A.

−$4 \mathrm{\pi }$

$$\begin{gathered} {{\mathrm{w}}_1}^0 = 12 \times 2\mathrm{\pi } \frac{\mathrm{rad}}{\mathrm{s}} \\ = 24 \mathrm{\pi } \frac{\mathrm{rad}}{\mathrm{s}} \\ {\mathrm{w}}_{\mathrm{f}} = 0 \\ ∆\mathrm{t} = 6\mathrm{s} \\ \mathrm{As} {\mathrm{w}}_{\mathrm{f}} = {\mathrm{w}}_{\mathrm{i}} + \mathrm{\alpha } ∆\mathrm{t} \\ \mathrm{We} \mathrm{have}, \mathrm{\alpha } = \frac{- {\mathrm{w}}_{\mathrm{i}}}{∆\mathrm{t}} \\ = - \frac{24 \mathrm{\pi }}{6} \\ = - 4\mathrm{\pi } \frac{\mathrm{rad}}{{\sec }^2} \end{gathered}$$

D.

$T\propto R^{\frac{n+1}{2}}$

$$\begin{gathered} m{\omega }^{2}R = Force \propto \frac{1}{R^n} \\ (Force = \frac{m{v}^2}{R}) \\ \Rightarrow {\omega }^2 \propto \frac{1}{R^{n+1}} \\ \Rightarrow \omega \propto \frac{1}{R^{{\displaystyle \frac{n+1}{2}}}} \\ Time Period T = \frac{2\pi }{\omega } \\ Time Period T\hspace{0.17em}\propto R^{\frac{n+1}{2}} \end{gathered}$$

B.

C.

7.1 N/m

Given frequency f  = 10¹²/sec

Angular frequency ω = 2πf = 2π x 10¹² /sec
Force constant k = mω²

= $$\begin{gathered} \frac{108 x {10}^{-3}}{6.02 x {10}^{23}} x 4{\pi }^2 x {10}^{24} \\ k = 7.1 N/m \end{gathered}$$

C.

4

The point P must be the centre of mass of the T-shaped object since the force F does not produce any rotational motion of the object. So, we have to find the distance of the centre of mass from the point C.

The horizontal part of the T-shaped object has length L. If the mass of the horizontal portion is ‘m’, the mass of the vertical portion of the T- shaped object is 2m since its length is 2L. For finding the centre of mass of the T shaped object, it is enough to consider two point masses m and 2m located respectively at the midpoints of the horizontal and vertical portions of the T.

Therefore, the T-shaped object reduces to two point masses m and 2m at distances 2L and L respectively from the point C. The distance ‘r’ of the centre of mass of the system from the point C is given by

r = (m₁r₁ + m₂r₂)/(m₁ + m₂) = (m×2L + 2m×L)/(m + 2m) = 4L/3

[ Note that we have used the equation, r = (m₁r₁ + m₂r₂)/(m₁ + m₂) for the position vector r of the centre of mass in terms of the position vectors r₁ and r₂ of the point masses m1 and m₂. We could use the simple equation involving the distances from C since the points are collinear].