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101.

Suppose we employ a system in which in which the unit of mass equals `100 kg` , the unit of length equals `1 km` and the unit of time `100 s` and call the unit of energy eluoj ( joule written in reverse order), then what is the relation between eluoj and joule?

Answer» `[E] = [ML^(2)T^(-2)]`
` = [ 100 kg] xx [ 1 km]^(2) xx [ 100 s]^(-2)`
` = 100 kg xx 10^(6) m^(2) xx 10^(-4) s^(-2)`
`= 10^(4) xx m^(2) s^(-2) = 10^(4) J`
Discussion
102.

Consider three quantities: `x = (E)/( b) , y = (1)/(sqrt( mu_(0) epsilon _(0))) , and z = (l) /( C R)`. Here , `l` is the length of a wire , `C` is the capacitance , and `R` is a resistance. All other symbols have usual meanings. ThenA. ` x and y ` have the same dimensions.B. `x and z` have the same dimensions.C. `y and z` have the same dimensions.D. None of the above three pairs have the same dimensions.

Answer» Correct Answer - A::B::C
Unit of x `= ("Uit of E")/("Unit of B")` = Unit of velocity
Because `E = vB`
` y = (1)/(sqrt( mu_(0) epsilon _(0)))` = c rarr velocity of light
unit of `z = ("Unit of l")/("Unit of RC") = ("Length")/("Time") = "Velocity"`
Discussion
103.

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

Answer» According to the provided information , `F prop m^(a) v^(b) r^( c ) `
rArr `F = km^(a) v^(b) r^(c ) ` ….(i)
Where `k` is the dimensionless constant of proportionality and `a , b, c` are the constant powers powers of `m , v , r`, respectively.
Now using the principle of homogenity , comparing the power of like quantities on both the sides , we have
` a = 1 ( ii) b + c = 1 (iii) and b = 2 (iv)
Using (ii) , (iii) , and (iv) , we have a = 1 , b = 2 , and `c = -1`.
Using (ii) , (iii) , and (iv), we have ` a = 1 , b = 2 , c = -1`.
Using these values in (i) , ` F = km^(1) v^(2) r ^(-2)`
`implies F = K (mv^(2))/(r)` which is the desired relation.
Discussion
104.

Which of the following pairs have different dimensions?A. `Frequency and angular velocity.B. Tension and surface tension.C. Density and energy density.D. Linearmomentum and angular momentum.

Answer» Correct Answer - B::C::D
Frequency and angular velocity have the same dimension `[T^(-1)]`.
Tension has dimensions of force and surface tension has dimensions of `"force"//"length"`. Density has dimensions of `"mass"//"volume"` and energy density has dimensions of `"energy"//"volume"`. Angular momentum has dimensions of linear momentum `xx` distance.
Discussion
105.

Of the following quantities , which one has the dimensions different from the remaining three?A. Energy densityB. Force per unit areaC. Product of charge per unit volume and voltageD. Angular momentum per unit mass

Answer» Correct Answer - D
Energy density `= ("Energy")/("Volume") = [ML^(-1)T^(-2)]`
`"Force//Area" = ML^(-1)T^(-2) ["Charge // Volume"] xx ["Voltage"]`
`= (Q)/( vol.) xx (W)/(Q) = ("Work")/("Volume") = ML^(-1)T^(-2)`
The dimensions of `(d)` are different , i.e.,
`(ML^(2)T^(-1))/(M) = M^(0)L^(2)T^(-1)`
Hence , correct choice is `(d)`.
Discussion
106.

The error in the measurement of the radius of a sphere is `0.5 %`. What is the permissible percentage error in the measurement of its (a) surface area and (b) volume ?

Answer» Correct Answer - (a). `% ; (b) 1.5%`
Given `Delta r // r = 0.5%`.
(a). The surface area of a sphere of radius `r is A = 4 pi r^(2)`.
Percentage error in `A = (Delta A)/(A) = ( 2 Delta r )/( r ) = 2 xx 0.5% = 1%`
(b). The volume of a sphere of radius `r is V = ( 4 pi)/(3) r^(3)`.
Percentage error in `V = ( Delta V) /( V) = ( 3 Delta r)/( r ) = 3 xx 0.5 % = 1.5%`
Discussion
107.

If all measurements in an experiment are taken up to the same number of significant figures , then mention two possible reasons for maximum error.

Answer» The maximum error will be due to `(a)` measurement , which is least accurate and `(b)` measurement of the quantity which has maximum power in formulas.
Discussion
108.

The quantities `A and B` are related by the relation `A//B = m`, where `m` is the linear mass density and `A` is the force , the dimensions of `B` will beA. Same as that of pressureB. Same as that of workC. That of momentumD. Same as that of learnt heat

Answer» Correct Answer - D
`(A)/(B) = m , B = (A)/(m) = ("Force")/("Linear density") = (MLT^(-2))/(ML^(-1))`
`:. B = [M^(0) L^(2) T^(-2)]`
Latent heat `= ("Heat energy")/("Mass") = (ML^(2)T^(-2))/(M) = [ M^(0) L^(2) T^(-2) ]`
Thus , `B` has dimensions as that of latent heat.
Discussion
109.

Given `R_(1) = 5.0 +- 0.2 Omega, and R_(2) = 10.0 +- 0.1 Omega`. What is the total resistance in parallel with possible `%` error?

Answer» In parallel , `R_(P) = (R_(1) R_(2))/( R_(1) + R_(2)) = ( 5.0 xx 10.0)/( 5.0 + 10.0) = ( 50)/(15) = 3.3 Omega`
Also `(Delta R_(P))/(R_(P)) xx 100 = (Delta R_(1))/(R_(1))xx100+(Delta R_(2))/(R_(2))xx100+(Delta(R_(1)+R_(2)))/(R_(1)+R_(2))xx100`
`= (0.2) /(5.0) xx 100 + (0.1)/(10.0) xx 100 + (0.3)/(15) xx 100 = 7%`
`:. R_(P) = 3.3 Omega +- 7%`
Discussion
110.

In the relation ` y = r sin ( omega t - kx)`, the dimensions of `omega//k` areA. `[M^(0) L^(0) T^(0)]`B. `[M^(0) L^(1) T^(-1)]`C. `[M^(0) L^(0) T^(1)]`D. `[M^(0) L^(1) T^(0)]`

Answer» Correct Answer - B
`y = r sin ( omega t - kx)`
Here `omega t = angle rArr omega = (1)/(T) = T^(-1)`
Similarly , `kx = angle rArr K = (1)/(x) = L^(-1)` :. (omega)/(k) = (T^(-1))/(L^(-1)) = LT^(-1)`
Or simply `omega//k` represents wave velocity.
`(omega)/(k) = ( 2 pi f)/( 2 pi//f) = f lambda = v` , where f is frequency.
Discussion
111.

If `S and V` are one main scale and one Vernier scale and ` n - 1` divisions on the main scale are equivalent to `n` divisions of the Vernier , thenA. The least count is `S//n`.B. The Vernier constant is `S//n`.C. The same Vernier constant can be used for circular Vernier also.D. The same vernier constant cannot be used for circular Verniers.

Answer» Correct Answer - A::B::C
Least count ` = 1 MSD - 1 VSD`
` = S - V = S - ((n-1)/( n)) S = ( S)/(n)`
`[ because nV = ( n-1) S]`
It is also called vernier constant .
So choices (a) and (b) are correct.
Choice (d) is wrong and choice ( c) is correct , since for all vernier scales similiar approach can be used.
Discussion
112.

In the relation `( dy)/( dt) = 2 omega sin ( omega t + phi_(0))`, the dimensional formula for ` omega t + phi_(0)` isA. `MLT`B. `MLT^(0)`C. `ML^(0) T^(0)`D. `M^(0) L^(0) T^(0)`

Answer» Correct Answer - D
Here `( omega t + phi_(0))` is dimensionless because it is argument of a trigonometric function.
Discussion
113.

Two resistances `R_(1) = 100 +- 3 Omega and R_(2) = 200 +- 4 Omega` are connected in series . Find the equivalent resistance of the series combination.

Answer» Correct Answer - `300+- 7 ohm`
The equivalent resistance
`R = R_(1) + R_(2) = (100 +- 3) ohm + ( 200 +- 4) ohm`
` = 300 +- 7 ohm`
Discussion