Dimensional Analysis

Q.

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: [position] = $\left[X^{\alpha }\right]$; [speed] = $\left[X^{\beta }\right]$; [acceleration] =$\left[X^p\right]$; [linear momentum] = $\left[X^q\right]$; [force] = $\left[X^r\right]$Then

1. $\alpha +p=2\beta$

2. $p+q-r=\beta$

3. $p-q+r=\alpha$

4. $p+q+r=\beta$

Q.

If the velocity of light (c), acceleration due to gravity (g), and atmospheric pressure (p) are chosen as fundamental units, then the dimensions of length will be?

Q. The wavelength (w) of matter waves man depends on Plancks constant (H) , mass(m) and velocity (v) of particle . Use the method of dimension to derive the formula

Q.

The heat produced in a wire carrying an eletric 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 $M{L}^2{I}^{-2}{T}^{-3}$and heat is a form of energy.

1. 

2. 

3. 

4. 

Q. If the length of a second pendulum is increased by 2% then in a day the pendulum will gain/lose how many seconds

Q.

The velocity of water waves (v) may depend on their wavelength ${}^{\prime }\lambda {}^{\prime }$, density of water 'p' and acceleration due to gravity 'g'. The method of dimensions gives the relation between these quantities as

1. $v^2\propto \lambda gp$
2. $v^2\propto {\lambda }^{-1}{g}^{-1}{p}^{-1}$
3. $v^2\propto p\lambda$
4. $v^2\propto g\lambda$

Q. From the equation $tan\theta =\frac{rg}{v^2}$ , one can obtain the angle of banking $\theta$ for a cyclist taking a curve (the symbols have their usual meanings). Then say, it is

1. Both dimensionally and numerically correct
2. Neither numerically nor dimensionally correct
3. Dimensionally correct only
4. Numerically correct only

Q.

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

1. 

2. 

3. 

4. 1000 ergs

Q. A small steel ball of radius $r$ is allowed to fall under gravity through a column of viscous liquid of coefficient $\eta$ . After some time the velocity of the ball attains a constant value known as terminal velocity, $V_T$. The terminal velocity depends on mass of the ball '$m$', coefficient of viscosity '$\eta$', the radius of the ball '$r$' and acceleration due to gravity $g$. The relationship between terminal velocity and other factors given is :

1. $V_T\propto \frac{\eta r}{mg}$
2. $V_T\propto$ $\eta rmg$
3. $V_T\propto \frac{mg}{\eta r}$
4. $V_T\propto \frac{mgr}{\eta }$

Q.

What is the unit of Density?

1. 

2. 

3. 

4. 

Q. The period of a body under SHM i.e. presented by T = $P^a{D}^b{S}^c$ ; where P is pressure, D is density and S is surface tension. The value of a, b and c are

1. 
2. 
3. 
4. -1, -2, 3

Q.

Can speed be expressed as $\frac{time}{length}$?

1. Yes

2. No

Q.

The velocity of water waves $v$ may depend upon their wavelength $\lambda$ the density of water $\rho$ and the acceleration due to gravity g. The method of dimensions gives the relation between these quantities as

1. 
2. 
3. 
4. ν² rg

Q. A physcial quantity x depends on quantities y and z as follows: x=Ay+B tan Cz , where A, B and C are constants. Which of the following have different dimensions

1. C and z^-1
2. x and B
3. y and B/A
4. x and A

Q. The position of a particle at time t is given by the relation x(t)= $\left(\frac{v_0}{\alpha }\right)(1-c^{-at})$ where $v_0$ and $\alpha$ are constants . The dimensions of $v_0$ and $\alpha$ are respectively

1. M^oL¹T^-1 and T^-1
2. M^oL¹T^o and T^-1
3. M^oL¹T^-1 and LT^-2
4. M^oL¹T^-1 and T

Q. In C.G.S. system the magnitude of the force is 100 dynes. In another system where the fundamental physical quantities are kilogram, metre and minute, the magnitude of the force is

1. 0.036
2. 0.36
3. 3.6
4. 36

Q. The velocity of a particle depends upon as v = a + bt + $c{t}^2$ ; if the velocity is in m/sec, the unit of a will be

1. m/sec²
2. m²/sec
3. m/sec³
4. m/sec

Q. Suppose speed of light $\left(c\right)$, force $\left(F\right)$ and kinetic energy $\left(K\right)$ are taken as the fundamental units, then the dimensional formula for mass will be:

1. ${KC}^{-2}$
2. ${KF}^{-2}$
3. ${CK}^{-2}$
4. ${FC}^{-2}$

Q. The velocity of water waves $\left(v\right)$ may depend on their wavelength $\lambda$, the density of water $\rho$ and the acceleration due to gravity $g$. The method of dimensions gives the relation between these quantities is

1. $v^2\propto g\lambda$
2. $v$
3. $v^2\propto g{\lambda }^2$
4. $v^2\propto g^{-1}{\lambda }^2$

Q.

In a direct impact, loss in kinetic energy is given by $△K=\frac{M_1{M}_2}{2(M_1+M_2)}.(V_1-V_2{)}^2(1-K^2)$ with usual

notations(except k). The quality k will have dimensional formula:

1. 

2. 

3. 

4. 

Q. E, m, l and G denote energy, mass, angular momentum and gravitational constant respectively, then the dimension of $\frac{E{l}^2}{m^2{G}^2}$ are

1. [M⁰L⁰T⁰]
2. [M⁰L⁰T¹]
3. [M⁰L¹T¹]
4. [M⁰L¹T^-1]

Q. The velocity of water waves may depend on their wavelength $\lambda$ , the density of water $\rho$ and the acceleration due to gravity $g$. The method of dimensional analysis gives the relation between these quantities as:

1. $V^2=K{\lambda }^{-1}{g}^{-1}{\rho }^{-1}$
2. $V^2=K\lambda \rho g$
3. $V^2=K{\lambda }^3{g}^{-1}{\rho }^{-1}$
4. $V^2=K\lambda g$

Q. What should be the velocity of an electron having de-brogles wave length 100 amstrong

Q. the least count of a stop watch is 1/5 seconds. the time of 20 oscillations of a pendulum is measured to be 25 seconds. the maximum percentage error in the measurement of time will be