5.74 g of a substance occupies \(1.2~\mathrm{cm^3}\). Its density by keeping the significant figures in view is:
1. \(4.7333~\mathrm{g/cm^3}\)
2. \(3.8~\mathrm{g/cm^3}\)
3. \(4.8~\mathrm{g/cm^3}\)
4. \(3.7833~\mathrm{g/cm^3}\)

The SI unit of energy is $\mathrm{J} = {\mathrm{kgm}}^2{\mathrm{s}}^{-2}$; that of speed v is ${\mathrm{ms}}^{-1}$ and of acceleration a is ${\mathrm{ms}}^{-2}$. Which of the formula for kinetic energy (K) given below can you rule out on the basis of dimensional arguments (m stands for the mass of the body)?

(a) K = ${\mathrm{m}}^2{\mathrm{v}}^3$

(b) K = $\left(1/2\right){\mathrm{mv}}^2$

(c) K = ma

(d) K = (3/16)${\mathrm{mv}}^2$

(e) K = (1/2)${\mathrm{mv}}^2$+ma

1.  (a), (c) & (d)

2.  (b) & (d)

3.  (a), (c), (d) & (e)

4.  (a), (c) & (e)

Let us consider an equation \(\frac{1}{2}mv^2=mgh\) where \(m\) is the mass of the body, \(v\) its velocity, \(g\) is the acceleration due to gravity and \(h\) is the height. The equation is:
 

Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (l), the mass of the bob (m) and acceleration due to gravity (g). Expression for its time period is:

1.  $\mathrm{T} = \frac{1}{2\mathrm{\pi }}\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$

2.  $\mathrm{T} = 2\mathrm{\pi }\left(\frac{\mathrm{l}}{\mathrm{g}}\right)$

3.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\mathrm{g}}}$

4.  $\mathrm{T} = 2\mathrm{\pi }\sqrt{\frac{\mathrm{g}}{\mathrm{l}}}$