A Uniform Rod

Q.

The density of a linear rod of length L varies as
$\rho$ = A + Bx where x is the distance from the left end.
Locate the centre of mass.

1. $\frac{2AL+3B{L}^2}{2(3A+BL)}$

2. $\frac{3AL+4B{L}^2}{2(3A+2BL)}$

3. $\frac{3AL+2B{L}^2}{3(2A+BL)}$

4. $\frac{3AL+B{L}^2}{(2A+BL)}$

Q. A solid object is generated by the rotation of a parabola as shown in the figure. Assuming that the height of object is ℎ as shown in figure, the location of centre of mass of such a paraboloid (from 𝑂) of uniform density formed by rotating a parabola $y=k{x}^2$ about 𝑥−𝑎𝑥𝑖𝑠 is

1. $y_{COM}=\frac{h}{6}$
2. $y_{COM}=\frac{h}{2}$
3. $y_{COM}=\frac{h}{3}$
4. $y_{COM}=\frac{2h}{3}$

Q. A uniform rod of length $5\text{m}$ is placed along $x$ - axis as shown in the figure. The position of centre of mass of the rod from the origin is

1. $2.5\text{m}$
2. $5\text{m}$
3. $7.5\text{m}$
4. $10\text{m}$

Q. A long thin bar of length $L$ is made of material whose density varies along the length of the bar. Let $x$ be the distance from one end of the bar. If mass density of bar is given by $P\left(kg/m\right)=a{x}^2,0\le x\le L;$ where $x$ is in meter, find the centre of mass of bar.

1. $\frac{2L}{3}$
2. $\frac{3L}{4}$
3. $\frac{L}{2}$
4. L

Q. A wire of mass $m\text{kg}$ with uniform cross section is bent in the shape as shown in figure. If origin is taken at $\text{O}$, find the co-ordinates of the center of mass of the given system (in metres).

1. $(\frac{15}{14},\frac{12}{14})$
2. $(\frac{15}{7},\frac{12}{7})$
3. $(2,2)$
4. $(\frac{12}{7},\frac{15}{7})$

Q. A rod of length $10\text{m}$ is inclined on the wall at an angle ${37}^{\circ }$ with horizontal.


Find out position of centre of mass of the rod assuming the wall to be along $y-$ axis and foot of the wall as the origin.

1. $(5\text{m},4\text{m})$
2. $(4\text{m},5\text{m})$
3. $(4\text{m},3\text{m})$
4. $(5\text{m},3\text{m})$

Q. A semi circular arc of radius r and a straight wire along the diameter, both are carrying same current i. Find out magnetic force per unit length on the small element P, which is at the centre of curvature?

1. $\left(\frac{2{\mu }_0{i}^2}{r}\right)$
2. $\left(\frac{{\mu }_0{i}^2}{2r}\right)$
3. $\left(\frac{{\mu }_0{i}^2}{4r}\right)$
4. $\left(\frac{{\mu }_0{i}^2}{r}\right)$

Q. In Fig. 9-37, three uniform thin rods, each of length $L=22cm,$ form an inverted U. The vertical rods each have amass of $14g$ ; the horizontal rod has amass of $42g.$ What are (a) the x coordinate and (b) the y coordinate of the system’s center of mass?

Q. A $YO-YO$ us a toy in which a string is wound round a central shaft as shown in figure. The string unwinds and rewinds itself alternately making the $YO-YO$ rises and fall. The ratio of the tensions in the string during descent and ascent is:

1. $1:1$
2. $r_2:r_1$
3. $r_1:r_2$
4. $r_1:1$

Q. A uniform rod of length $5\text{m}$ is placed along $x$ - axis as shown in the figure. The position of centre of mass of the rod from the origin is

1. $2.5\text{m}$
2. $5\text{m}$
3. $7.5\text{m}$
4. $10\text{m}$