The coordinates of the position of masses \(m_1=7\) gm, \(m_2=4\) gm, \(m_3=10\) gm are \(\vec r_1=(\hat i+5\hat j-3\hat k),\) \(\vec r_2=(2\hat i+5\hat j+7\hat k),\) \(\vec r_3=(3\hat i+3\hat j-\hat k)\) respectively in cm. The position of the centre of mass of the system would be:
1. \(\left(-\frac{15}{7}, \frac{85}{17}, \frac{1}{7}\right) \text{cm}\)
2. \(\left(\frac{15}{7},-\frac{85}{17}, \frac{1}{7}\right) \text{cm}\)
3. \(\left(\frac{15}{7}, \frac{85}{21},-\frac{1}{7}\right)\text{cm}\)
4. \(\left(\frac{15}{7}, \frac{85}{21}, \frac{7}{3}\right)\text{cm}\)

A horizontal heavy uniform bar of weight \(W\) is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to:

The moment of inertia of a thin uniform circular disc about one of its diameter is I. Its moment of inertia about an axis perpendicular to the circular surface and passing through its center will be:

1. $\sqrt{2}I$

2. 2 l

3. $\frac{I}{2}$

4. $\frac{I}{\sqrt{2}}$

A child is standing on the edge of a merry-go-round that has the shape of a disk, as shown in the figure. The mass of the child is 40 kilograms. The merry-go-round has a mass of 200 kilograms and a radius of 2.5 meters, and it is rotating with an angular velocity of $\omega =2.0$ radians per second. The child then walks slowly towards the center of the merry-go-round. When the child reaches the center, what is the angular velocity of the disc? (The size of the child can be neglected.)
     
1. 2.0 rad/s

2. 2.2 rad/s

3. 2.4 rad/s

4. 2.8 rad/s

A hollow cylinder with an inner radius of R, an outer radius of 2R, and a mass M is rolling without slipping at the speed of its centre v. Its kinetic energy will be:
  

1. $\frac{11}{16}M{v}^2$

2. $\frac{7}{4}M{v}^2$

3. $\frac{13}{16}M{v}^2$

4. None of these

A uniform rod of mass 2M is bent into four adjacent semicircles, each of radius r, all lying in the same plane. The moment of inertia of the bent rod about an axis through one end A and perpendicular to the plane of the rod is:
      

1. 22 ${\mathrm{Mr}}^2$

2. 88 ${\mathrm{Mr}}^2$

3. 44 ${\mathrm{Mr}}^2$

4. 66 ${\mathrm{Mr}}^2$

A man '\(A\)', mass \(60\) kg, and another man '\(B\)', mass \(70\) kg, are sitting at the two extremes of a \(2\) m long boat, of mass \(70\) kg, standing still in the water as shown. They come to the middle of the boat. (Neglect friction). How far does the boat move on the water during the process?

A uniform rod of length 1 m and mass 2 kg is suspended by two vertical inextensible strings as shown in following figure. Calculate the tension T (in newtons) in the left string at the instant when the right string snaps (g = 10 m/$s^2$).
       
1. 2.5 N

2. 5 N

3. 7.5 N

4. 10 N

A ring and a disc with the same moment of inertia roll along a plane surface at the same speed. If ${\mathrm{E}}_{\mathrm{r}}$ be the rotational kinetic energy of the ring and ${\mathrm{E}}_{\mathrm{d}}$ be that of the disc, then:

1. ${\mathrm{E}}_{\mathrm{r}}$>${\mathrm{E}}_{\mathrm{d}}$

2. ${\mathrm{E}}_{\mathrm{r}}$<${\mathrm{E}}_{\mathrm{d}}$

3. ${\mathrm{E}}_{\mathrm{r}}$=${\mathrm{E}}_{\mathrm{d}}$

4. The relation depends upon the radii of the ring and disc