Question

A particle executing simple harmonic motion of amplitude $5cm$ has a maximum speed of $31.4cm{s^{ - 1}}$. The frequency of the oscillation is:
A) $1Hz$
B) $3Hz$
C) $2Hz$
D) $4Hz$

Answer 1

A particle executing simple harmonic motion has several defining quantities associated with it like amplitude, frequency, angular velocity etc. We will use the relations between them to calculate the frequency of oscillation.

Complete step by step solution:
Let’s start by defining amplitude. In simple harmonic motion, the amplitude is the maximum amount of displacement of a particle on the medium from its rest position or the starting point of oscillation. It is denoted by $a$.
The maximum speed of oscillation means the maximum linear velocity of the particle with which the particle executes simple harmonic motion. It is denoted by ${V_{\max }}$.
The maximum velocity of oscillation is related to the amplitude of oscillation by the following formula,
${V_{\max }} = a\omega$ ………. (1)
Where $\omega$ is the angular velocity of the particle.
The formula for angular velocity as we know it is,
$\omega = 2\pi f$ ………. (2)
Where $f$ is the frequency of oscillation.
Frequency of oscillation is defined as the number oscillations per unit time and has the unit of Hertz $\left( {Hz} \right)$, or ${s^{ - 1}}$.
Putting the value of $\omega$ in equation (1), we get,
$\Rightarrow {V_{\max }} = 2\pi f \times a$
$\Rightarrow f = \dfrac{{{V_{max}}}}{{2\pi a}}$
Substituting the respective values in the above equation we get,
$\therefore f = \dfrac{{31.4}}{{2 \times 3.14 \times 5}} = 1Hz$

So our correct answer is option (A).

Additional information:
For a spring with spring constant $k$ in which a particle is exhibiting SHM with a mass $m$, the value of $\omega$ becomes $\omega = \sqrt {\dfrac{k}{m}}$ so the formula for maximum velocity becomes ${V_{\max }} = A\sqrt {\dfrac{k}{m}}$.

Note: After looking at the question, analyse it and apply required relations between the physical quantities. Take care of the system of units. This question was asked in c.g.s. system and the c.g.s unit of frequency is also $Hz$. The examiner can also provide the value of angular acceleration, in that case use the relation between angular velocity and acceleration.