Question

A simple accelerometer (an instrument for measuring acceleration) can be made in the form of a tube filled with a liquid and bent as shown. During motion, the level of the liquid in the left arm will be at height $h_1$ and in the right arm at a height $h_2$. Determine the acceleration a of a carriage in which the accelerometer is installed, assuming that the diameter of the tube is much smaller than $h_1$ and $h_2$

• A. $\frac{g(h_1-h_2)}{h_1+h_2}$
• B. $\frac{g(h_1+h_2)}{h_1-h_2}$
• C. $\frac{g(h_2-h_1)}{h_1+h_2}$
• D. $\frac{g\left(h_2\right)}{h_2-h_1}$

Answer 1

The correct option is A $\frac{g(h_1-h_2)}{h_1+h_2}$

Let the pressure at points A and B be $P_A$ and $P_B$ respectively and the acceleration of the tube be $a$

$\therefore$ $P_A-P_B=\rho al$ where $l=x_1+x_2$

Using trignometry, $\frac{x_1}{h}=tan\frac{\pi }{4}$ $⟹x_1=h_1$

Similarly $x_2=h_2$

$⟹$ $P_A-P_B=\rho a(h_1+h_2)$

Also $P_A-P_B=\rho g(h_2-h_1)$

$\therefore$ $\rho g(h_2-h_1)=\rho a(h_1+h_2)$ $⟹a=\frac{g(h_2-h_1)}{h_1+h_2}$