The quantities \(A\) and \(B\) are related by the relation, \(m= \frac{A}{B}\), where \(m\) is the linear density and \(A\) is the force. The dimensions of \(B\) are of:
1. | Pressure | 2. | Work |
3. | Latent heat | 4. | None of the above |
A small steel ball of radius \(r\) is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity \(\eta\). After some time the velocity of the ball attains a constant value known as terminal velocity \(v_T\). The terminal velocity depends on \((\text{i})\) the mass of the ball \(m\) \((\text{ii})\) \(\eta\) \((\text{iii})\) \(r\) and \((\text{iv})\) acceleration due to gravity \(g\). Which of the following relations is dimensionally correct:
1. | \(v_T \propto \frac{mg}{\eta r}\) | 2. | \(v_T \propto \frac{\eta r}{mg}\) |
3. | \(v_T \propto \eta rmg\) | 4. | \(v_T \propto \frac{mgr}{\eta }\) |
If energy (\(E\)), velocity (\(v\)) and time (\(T\)) are chosen as the fundamental quantities, the dimensional formula of surface tension will be:
1. \( {\left[E v^{-2} T^{-1}\right]} \)
2. \( {\left[E v^{-1} T^{-2}\right]} \)
3. \( {\left[E v^{-2} T^{-2}\right]} \)
4. \({\left[E^{-2} v^{-1} T^{-3}\right]}\)
If the dimensions of a physical quantity are given by \([M^aL^bT^c],\)
1. | pressure if \(a=1, ~b=-1,~c=-2\) |
2. | velocity if \(a=1,~b=0,~c=-1\) |
3. | acceleration if \(a=1,~b=1,~c=-2\) |
4. | force if \(a=0, ~b= -1,~c=-2\) |
The pitch of a screw gauge is \(1~\)mm and there are \(100\) divisions on the circular scale. While measuring the diameter of a wire, the linear scale reads \(1\) mm and \(47\)th division on the circular scale coincides with the reference line. The length of the wire is \(5.6\) cm. Curved surface area (in cm2) of the wire in appropriate number of significant figures will be:
1. \(2.4\) cm2
2. \(2.56\) cm2
3. \(2.6\) cm2
4. \(2.8\) cm2
Find the thickness of the wire. The least count is \(0.01\) mm. The main scale reads: (in mm)
1. \(7.62\)
2. \(7.63\)
3. \(7.64\)
4. \(7.65\)
The frequency of vibration \(f\) of a mass \(m\) suspended from a spring of spring constant \(k\) is given by a relation of type \(f= Cm^{x}k^{y}\); where \(C\) is a dimensionless quantity. The values of \(x\) and \(y\) will be:
1. \(x=\frac{1}{2},~y= \frac{1}{2}\)
2. \(x=-\frac{1}{2},~y= -\frac{1}{2}\)
3. \(x=\frac{1}{2},~y= -\frac{1}{2}\)
4. \(x=-\frac{1}{2},~y= \frac{1}{2}\)
If \(\int \frac{d x}{\sqrt{a^2-x^2}}=a^n \sin ^{-1} \frac{x}{a}\) is dimensionally correct, then the value of \(n\) will be:
1. | \(1\) | 2. | \(\text{zero}\) |
3. | \(\text-1\) | 4. | \(2\) |
When units of mass, length, and time are taken as \(10~\text{kg}, 60~\text{m}~\text{and}~60~\text{s}\) respectively, the new unit of energy becomes \(x\) times the initial SI unit of energy. The value of \(x\) will be:
1. \(10\)
2. \(20\)
3. \(60\)
4. \(120\)
Which of the following relations is dimensionally wrong? [The symbols have their usual meanings]
1. \(s= ut+\frac{1}{6}at^2\)
2. \(v^2= u^2+\frac{2as^2}{\pi}\)
3. \(v= u-2at\)
4. All of these