For the damped oscillator system shown in above figure, the block has a mass of $1.50 k g$ and the spring constant is $8.00 N / m$ . The damping force is given by $- b (d x / d t)$ , where $b = 230 g / s$ . The block is pulled down $12.0 c m$ and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?

Open in App

Solution

(a) We want to solve $e^{- b t / 2 m} = 1 / 3$ for $t$ . We take the natural logarithm of both sides to obtain $- b t / 2 m = l n (1 / 3)$ . Therefore, $t = - (2 m / b) l n (1 / 3) = (2 m / b) l n 3$ . Thus,

$t = \frac{2 (1.50 k g)}{0.230 k g / s} ln 3 = 14.3 s$ .

(b) The angular frequency is

$ω^{'} = \sqrt{\frac{k}{m} - \frac{b^{2}}{4 m^{2}}} = \sqrt{\frac{8.0 N / m}{1.50 k g} - \frac{{(0.230 k g / s)}^{2}}{4 {(1.50 k g)}^{2}}} = 2.31 r a d / s$ .

The period is $T = 2 π / ω ´ = (2 π) / (2.31 r a d / s) = 2.72 s$ and the number of oscillations is

$t / T = (14.3 s) / (2.72 s) = 5.27$ .

The displacement $x (t)$ as a function of time is shown below. The amplitude, $x_{m} e^{- b t / 2 m}$ , decreases exponentially with time.

Suggest Corrections

Similar questions

Q. For the damped oscillator shown in Fig, the mass of the block is

200 g, k = 80 N m^{- 1}

and the damping constant

b

40 g s^{- 1}

Calculate.
(a) The period of oscillation,
(b) Time period for its amplitude of vibrations to drop to half of its initial value
(c) The time for the mechanical energy to drop to half initial value.

Q. Solve this:
Example 14.10 For the damped oscillator the mass m of the block is 200 g, k = 90 N

m^{- 1}

and the damping constant b is 40 g

s^{- 1}

. Calculate (a) the period of oscillation, (b) time taken for its amplitude of vibrations to drop to half of its initial value and (c) the time taken for its mechanical energy to drop to half its initial value.

For the damped oscillator shown, the mass m of the block is 200 g, k = 90 N m $^{- 1}$ and the damping constant b is 40 g s $^{- 1}$ . What is the period of oscillation, time taken for its amplitude of vibrations to drop to half of its initial value.

Q. The block shown in the figure, is acted on by a spring with spring constant

k

and a weak frictional force of constant magnitude

f

. The block is pulled a distance

x_{0}

from equilibrium position and then released. It oscillates many times and ultimately comes to rest. Show that the decrease of amplitude is the same for each cycle of oscillation.

Q. for a damped oscillator ,the mass m of the block is 400g, k=45N/m and the damping cons†an t b is 80g/s.calculate time taken for its amplitude if vibrations to drop to half if its initial value

Join BYJU'S Learning Program