Chapter 5

The centre of mass of a systems of two particles is (A) on the line joining them and midway between them (B) on the line joining them at a point whose distance from each particle is proportional to the square of the mass of that particle. (C) on the line joining them at a point whose distance from each particle inversely proportional to the mass of that particle. (D) On the line joining them at a point whose distance from each particle is proportional to the mass of that particle.

Particles of $1\mathrm{gm},1\mathrm{gm},2\mathrm{gm},2\mathrm{gm}$ are placed at the corners $A,B,C,D$, respectively of a square of side $6\mathrm{cm}$ as shown in figure. Find the distance of centre of mass of the system from geometrical centre of square. (A) $1\mathrm{cm}$ (B) $2\mathrm{cm}$ (C) $3\mathrm{cm}$ (D) $4\mathrm{cm}$

Three particles of the same mass lie in the $(\mathrm{X},\mathrm{Y})$ plane, The $(X,Y)$ coordinates of their positions are $(1,1),(2,2)$ and $(3,3)$ respectively. The $(X,Y)$ coordinates of the centre of mass are $$\begin{gathered} \mathrm{A} \\ (1,2) \end{gathered}$$ $$\begin{gathered} \mathrm{B} \\ (2,2) \end{gathered}$$ $$\begin{gathered} \mathrm{C} \\ (1.5,2) \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ (2,1.5) \end{gathered}$$

Consider a two-particle system with the particles having masses ${\mathrm{M}}_1$, and ${\mathrm{M}}_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved so as to keep the centre of mass at the same position? $$\begin{gathered} \mathrm{A} \\ \left[\left({\mathrm{M}}_1\mathrm{d}\right)/({\mathrm{M}}_1+{\mathrm{M}}_2)\right] \end{gathered}$$ $$\begin{gathered} \mathrm{B} \\ \left[\left({\mathrm{M}}_2\mathrm{d}\right)/({\mathrm{M}}_1+{\mathrm{M}}_2)\right] \end{gathered}$$ $$\begin{gathered} \mathrm{C} \\ \left[\left({\mathrm{M}}_1\mathrm{d}\right)/\left({\mathrm{M}}_2\right)\right] \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ \left[\left({\mathrm{M}}_2\mathrm{d}\right)/\left({\mathrm{M}}_1\right)\right] \end{gathered}$$

From a uniform circular disc of radius $\mathrm{R}$, a circular disc of radius $\mathrm{R}/6$ and having centre at a distance $+\mathrm{R}/2$ from the centre of the disc is removed. Determine the centre of mass of remaining portion of the disc. $$\begin{gathered} \mathrm{A} \\ [(-\mathrm{R})/70] \end{gathered}$$ $$\begin{gathered} \mathrm{B} \\ [(+\mathrm{R})/70] \end{gathered}$$ $$\begin{gathered} \mathrm{C} \\ [(-\mathrm{R})/7] \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ [(+\mathrm{R})/7] \end{gathered}$$

A circular plate of uniform thickness has a diameter of $56\mathrm{cm}$. A circular portion of diameter $42\mathrm{cm}.$ is removed from tve $\mathrm{x}$ edge of the plate. Find the position of centre of mass of the remaining portion with respect to centre of mass of whole plate. $$\begin{gathered} \mathrm{A} \\ -7\mathrm{cm} \\ \mathrm{B} \\ +9\mathrm{cm} \\ \mathrm{C} \\ -9\mathrm{cm} \\ \mathrm{D} \\ +7\mathrm{cm} \end{gathered}$$

Two blocks of masses $10\mathrm{kg}$ an $4\mathrm{kg}$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives velocity of $14\mathrm{m}/\mathrm{s}$ to the heavier block in the direction of the lighter block. The velocity of the centre of mass is : $$\begin{gathered} \mathrm{A} \\ 30\mathrm{m}/\mathrm{s} \end{gathered}$$ $$\begin{gathered} \mathrm{B} \\ 20\mathrm{m}/\mathrm{s} \end{gathered}$$ $$\begin{gathered} \mathrm{C} \\ 10\mathrm{m}/\mathrm{s} \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ 5\mathrm{m}/\mathrm{s} \end{gathered}$$

A particle performing uniform circular motion has angular momentum $L$., its angular frequency is doubled and its $K.E$. halved, then the new angular momentum is $$\begin{gathered} \mathrm{A} \\ 1/2 \end{gathered}$$ \{B $\text{Extra close brace or missing open brace}$ $$\begin{gathered} \mathrm{C} \\ 2\mathrm{L} \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ 4\mathrm{L} \end{gathered}$$

A circular disc of radius $\mathrm{R}$ is removed from a bigger disc of radius $2\mathrm{R}$. such that the circumferences of the disc coincide. The centre of mass of the remaining portion is $\alpha R$ from the centre of mass of the bigger disc. The value of $\alpha$ is. $$\begin{gathered} \mathrm{A} \\ 1/2 \end{gathered}$$ \{B \} $1/6$ \{C\} $1/4$ $$\begin{gathered} \mathrm{D} \\ [(-1)/3] \end{gathered}$$

A body of mass $\mathrm{m}$ is tied to one end of spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is one centimetre. If the angular velocity is doubted, the elongation in the spring is $5\mathrm{cm}$. The original length of spring is... $$\begin{gathered} \mathrm{A} \\ 16\mathrm{cm} \end{gathered}$$ $$\begin{gathered} B \\ 15\mathrm{cm} \end{gathered}$$ $$\begin{gathered} \mathrm{C} \\ 14\mathrm{cm} \end{gathered}$$ $$\begin{gathered} \mathrm{D} \\ 13\mathrm{cm} \end{gathered}$$