COM of solid bodies

Q.

Find the center of mass with respect to 0, of uniform semicircular wire of mass M and Radius R (as shown in figure)

1. $(0,\frac{2R}{\pi })$

2. $(0,\frac{3R}{\pi })$

3. $(0,\frac{R}{\pi })$

4. $(0,\frac{2R}{3\pi })$

Q.

We know centre of mass of a body can lie outside of the body, like in case of a ring. Now let’s take a semicircular ring whose centre is at 0, 0. Find the coordinate of it's center of mass.

1. $\frac{4R}{\pi }$

2. $\frac{8R}{\pi }$

3. $\frac{2R}{\pi }$

4. $\frac{R}{2}$

Q.

A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surfaced of a fixed table. Initially the right edge of the block is at x = 0, in a coordinate 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 option is/are correct ?

1. The velocity of the point mass m is $v=\sqrt{\frac{2gR}{1+\frac{m}{M}}}$.

2. The x component of displacement of the centre of mass of the block M is $-\frac{mR}{M+m}$.

3. The position of the point mass is x= $-\sqrt{2}\frac{mR}{M+m}$.

4. The velocity of the block M is V= $-\frac{m}{M}\sqrt{2gR}$.

Q.

Seven homo genous bricks , each of length $L$ , are arranged as shown below. Each brick is displaced with respect to the one in contact by $\frac{L}{10}$. Find $X$ coordinate of the centre of mass relative to the origin shown

1. $\frac{11L}{60}$

2. $\frac{7}{10}L$

3. $\frac{4L}{35}$

4. None of these

Q.

lf all the particles of a system lie in a cube, is it necessary that the centre of mass be in the cube?

1. Yes

2. No

3. Sometimes

4. Maybe

Q.

Half of the rectangular plate shown in figure is made of a material of density $p_1$ and the other half of density $p_2$. The length of the plate is $L$. Locate the centre of mass of the plate.

1. ${\left[\frac{2p_1+3p_2}{5(p_1+p_2)}\right]}^2$

2. ${\left[\frac{p_1+3p_2}{4(p_1+p_2)}\right]}^2$

3. ${\left[\frac{2p_1+5p_2}{3(p_1+2p_2)}\right]}^2$

4. ${\left[\frac{p_1+4p_2}{3(p_1+p_2)}\right]}^2$

Q.

Figure shows a uniform disc of radius R, from which a hole of radius $\frac{R}{2}$ has been cut out from left of the center and is placed on right of the center of disc. Find the centre of mass of the resulting disc.

1. $(\frac{R}{4},0)$

2. $(\frac{-R}{4},0)$

3. $(\frac{-R}{6},0)$

4. $(\frac{R}{6},0)$

Q.

Locate the COM of a uniform hemisherical solid with respect to 0 as shown in figure. (Radius = r)

1. $(0,\frac{4r}{3})$

2. $(0,\frac{3r}{8})$

3. $(0,\frac{3r}{5})$

4. $(0,\frac{3r}{7})$