Centre of mass

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is ________ m. (Take $g=10$ m/s²)

The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect to mid point ($p$) of normal bisector, as shown in the figure is ________.

Two identical bodies A and B of equal masses have initial velocities $\vec v_1=4\hat i$ m/s and $\vec v_2=4\hat j$ m/s respectively. The body A has acceleration $\vec a_1=6\hat i+6\hat j$ m/s² while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in ________ path.

In a scattering experiment, a particle of mass $2m$ collides with another particle of mass $m$, which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation $\theta$ of the heavier particle, as shown in the figure, in radians is:

If the collision occurs at time $t_0 = 0$, the value of $v_{\text{cm}}/(a\omega)$ will be _____.

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times10^{-8}\ \mathrm{N\,m\,rad^{-1}}$. The angular frequency of the oscillations in $n\times10^{-3}\ \mathrm{rad\,s^{-1}}$. The value of $n$ is _____.

Two blocks of masses 10 kg and 30 kg are placed on the same straight line with coordinates $(0,0)$ cm and $(x,0)$ cm respectively. The block of 10 kg is moved on the same line through a distance of 6 cm towards the other block. The distance through which the block of 30 kg must be moved to keep the position of centre of mass of the system unchanged is :

Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 3 m each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $\sqrt x$ m. The value of $x$ is ______.

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

Two bodies of mass 1 kg and 3 kg have position vectors $\hat i+2\hat j+\hat k$ and $-3\hat i-2\hat j+\hat k$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :

A block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x = 0$, in a co-ordinate system fixed to the table. A point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is $x$ and the velocity is $v$. At that instant, which of the following options is/are correct?

Consider regular polygons with number of sides $n = 3, 4, 5\ldots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

A rigid uniform bar AB of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\theta$. Which of the following statements about its motion is/are correct?

Two thin circular discs of mass $m$ and $4m$, having radii of $a$ and $2a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24}\,a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point 'O' is $\vec L$ (see the figure). Which of the following statement(s) is(are) true?

In the figure shown ABC is a uniform wire. If centre of mass of wire lies vertically below point A, then $\dfrac{BC}{AB}$ is close to :

Distance of the centre 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 $\mu(x)=a+\dfrac{bx}L$, where $x$ is measured from A. If the CM of the rod lies at a distance of $\left(\dfrac7{12}L\right)$ 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 centre of mass from the lighter end is :

A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg mass reverses its direction and moves with a speed of 2 m s⁻¹. Which of the following statement(s) is (are) correct for the system of these two masses?