Continuous Bodies: Standard Results & Variable Density

The center of mass of a thin rectangular plate (fig - x) with sides of length $a$ and $b$, whose mass per unit area $(\sigma)$ varies as $\sigma = \frac{\sigma_0 x}{ab}$ (where $\sigma_0$ is a constant), would be __________.

The distance of centre of mass from end A of a one dimensional rod (AB) having mass density $\rho = \rho_0\left(1 - \frac{x^2}{L^2}\right)\text{ kg/m}$ and length L (in meter) is $\frac{3L}{\alpha}\text{ m}$. The value of $\alpha$ is ___________. (where x is the distance from end A)

The position of the centre of mass of a uniform semi-circular wire of radius 'R' placed in x-y plane with its centre at the origin and the line joining its ends as x-axis is given by $\left(0, \frac{xR}{\pi}\right)$. Then, the value of | x | is ______________.

The disc of mass M with uniform surface mass density $\sigma$ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position $\frac{x}{3}\frac{a}{\pi}, \frac{x}{3}\frac{a}{\pi}$ where x is _____________. (Round off to the Nearest Integer).

[a is an area as shown in the figure]

The centre of mass of solid hemisphere of radius 8 cm is x from the centre of the flat surface. Then value of x is __________.

A rod of length L has non-uniform linear mass

density given by $\rho$(x) = $a + b\left(\frac{x}{L}\right)^2$, where a

and b are constants and 0 $\leq$ x $\leq$ L. The value

of x for the centre of mass of the rod is at:

Distance of the center of mass of a solid uniform cone from its vertex is $z_0$. If the radius of its base is $R$ and its height is $h$ then $z_0$ is equal to:

A uniform thin rod AB of length L has linear mass density $\mu$ (x) = a + $\frac{bx}{L},$ where x is measured from A. If the CM of the rod lies at a distance of $\left(\frac{7}{12}L\right)$ from A, then a and b are related as:

A thin bar of length L has a mass per unit length $\lambda$, that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is $\lambda$o, then the distance of the center of mass from the lighter end is:

A thin rod of length $'L'$ is lying along the $x$-axis with its ends at $x = 0$ and $x = L$. Its linear density (mass/length) varies with $x$ as $k\left(\frac{x}{L}\right)^n,$ where $n$ can be zero or any positive number. If the position $X_{CM}$ of the center of mass of the rod is plotted against $'n',$ which of the following graphs best approximates the dependence of $X_{CM}$ on $n$?

A rod of length is 3 m and its mass acting per unit length is directly proportional to distance x from one of its end then its centre of gravity from that end will be at: