A Uniform Rod

Q. On applying force on a rod along its length, the rod gets elongated by $0.04\text{m}$. If the length of rod is doubled and diameter is also doubled and same force is applied along the length, and it is found that the rod gets elongated by $x\times {10}^{-2}\text{m}$. Find the value of $x$.

1. $2$
2. $3$
3. $4$
4. $5$

Q. A uniform ring having mass $m$, radius $R$, cross sectional area of the wire $A$ and Young's modulus $Y$ is rotating with an angular speed $\omega$ (where $\omega$ is small) about an axis passing through centre and perpendicular to plane of ring on a smooth horizontal surface. Which of the following options are correct?

1. Tension in the wire is $\frac{mR{\omega }^2}{2\pi }$.
2. Change in length of the wire is $\frac{m{R}^2{\omega }^2}{2A.Y}$.
3. Change in radius of the ring is $\frac{m{R}^2{\omega }^2}{2\pi AY}$.
4. Elastic potential energy stored is $\frac{1}{4\pi }\left(\frac{m^2{\omega }^4{R}^3}{AY}\right)$.

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 thin bar of length $L$ has a mass per unit length $\lambda$, that increases linearly with distance form one end. If its total mass is $M$ and its mass per unit legth at the lighter end is ${\lambda }_0$, then the distance of the center of mass from the lighter end is:

1. $\frac{L}{2}-\frac{{\lambda }_0{L}^2}{4M}$
2. $\frac{2L}{3}-\frac{{\lambda }_0{L}^2}{6M}$
3. $\frac{L}{3}+\frac{{\lambda }_0{L}^2}{8M}$
4. $\frac{L}{3}+\frac{{\lambda }_0{L}^2}{4M}$

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 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. Calculate the centre of mass of a non-uniform rod whose linear mass density $\left(\lambda \right)$ varies as $\lambda =\frac{{\lambda }_o}{L}{x}^2$, where ${\lambda }_0$ is a constant, $L$ is the length of the rod and $x$ distance is measured from one end of the rod..

Q. Moon is revolving round the earth as well as it is rotating about its own axis. The ratio of its angular momentum in two cases will be (orbital radius of moon $=3.82\times {10}^{8}m$ and radius of moon $=1.74\times {10}^{6}m$):

1. $1.2\times {10}^{5/4}$
2. $1.2\times {10}^{5/3}$
3. $1.22\times {10}^{5/2}$
4. $1.2\times {10}^5$

Q. An instantaneous displacement of a simple harmonic oscillator is $x=A\cos (\omega t+\frac{\pi }{4})$. Its speed will be maximum at

1. $\frac{\pi }{4\omega }$
2. $\frac{\pi }{2\omega }$
3. $\frac{\pi }{\omega }$
4. $\frac{2\pi }{\omega }$

Q. For the one-dimensional motion, described by x = t - sint

1. x(t) > 0 for all t > 0
2. v(t) > 0 for all t > 0
3. a(t) > 0 for all t > 0
4. all of these

Q. The centre of mass of a non uniform rod of length $L$ whose mass per unit length varies as $\rho =k{x}^2/L$ (where $k$ is a constant and $x$ is the distance measured from one end) is at what distance from the same end?

1. $3L/4$
2. $L/4$
3. $2L/3$
4. $L/3$

Q. A wheel has a speed of 1200 revolutions per minute and is made to slow down at a rate of 4 rad/s${}^2$. The number of revolutions it makes before coming to rest is:

1. 272
2. 314
3. 722
4. 143

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

Q. A car weighing $1000kg$ is going up an incline with a slope of ${\sin }^{-1}\left(2/25\right)$ at a steady speed of $18kmph$. If $g=10m{s}^{-2}$, the power of its engine is:

1. $4kW$
2. $50kW$
3. $625kW$
4. $25kW$

Q. A barometer reads 76 cm of mercury. If the tube is gradually inclined at an angle of ${60}_0$ with gertical, keeping the open end immersed in the reservoir, the length of the mercury column will be :

1. 152 cm
2. 76 cm
3. 38 cm
4. $38\sqrt{3}cm$

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. From the uniform disc of radius $R$, an equilateral triangle of side $\sqrt{3}R$ is cut as shown in the figure. The new position of the centre of mass is :

1. $(0,0)$
2. $(0,R)$
3. $(0,\frac{\sqrt{3}R}{2})$
4. $Noneofthese$

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 smooth light horizontal rod $AB$ can rotate about a vertical axis passing through its end $A$. The rod is fitted with a small sleeve of mass $m$ attached to the end $A$ by a weightless spring of length $l_0$ and stiffness $\kappa$. What work must be performed to slowly get this system going and reaching the angular velocity $\omega$?

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. Find the centre of mass of a non uniform rod of length $L$, whose mass per unit length varies as $\rho =\frac{k\cdot x^2}{L}$ (where $k$ is a constant and $x$ is the distance of any point from one end) from the same end.

1. $\frac{3}{4}L$
2. $\frac{1}{4}L$
3. $\frac{2}{3}L$
4. $\frac{1}{6}L$

Q. Figure $11-45$ shows a rigid structure consisting of a circular hoop of radius $R$ and mass $m$ and a square made of four thin bars, each of length $R$ and mass $m$. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of $2.5s$. Assuming $R=0.50m$ and $m=2.0kg$, calculate its angular momentum about that axis.

Q. Where is the center of mass, $x$?

1. $\frac{3\ell }{5}$
2. $\frac{6\ell }{5}$
3. $\frac{4\ell }{5}$
4. $\frac{7\ell }{5}$

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