14.19 One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

2 Views | Posted 5 months ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

5 months ago

Area of cross section of the U tube = A

Density of the mercury column = $ρ$

Acceleration due to gravity = g

Restoring force, F = Weight of the mercury column of a certain height = - (Volume $\times d e n s i t y \times g)$

F = -(A $\times 2 h \times ρ \times g)$ = -2A $ρ g$ = -k $\times d i s p l a c e m e n t i n o n e o f t h e a r m s (h)$

Where, 2h is the height of the mercury columns in two arms

The constant k is given by k = $- \frac{F}{h}$ = 2A $ρ g$

Time period, T = 2 $π \sqrt{\frac{m}{k}}$ = 2 $π \sqrt{\frac{m}{2 A ρ g}}$ , where m is the mass of the mercury column

Let l be the length of the total mercury in the U tube

Mass of the mercury, m = Volume of the mercury $\times$ density of mercury = Al $ρ$

Hence T = 2 $π \sqrt{\frac{A l ρ}{2 A ρ g}}$ = 2 $π \sqrt{\frac{l}{2 g}}$

Area of cross section of the U tube = ADensity of the mercury column = <math><mi>ρ</mi></math>Acceleration due to gravity = gRestoring force, F = Weight of the mercury column of a certain height = - (Volume <math><mo>×</mo><mi>d</mi><mi>e</mi><mi>n</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>y</mi><mi></mi><mo>×</mo><mi>g</mi><mo>)</mo></math>F = -(A <math><mo>×</mo><mn>2</mn><mi>h</mi><mo>×</mo><mi></mi><mi>ρ</mi><mo>×</mo><mi>g</mi><mo>)</mo></math> = -2A <math><mi></mi><mi>ρ</mi><mi>g</mi></math> = -k <math><mo>×</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">s</mi><mi></mi><mo>(</mo><mi>h</mi><mo>)</mo></math>Where, 2h is the height of the mercury columns in two armsThe constant k is given by k = <math><mo>-</mo><mfrac><mrow><mrow><mi>F</mi></mrow></mrow><mrow><mrow><mi>h</mi></mrow></mrow></mfrac></math> = 2A <math><mi></mi><mi>ρ</mi><mi>g</mi></math>Time period, T = 2 <math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mi>k</mi></mrow></mrow></mfrac></mrow></msqrt></math> = 2 <math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>A</mi><mi></mi><mi>ρ</mi><mi>g</mi></mrow></mrow></mfrac></mrow></msqrt></math> , where m is the mass of the mercury columnLet l be the length of the total mercury in the U tubeMass of the mercury, m = Volume of the mercury <math><mo>×</mo><mi></mi></math> density of mercury = Al <math><mi></mi><mi>ρ</mi></math>Hence T = 2 <math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">l</mi><mi></mi><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>A</mi><mi></mi><mi>ρ</mi><mi>g</mi></mrow></mrow></mfrac></mrow></msqrt></math> = 2 <math><mi>π</mi><msqrt><mrow><mfrac><mrow><mrow><mi mathvariant="normal">l</mi><mi></mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>g</mi></mrow></mrow></mfrac></mrow></msqrt></math>

0 Upvotes 0 Downvotes

14.19 One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

1 Answer

Similar Questions for you

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question