Question

Which of the following equations is not dimensionally consistent? the symbols have their usual meaning.


s=ut−gt2

Ft=mv−m0

−kx+F0sinωt=ma

ω2=ω20+αθ

Answer 1

Answer on Question 52044, Physics, Mechanics | Kinematics | Dynamics

Question:

Which of the following equations is not dimensionally consistent? The symbols have their usual meaning:

a) $s = ut - gt^2$

b) $Ft = mv - m_0$

c) $-kx + F_0\sin(\omega t) = ma$

d) $\omega^2 = \omega_0^2 + \alpha \theta$

Solution:

Let's check the dimensions. If the dimensions on both side of the equation is equal, the equation is dimensionally consistent:

a) $[m] = \left[\frac{m}{s}\right] \cdot [s] - \left[\frac{m}{s^2}\right] \cdot [s^2]$,

$$[m] = [m]$$

Therefore, this equation is dimensionally consistent.

b) $\left[kg\cdot \frac{m}{s^2}\right]\cdot [s] = [kg]\cdot \left[\frac{m}{s}\right] - [kg],$

$$\left[ k g \cdot \frac {m}{s} \right] \neq [ k g ] \cdot \left[ \frac {m}{s} \right] - [ k g ].$$

Therefore, this equation is not dimensionally consistent.

c) $-\left[\frac{kg\cdot\frac{m}{s^2}}{m}\right]\cdot [m] = \left[kg\cdot \frac{m}{s^2}\right]\cdot \sin \left[\frac{rd}{s}\cdot s\right] = [kg]\cdot \left[\frac{m}{s^2}\right],$

$$\begin{array}{l} - \left[ k g \cdot \frac {m}{s ^ {2}} \right] = \left[ k g \cdot \frac {m}{s ^ {2}} \right] \cdot \sin [ r d ] = \left[ k g \cdot \frac {m}{s ^ {2}} \right], \\ \left[ k g \cdot \frac {m}{s ^ {2}} \right] = \left[ k g \cdot \frac {m}{s ^ {2}} \right]. \end{array}$$

Therefore, this equation is dimensionally consistent.

d) $\left[\frac{rd^2}{s^2}\right] = \left[\frac{rd^2}{s^2}\right] + \left[\frac{rd}{s^2}\right]\cdot [rd],$

$$\left[ \frac {r d ^ {2}}{s ^ {2}} \right] = \left[ \frac {r d ^ {2}}{s ^ {2}} \right] + \left[ \frac {r d ^ {2}}{s ^ {2}} \right],$$ $$\left[ \frac {r d ^ {2}}{s ^ {2}} \right] = \left[ \frac {r d ^ {2}}{s ^ {2}} \right].$$

Therefore, this equation is dimensionally consistent.

So, the equation b) $Ft = mv - m_0$ is not dimensionally consistent.

Answer:

b) $Ft = mv - m_0$.

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