If \(E\) and \(G\), respectively, denote energy and gravitational constant, then \(\frac{E}{G}\) has the dimensions of:
1. \([ML^0T^0]\)
2. \([M^2L^{-2}T^{-1}]\)
3. \([M^2L^{-1}T^{0}]\)
4. \([ML^{-1}T^{-1}]\)

A screw gauge gives the following readings when used to measure the diameter of a wire:
Main scale reading: \(0\) mm
Circular scale reading: \(52\) divisions 
Given that \(1\) mm on the main scale corresponds to \(100\) divisions on the circular scale, the diameter of the wire that can be inferred from the given data is:
1. \(0.26\) cm 
2. \(0.052\) cm 
3. \(0.52\) cm 
4. \(0.026\) cm 

If force \([F]\), acceleration \([A]\) and time \([T]\) are chosen as the fundamental physical quantities, then find the dimensions of energy:
1. \(\left[FAT^{-1}\right]\)
2. \(\left[FA^{-1}T\right]\)
3. \(\left[FAT\right]\)
4. \(\left[FAT^{2}\right]\)

The dimension of Planck's constant equals to that of:
1.  Energy
2.  Momentum
3.  Angular momentum
4.  Power

The universal gravitational constant is dimensionally represented as:

1. ${\mathrm{ML}}^2{\mathrm{T}}^{-1}$

2. ${\mathrm{M}}^{-2}{\mathrm{L}}^3{\mathrm{T}}^{-2}$

3. ${\mathrm{M}}^{-2}{\mathrm{L}}^2{\mathrm{T}}^{-1}$

4. ${\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}$

If the error in measurement of the radius of a sphere is 0.1%, then the error in its volume will be:
1. 0.3%
2. 0.4%
3. 0.5%
4. 0.6%

Which, of the following pair does not have equal dimensions?

1. Energy and torque

2. Force and impulse

3. Angular momentum and Planck's constant

4. Modulus of Elasticity and pressure

The ratio of the dimension of Planck's constant and that of the moment of inertia is the dimension of:

1. Velocity

2. Angular momentum

3. Time

4. Frequency

A student measures the distance traversed in free fall of a body, initially at rest in a given time. He uses this data to estimate \(g,\) the acceleration due to gravity. If the maximum percentage errors in the measurement of the distance and the time are \(e_1\) and \(e_2\) respectively, the percentage error in the estimation of \(g\) is:
1. $e_1+2e_2$
2. $e_1+e_2$
3. $e_1-2e_2$
4. $e_2-e_1$

The density of a material in a CGS system of units is \(4~\text{g/cm}^3\). In a system of units in which the unit of length is \(10~\text{cm}\) and the unit of mass is \(100~\text{g}\), the value of the density of the material will be:
1. \( 0.04 \)
2. \( 0.4 \)
3. \( 40 \)
4. \(400\)