphi - ?T - ?f - ?$

Given:

$$m = 100 \times 10^{-3} \mathrm{kg}$$ $$x = 2 \cos \left(2 \pi t + \frac {\pi}{4}\right)$$

Solution:

The equation of harmonic oscillations:

$$x = x_{\max} \cos (\omega t + \varphi_0) \quad (1),$$

where $\varphi = \omega t + \varphi_0$ - phase constant

With the condition of the problem:

$$x = 2 \cos \left(2 \pi t + \frac {\pi}{4}\right) \quad (2)$$

Of (1) and (2) $\Rightarrow \varphi = 2\pi t + \frac{\pi}{4}$

The cyclic frequency:

$$\omega = \frac {2 \pi}{T} \quad (3),$$

where $T$ - period of oscillatory motion

$$\text{Of (3)} \Rightarrow T = \frac {2 \pi}{\omega} \quad (4)$$

Of (1) and (2) $\Rightarrow \omega = 2\pi$ \quad (5)

(5) in (4): T=1 s

The equation of force:

$$f = f_{\max} \cos (\omega t + \varphi_0) \quad (6),$$

where $f_{\max}$ - peak value of force

Newton's Second Law in scalar form:

$$f = m a \quad (7),$$

where $a$ - acceleration of body

$$\text{Of (7)} \Rightarrow f_{\max} = m a_{\max} \quad (8),$$

Acceleration of body:

$$a = x_{t}^{"} \quad (9),$$

where $x_{t}^{c}$ – the second derivative of the coordinates of time

Of (1) and (9) $\Rightarrow$

$$a = \left(- \omega x _ {\max } \sin \left(\omega t + \varphi_ {0}\right)\right) ^ {\prime} = - \omega^ {2} x _ {\max } \cos \left(\omega t + \varphi_ {0}\right) = - a _ {\max } \cos \left(\omega t + \varphi_ {0}\right) (1 0)$$

Of (10) $\Rightarrow$ $a_{\max} = \omega^2 x_{\max}$ (11)

Of (2) $\Rightarrow$ $x_{\max} = 2 \, \text{m}$ (12)

(5) and (12) in (11): $a_{\max} = 8\pi^2 \, \text{m} / \text{s}^2$ (13)

(13) in (8): $f_{\max} = 0,8\pi^2 \, \text{N}$ (14)

(14) in (6): $f = 0,8\pi^2 \cos \left(2\pi t + \frac{\pi}{4}\right)$

**Answer:**

$$\varphi = 2 \pi t + \frac {\pi}{4}$$

T=1 s

$$f _ {\max } = 0, 8 \pi^ {2} \, \text{N}$$ $$f = 0, 8 \pi^ {2} \cos \left(2 \pi t + \frac {\pi}{4}\right)$$

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