The mass per unit length of a rod of length $l$ is given by $\lambda = \frac{M_0 x}{l}$,where $M_0$ is a constant and $x$ is the distance from one end of the rod. The position of the center of mass of the rod is:

On a horizontal frictionless frozen lake,a girl of mass $36 \,kg$ and a box of mass $9 \,kg$ are connected to each other by means of a rope. Initially,they are $20 \,m$ apart. The girl exerts a horizontal force on the box,pulling it towards her. How far has the girl travelled when she meets the box?

Write the difference between the centre of gravity and the centre of mass of a body.

Three circular discs of same material and same thickness of radii $r, 2r$ and $3r$ are placed on a horizontal plane such that their centres lie along a straight line. The radius of the middle disc is $2r$ and it touches the other two discs. The distance of the centre of mass of the system from the centre of the smaller disc is . . . . . . . (in $r$)

The linear mass density $(\lambda)$ of a rod of length $L$ kept along the $x$-axis varies as $\lambda = \alpha + \beta x$,where $\alpha$ and $\beta$ are positive constants. The centre of mass of the rod is at ..........

Two smooth and similar right-angled prisms are arranged on a smooth horizontal plane as shown in the figure. The lower prism has a mass $3$ times the upper prism. The prisms are held in an initial position as shown and are then released. As the upper prism touches the horizontal plane,the distance moved by the lower prism is