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i want some imp questions on the chapter units and measuerments

1. What is the basic principle of alpha particle scattering method for estimating the size of the nucleus ?

Both the ?- particle and nucleus are positively charged. When an ? - particle approaches a nucleus, its kinetic energy gradually changes into potential energy due to repulsive forces. At the distance of closest approach r_0, the entire energy changes into potential energy. This concept can be used to calculate r_0, which gives the order of the size of the nucleus.

2. If  the velocity of light is taken as the unit of velocity and one year as the unit of time, what must be the unit of length ? What is it called ?

Unit of length = unit of velocity X unit of time 

              = 3 X 10⁸ ms^-1 X 1 year

              = 3 X 10⁸ ms^-1 X 365 X 60 X 60s

              = 9.45 X 10¹⁵ ms^-1

              = 1 light year.

3. What is common between bar and torr ?

Both bar and torr are the units of pressure.

               1 bar = 1 atmospheric pressure

                       = 76 mm of Hg column

              1 torr =  1 mm of Hg column

           1 bar = 760 torr.

4. Distinguish between accuracy and precision.

By accuracy of a measurement we mean that the measured value of a physical quantity is as close to the true value as possible. On the other hand, a measurement is said to be precise, if same value of the quantity is obtained in each of the various measurements carried out with the given apparatus.

5.Two clocks are tested against a standard clock located in a national laboratory. At 12:00::00 noon by the standard clock, the readings of the two clocks are :

If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer ? 

The range of variation over the seven days of observations is 162 s for clock 1, and 31 s for clock 2. The average reading of clock 1 is much closer to the standard time than the average reading of clock 2. The important point is that a clock's zero error is not as significant for precision work as its variation, because a ' zero - error' can always be easily corrected. Hence clock 2 is to be preferred clock 1.

6. For the determination of 'g' using a simple pendulum, measurements of l and T are required. Error in the measurement of which of these will have larger effect on the value of 'g' thus obtained and why ? What is done to minimize this error?

Time period of a simple pendulum, 

Clearly, the error in the measurement of time period T has larger effect on the value of g then the error in the measurement of length l. 

(i) T is very small.

(ii) In contrast to l, T² appears in the formula for g.

To minimize the error, time period for a large number of oscillations is measured.

7. Magnitude of force F experienced by a certain object moving with speed u is given by F = Ku², where K is a constant. Find the dimensions of K. 

8. Using the principle of homogeneity of dimensions, find which of the following is correct:

Where T is time period, G is graviotional constant,M is mass and r is radius of orbit.

9. The mean value of period of oscillation of a simple pendulum in an experiment is 2.285 s. The asthmatic mean of all the absolute errors is 0.11s. Round off the period of simple pendulum to appropriate number of significant figures. Give reasons.

The absolute error 0.11 s has only two significant figures.

 Period of simple pendulum = 2.9 s

10. If nth division of main scale coincides with (n + 1) division scale, find the least count of the vernier. Given main scale division is equal to 'a' units. 

(n + 1)  divisions of vernier scale 

                 = n divisions of main scale

11. If the velocity of light c, the constant of gravitation G and Planck's constant h be chosen as fundamental units, find the dimensions of mass, length and time in terms of c, G and h.

We have,

12. The velocity of a body which has fallen freely under gravity varies as g ^p h ^q, where g is the acceleration due to gravity at the place and h is the height through which the body has fallen. Determine the values of p and q .

Let u = K g ^p h^q,

Where k = a dimensionless constant. 

putting the dimensions of various quantities, we get 

    LT^-1 = [LT^-2]^p [L]^q

or L¹T^-1 = L^p +q T ^-2p

Equating the powers of L and T on both sides, we get: 

p+q - 1,  -2p =-1

On solving, p = _^, q =  .

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

Let   T   = K p^a d^b E^c,

where K = a dimensionless constant. 

Putting the dimensions of various quantities, 

          T [Ml^-1T^-2]^a [ML^-3]^b [ML² T^-2]c

or M⁰ L⁰ T = M^{a+ b+ c} L ^{-a -3b+2c} T ^{-2a -2c}

Equating the powers of M, L and T on both sides, we get

a + b + c = 0, -a -3b + 2c = 0, -2a -2c =1

14. A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid  of coefficient of viscosity ?. After some time the velocity of the body attains a constant value u_T. The terminal velocity depends upon (i) the weight of the ball mg (ii) the coefficient of viscosity ? and (iii) the radius of the ball r. By the method of dimensions, determine the relation expressing terminal velocity.

Let  u_T = K (mg)^a ?^b r^c,

Where K = a dimensionless constant. 

 Putting the dimensions of various quantities, 

           LT^-1 = [MLT^-2]^a [ML^-1 T^-1]^b[L]^c

or   M⁰L¹T^-1 = M^a+b L^a-b+c T^{-2a -b}

Equating the powers of M, L and T on both sides, we get

     a+b = o,  a - b + c = 1,  -2a - b = -1 

On solving,  a= 1, b = - 1, c = -1

  u_T = K (mg)¹ ?^-1 r ^-1   or  

15. Derive dimensionally the relation : 

               S = ut +  _at².

Let S = Ku^x a^y t^z,

Where K = a dimensionless constant.

Putting dimensions of various quantities, 

              L  =  [LT^-1]^x [LT^-2]^y [T]^z

or           L¹T⁰ = L^{x + y} T ^{-x -2y +z}

Equating the powers of M, L and T, we get 

                     x+y=1

              - x - 2y + z = 0

These two equations cannot be solved for three unknowns x, y and z. The problem is spit into two parts. 

  (i)  Suppose the body has no acceleration. Then 

                ^{S = K1 ux tz}

                L =  [LT^-1]^x [T]^{z =} L^xT^{- x +z}

Equating the powers of L and T, x = 1, -x + z = 0

On solving, x = 1, z = 1

Hence    S = k₁ ut

  (ii) Suppose the body has no initial velocity. Then 

               S  K2 a^yt^z

              L = [LT^-2]^y [T]^{z  =} L^yT^{-2y +z}

Equating the powers of L and T, y = 1, -2y + z = 0

On solving,             y = a, z = 2

Hence                    S = K₂ at²

  (iii) Suppose the body has both acceleration and initial velocity. Then

           S  = k¹ ut + K₂ at ²

It is found that K₁ = 1 and K₂ = 1/2. Therefore, 

            S = ut + _at².z

16. The specific heats of a gas are measured as  C_^p = (12.28 ± 0.2) unis and C_u = (3.97 ± 0.3) units. Find the value of real gas constant R and percentage error in R.

                GAs constant,

                R = C_{^{p -}} C_u

                    = (12.28 ± 0.2) - (3.97 ± 0.3)

                    = (8.31 ± 0.5) units

% Error in R,

17. The heat dissipated in a resistance can be determined from the relation : 

If the maximum errors in the measurement of current, resistance and time are 2%, 1% and  1%  respectively, what would be the maximum error in the dissipated heat ?