B M Sharma Solutions for Chapter: Center of Mass, Exercise 1: DPP 1.1

A wire of uniform cross-section is bent in a shape as shown in the figure. The coordinates of the center of mass of each side are shown. The coordinates of the center of mass of the system is,

The $y$-coordinate of the center of mass of the system of three rods of length $2a$ and two rods of length $a$ as shown in the figure is: (Assume all rods to be of uniform density.)

A homogeneous plate, $PQRST$ is as shown in the figure. The centre of mass of plate lies at mid-point $A$ of segment $QT$. Then, the ratio of $\frac{b}{a}$ is:$\left(PQ=PT=b, QR=RS=ST=a\right)$,

Eight solid uniform cubes of edge $l$ are stacked together to form a single cube with center $\mathrm{O}$. One cube is removed from this system. The distance of the center of mass of remaining seven cubes from $\mathrm{O}$ is,

In the figure, the $L$-shaped shaded piece is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the $L$-shaped piece is,

From the circular disc of radius $4R$, two small discs of radii$R$are cut-off. The center of mass of the new structure will be,

From the circular disc of radius$4R$, two small discs of radii $R$ are cut off. The center of mass of the new structure will be: (Centre of lower circular cavity lies on$x-$axis and center of upper circular cavity lies on$y-$axis.)

A disc (of radius $r \mathrm{cm}$) of uniform thickness and uniform density $\rho$ has a square hole with sides of length, $l=\frac{r}{\sqrt{2}} \mathrm{cm}$. One corner of the hole is located at the center of the disc and center of the hole lies on $y$-axis as shown. Then, the $y$-coordinate of position of center of mass of disc with hole (in $\mathrm{cm}$) is,