WORKSHEET-1- (DIMENSIONAL ANALYSIS)

1. Find the dimensional formulae of the following quantities :

(a) the universal constant of gravitation G     (b) the surface tension S,       

(c) the thermal conductivity k and                  (d) the coefficient of viscosity

2. Find the dimensional formulae of

(a) the charge Q,         (b) the potential V,      (c) the capacitance C, and       (d) the resistance R.

Some of the equations containing these quantities are

Q = It, U = VIt, Q= CV and V = RI; where I denotes the electric current, t is time and U is energy.

3. The SI and CGS units of energy are joule and erg respectively. How many ergs are equal to one  

     joule? Young's modulus of steel is 19 x 10¹⁰ N/m². Express it in dyne/cm². Here dyne is the    

     CGS unit of force.

4. If velocity, time and force were chosen as basic quantities, find the dimensions of mass.

5. Test dimensionally if the equation v² =u² + 2ax may be correct.

6. The distance covered by a particle in time t is given by x = a + bt + ct² + dt³; find the dimensions of     

     a, b, c and d.

6.  If the centripetal force is of the form m^a v^b r^c, find the values of a, b and c.

7. When a solid sphere moves through a liquid, the liquid opposes the motion with a force F. The magnitude of F depends on the coefficient of viscosity η of the liquid, the radius r of the sphere and the speed v of the sphere. Assuming that F is proportional to different powers of these quantities, guess a formula for F using the method of dimensions.

8. The heat produced in a wire carrying an electric current depends on the current, the resistance and the time. Assuming that the dependence is of the product of powers type, guess an equation between these quantities using dimensional analysis. The dimensional formula of resistance is [ML² I^-2T ^-3] and heat is a form of energy.

9.  Taking force, length and time to be the fundamental quantities find the dimensions of (a) density, (b) pressure, (c) momentum and (d) energy.

10. Suppose the acceleration due to gravity at a place is 10 m/s². Find its value in cm/ (minute)².

11. Express the power of a 100 watt bulb in CGS unit.

12. The surface tension of water is 72 dyne/cm. Convert it in SI unit.

13. The volume of a liquid flowing out per second of a pipe of length land radius r is written by a student as  

where P is the pressure difference between the two ends of the pipe and η is coefficient of viscosity of the liquid having dimensional formula [ML^–1 T^–1]. Check whether the equation is dimensionally correct.

14. Young’s modulus of steel is 19 x10¹⁰ N/m². Express it in dynes cm². Here dynes are the C.G.S unit       

      of force.

15. The velocity u of water waves may depend on their wavelengthl density of water r and the acceleration due to gravity g. Find relation between these quantities by the method of dimension?

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