Question 161: Oscillations

Question

An object of mass $$0.2 \mathrm{~kg}$$ executes simple harmonic motion along $$x$$ axis with frequency of $$\left(\frac{25}{\pi}\right) \mathrm{Hz}$$. At the position $$x=0.04 \mathrm{~m}$$ the object has kinetic energy $$0.5 \mathrm{~J}$$ and potential energy $$0.4 \mathrm{~J}$$. The amplitude of oscillation is ________ $$\mathrm{cm}$$.

Accepted Answer

To solve for the amplitude of oscillation, we start by using the properties of simple harmonic motion (SHM). In SHM, the total energy of the system is conserved and is given by the sum of kinetic energy (KE) and potential energy (PE).
Given:
Mass $$m = 0.2 \ \mathrm{kg}$$
Frequency $$f = \left(\frac{25}{\pi}ight) \ \mathrm{Hz}$$
Position $$x = 0.04 \ \mathrm{m}$$
KE at $$x = 0.04 \ \mathrm{m}$$ is $$0.5 \ \mathrm{J}$$
PE at $$x = 0.04 \ \mathrm{m}$$ is $$0.4 \ \mathrm{J}$$
The total mechanical energy (E) of the SHM system can be found by summing the given kinetic and potential energies:
$$ E = KE + PE = 0.5 \ \mathrm{J} + 0.4 \ \mathrm{J} = 0.9 \ \mathrm{J} $$
For simple harmonic motion, the total energy (E) is also related to the amplitude (A) by the following formula:
$$ E = \frac{1}{2} k A^2 $$
where $$k$$ is the spring constant. First, we need to find the angular frequency $$\omega$$:
$$ \omega = 2 \pi f = 2 \pi \left(\frac{25}{\pi}ight) \ \mathrm{Hz} = 50 \ \mathrm{rad/s} $$
The spring constant $$k$$ can be calculated using the relationship between $$m$$, $$\omega$$, and $$k$$:
$$ \omega = \sqrt{\frac{k}{m}} \Rightarrow k = m \omega^2 = 0.2 	imes (50)^2 = 500 \ \mathrm{N/m} $$
Now, substituting $$k$$ back into the energy equation, we solve for the amplitude $$A$$:
$$ 0.9 = \frac{1}{2} 	imes 500 	imes A^2 \Rightarrow A^2 = \frac{0.9 	imes 2}{500} \Rightarrow A^2 = \frac{1.8}{500} \Rightarrow A^2 = 0.0036 \Rightarrow A = \sqrt{0.0036} = 0.06 \ \mathrm{m} $$
Converting $$A$$ from meters to centimeters:
$$ A = 0.06 \ \mathrm{m} 	imes 100 = 6 \ \mathrm{cm} $$
Thus, the amplitude of oscillation is $$6 \ \mathrm{cm}$$.

Oscillations

Integer / Numeric Answer Type