Question

Which of the following is the correct unit of k in the equation of a damped harmonic oscillator given as
−bv−kx=ma
, where b is the damping factor and all the symbols have their usual meaning?

Answer 1

Answer on Question 52104, Physics, Mechanics | Kinematics | Dynamics

Question:

Which of the following is the correct unit of $k$ in the equation of a damped harmonic oscillator given as $-bv - kx = ma$, where $b$ is the damping factor and all the symbols have their usual meaning?

a) $kgms^{-2}$

b) $kgs^{-1}$

c) $kgms^{-1}$

d) $kgs^{-2}$

Solution:

Let's consider the equation of a damped harmonic oscillator:

$$- bv - kx = ma.$$

The dimensions of both side of equation must be equal. We can see, that the term $ma$ have the dimension of force:

$$[ma] = \left[ kg \cdot \frac{m}{s^2} \right] = [N].$$

Then, the term $kx$ must have the same dimension as $ma$:

$$[kx] = [ma] = \left[ kg \cdot \frac{m}{s^2} \right].$$

Assuming that $[x] = [m]$ we finally get the dimension of unit of $[k]$:

$$[k] = \left[ \frac{kg}{s^2} \right].$$

Answer:

d) $kgs^{-2}$

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