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 centre of mass of this system has a position vector
$-\hat i\,\, + \,\,\hat j\,\, + \,\,\hat k$
$-2\hat i\,\,+ \,\,2\hat k$
$-2\hat i\,\, - \,\,\hat j\,\, + \,\,\hat k$
$2\hat i\,\, - \,\,\hat j\,\, - \,\,2\hat k$
The linear mass density of a rod of length $L$ varies as $\lambda = kx^2$, where $k$ is a constant and $x$ is the distance from one end. The position of centre of mass of the rod is
A thin rod of length $L$ and mass $M$ is bent at its mid-point into two halves so that the angle between them is $90^o$. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is
A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega $ . Two objects each of mass $m$ are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity
A constant torque acting on a uniform circular wheel changes its angular momentum from $L_0$ to $4L_0$ in $4\,s$ . The magnitude of this torque is
A mass $‘m’$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?