A gas bubble from an explosion under water, oscillates with period T proportional to p^ad^bE^c, where p is pressure, d is density and E is energy of an explosion. Find the value of a, b and c.

Solution

Given, time period is proportional to $straight T space alpha space straight p to the power of straight a straight d to the power of straight b space straight E to the power of straight c$
Equating both sides of above equation dimensionally, we get
$open square brackets straight M to the power of 0 straight L to the power of 0 straight T to the power of 1 close square brackets space equals open square brackets straight M to the power of 1 straight L to the power of negative 1 end exponent straight T to the power of negative 2 end exponent close square brackets to the power of straight a open square brackets straight M to the power of 1 straight L to the power of negative 3 end exponent straight T to the power of 0 close square brackets to the power of straight b open square brackets straight M to the power of 1 straight L squared straight T squared close square brackets to the power of straight c space space open square brackets straight M to the power of 0 straight L to the power of 0 straight T to the power of 1 close square brackets space equals space open square brackets straight M to the power of straight a plus straight b plus straight c end exponent straight L to the power of negative straight a minus 3 straight b plus 2 straight c end exponent straight T to the power of negative 2 straight a minus 2 straight c end exponent close square brackets$

Now, omparing the dimensions of M, L and T on both sides, we get
a + b + c = 0 ....(1)
- a - 3b + 2c = 0 ....(2)
- 2a - 2c = 0 ....(3)
Solving (1), (2) and (3), we get

straight a equals negative 5 over 6 comma space straight b equals 1 half space and space straight c space equals 1 third

Some More Questions From Units and Measurement Chapter

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Units And Measurement

A gas bubble from an explosion under water, oscillates with period T proportional to p^ad^bE^c, where p is pressure, d is density and E is energy of an explosion. Find the value of a, b and c.

Some More Questions From Units and Measurement Chapter

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

Can a physical quantity have two or more units? Give example.

What are fundamental quantities?

How many fundamental quantities are there in mechanics?

How many fundamental quantities are there in Physics?

Name the fundamental quantities of mechanics.

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Units And Measurement

A gas bubble from an explosion under water, oscillates with period T proportional to padbEc, where p is pressure, d is density and E is energy of an explosion. Find the value of a, b and c.

Some More Questions From Units and Measurement Chapter

To measure a physical quantity X, the unit U is repeatedly used for n times. What is the magnitude of physical quantity?

Does the numerical factor in the quantitative measurement depend on the system of unit used?

Is one unit sufficient to measure a physical quantity?

Can a physical quantity have two or more units? Give example.

What are fundamental quantities?

How many fundamental quantities are there in mechanics?

How many fundamental quantities are there in Physics?

Name the fundamental quantities of mechanics.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

A gas bubble from an explosion under water, oscillates with period T proportional to p^ad^bE^c, where p is pressure, d is density and E is energy of an explosion. Find the value of a, b and c.