Expression for Torsional Pendulum

Q. Two masses $m$ and $m/2$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $K$ at the centre of mass of the rod - mass system as shown in the figure.
Because of torsional constant $K$, the restoring torque is $\tau =K\theta$ for angular displacement $\theta .$
If the rod is rotated by ${\theta }_{\circ }$ and released so that rod oscillates. The tension in the rod, when it passes through its mean position will be

1. $\frac{3K{\theta }_{\circ }^2}{l}$
2. $\frac{K{\theta }_{\circ }^2}{l}$
3. $\frac{2K{\theta }_{\circ }^2}{l}$
4. $\frac{K{\theta }_{\circ }^2}{2l}$

Q. A pendulum clock keeps correct time at $0^{o}C$. Its mean coefficient of linear expansions is $\frac{\alpha }{{}^{o}C}$, then the loss in seconds per day by the clock if the temperature rises by $t^{o}C$ is

1. 

2. 

3. 

4. 

Q.

A uniform rod of length $l$ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed$\omega$ the rod makes an angle $\theta$ with it (see figure). To find$\theta$ equate the rate of change of angular momentum (direction going into the paper) $\left(\raisebox{1ex}{$m{l}^2$}\!\left/ \!\raisebox{-1ex}{$12$}\right.\right){\omega }^2\sin \theta \cos \theta$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces $F_H$ and $F_v$ about the CM. The value of $\theta$is then such that:

1. $\cos \theta =\frac{2g}{3l{\omega }^2}$

2. $\cos \theta =\frac{3g}{2l{\omega }^2}$

3. $\cos \theta =\frac{g}{2l{\omega }^2}$

4. $\cos \theta =\frac{g}{l{\omega }^2}$

Q.

Unit of torsional rigidity?

Q. Torsional stiffness is defined as the torque required to produce a unit angle of twist.

1. False
2. True

Q. A uniform disc of radius $5.0\text{cm}$ and mass $200\text{g}$ is fixed at its centre to a metal rod, the other end of which is fixed to a celling. The hanging disc is rotated about the rod through an angle and is released. If the disc makes torsional oscillations with time period $0.20\text{s}$, find the torsional constant of the rod.

1. $0.50{\text{kg m}}^2{\text{/s}}^2$
2. $0.75{\text{kg m}}^2{\text{/s}}^2$
3. $0.125{\text{kg m}}^2{\text{/s}}^2$
4. $0.25{\text{kg m}}^2{\text{/s}}^2$

Q. The given figure shows the cross section of a uniform pencil placed on a rough platform. The cross-section of the pencil is a hexagon of side $‘a^{\prime }$. The platform starts performing S.H.M. perpendicular to the length of the pencil in horizontal plane with angular frequency $‘{\omega }^{\prime }$. There is sufficient friction between the pencil and the platform such that there is no slipping between them. The maximum amplitude of oscillations so that the pencil does not topple is $\frac{g}{\sqrt{\alpha }{\omega }^2}$. Find $\alpha$

Q. A thin uniform rod $AB$ of mass $m$ undergoes pure translation with an acceleration $a$ towards right when two antiparallel forces act on it as shown in the figure. If the distance between $F_1$ and $F_2$ is $b$, then the length of the rod is

1. $\frac{2F_{2}b}{ma}$
2. $\frac{2F_{1}b}{ma}$
3. $\frac{4F_{2}b}{ma}$
4. $\frac{4F_{1}b}{ma}$

Q. Two blocks of masses $m_1$ and $m_2$ ($m_1>m_2$), are performing SHM together with the same amplitude and same time period as shown. Surface between $m_1$ and the ground is smooth, while between $m_1$ and $m_2$, the coefficient of friction is $\mu$. Given that $\frac{k_1}{m_1}>\frac{k_2}{m_2}$. Choose the correct option:

1. Time period of SHM is $2\pi \sqrt{\frac{m_1{m}_2}{(m_1+m_2)(k_1+k_2)}}$.
2. Time period of SHM is $2\pi \sqrt{\frac{(m_1+m_2)(k_1+k_2)}{k_1{k}_2}}$.
3. Time period of SHM is $2\pi \sqrt{\frac{m_1+m_2}{k_1+k_2}}$.
4. Maximum possible amplitude of this SHM is $\frac{2\mu m_{2}g(m_1+m_2)}{k_1{m}_2-k_2{m}_1}$

Q. A rod is released from the position as shown in figure. If there is sufficient friction between the rod and the plane to prevent slipping, the minimum value of friction co-efficient between rod and ground is

1. $0.6$
2. $0.4$
3. $0.5$
4. $0.3$

Q. A spring with no mass attached to it hangs from a rigid support. A mass m is now hung on the lower end to the spring. The mass is supported on a platform so that the spring remains relaxed. The supporting platformis then suddenly removed and the mass begins to oscillate. The lowest position of the mass during the oscillation is 5 cm below the place where it was resting on the platform. What is the angular frequency of oscillation? Take g = 10 $m{s}^{-2}$

1. $10rad{s}^{-1}$
2. $20rad{s}^{-1}$
3. $30rad{s}^{-1}$
4. $40rad{s}^{-1}$

Q. A uniform disc of radius $5.0\text{cm}$ and mass $200\text{g}$ is fixed at its centre to a metal rod, the other end of which is fixed to a celling. The hanging disc is rotated about the rod through an angle and is released. If the disc makes torsional oscillations with time period $0.20\text{s}$, find the torsional constant of the rod.

1. $0.50{\text{kg m}}^2{\text{/s}}^2$
2. $0.75{\text{kg m}}^2{\text{/s}}^2$
3. $0.125{\text{kg m}}^2{\text{/s}}^2$
4. $0.25{\text{kg m}}^2{\text{/s}}^2$

Q. The time period of oscillation of a torsional pendulum of moment of inertia I is

1. $T=2\pi \sqrt{I/k}$
2. $T=2\pi \sqrt{I/2k}$
3. $T=2\pi \sqrt{2I/k}$
4. $T=2\pi \sqrt{I/4k}$

Q. Two masses $m$ and $m/2$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $K$ at the centre of mass of the rod - mass system as shown in the figure.
Because of torsional constant $K$, the restoring torque is $\tau =K\theta$ for angular displacement $\theta .$
If the rod is rotated by ${\theta }_{\circ }$ and released so that rod oscillates. The tension in the rod, when it passes through its mean position will be

1. $\frac{3K{\theta }_{\circ }^2}{l}$
2. $\frac{2K{\theta }_{\circ }^2}{l}$
3. $\frac{K{\theta }_{\circ }^2}{l}$
4. $\frac{K{\theta }_{\circ }^2}{2l}$

Q. If $I^{}$ is the moment of inertia of a thin rod about an axis perpendicular to its length and passing through its centre of mass, and $I_2$ is the moment of inertia (about central axis) of the ring formed by bending the rod, then

1. $I_1:I_2=1:1$
2. $I_1:I_2=\pi :4$
3. $I_1:I_2={\pi }^2:3$
4. $I_1:I_2=3:5$

Q. A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its cross- sectional area is $4.9\times {10}^{-7}{m}^2$ If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad $s^{-1}$ If the Young's modulus of the material of the wire is $n\times {10}^{9}N{m}^{-2}$, the value of n is

1. 4
2. 1
3. 2
4. 8

Q. A thin rod of length $l$ in the shape of a semicircle is pivoted at one of its ends such that it is free to oscillate in its own plane. Find the frequency $f$ of small oscillations of the semicircular rod.

Q. A uniform disc of radius $5.0\text{cm}$ and mass $200\text{g}$ is fixed at its centre to a metal rod, the other end of which is fixed to a celling. The hanging disc is rotated about the rod through an angle and is released. If the disc makes torsional oscillations with time period $0.20\text{s}$, find the torsional constant of the rod.

1. $0.50{\text{kg m}}^2{\text{/s}}^2$
2. $0.75{\text{kg m}}^2{\text{/s}}^2$
3. $0.125{\text{kg m}}^2{\text{/s}}^2$
4. $0.25{\text{kg m}}^2{\text{/s}}^2$

Q. Suppose the amplitude of a simple pendulum having a bob of mass $m$ is ${\theta }_0$. Find the tension in the string when the bob is at its extreme position.

Q. A small hole is made in a disc of mass $M$ and radius $R$ at a distance $R/4$ from the centre. The disc is supported on a horizontal peg through this hole. The moment of inertia of the disc about horizontal peg is

1. $\frac{5}{16}M{R}^2$
2. $\frac{M{R}^2}{9}$
3. $\frac{9}{16}M{R}^2$
4. $\frac{5}{4}M{R}^2$

Q. In the figure shown, the springs are connected to the rod at one end and at the midpoint. The rod is hinged at its lower end. Rotational SHM of the rod (Mass $m$, length $L$) will occur only if

1. $k>mg/3L$
2. $k>2mg/3L$
3. $k>0$
4. $k>2mg/5L$

Q. Moment of inertia of a rod of mass M and length L about an axis passing through a point midway between centre and end is.

1. $\frac{M{L}^2}{6}$
2. $\frac{M{L}^2}{12}$
3. $\frac{7M{L}^2}{24}$
4. $\frac{7M{L}^2}{48}$

Q. An uniform rod of mass $3\text{kg}$ is hinged at the wall and connected through a light string as shown in the figure. Find the tension in the string.(Take $g=10{\text{m/s}}^2$).

1. $30\text{N}$
2. $15\text{N}$
3. $5\text{N}$
4. $10\text{N}$

Q. A $0.1$ kg mass is suspended from a wire of negligible mass. The length of the wire is $1$m and its cross-sectional area is $4.9\times {10}^{-8}{m}^2$. If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency $140$ rad $s^{-1}$. If the Young's modulus of the material of the wire is $n\times {10}^9$ $N{m}^{-2}$, the value of n is?

1. 4
2. 5
3. 6
4. 7

Q. A uniform stick of length $l$ is mounted so as to rotate about a horizontal axis perpendicular to the stick and at a distance $d$ from the centre of mass. The time period of small oscillations has a minimum value when $\frac{d}{l}$ is:

1. $\frac{1}{\sqrt{2}}g$
2. $\frac{1}{\sqrt{6}g}$
3. $\frac{1}{\sqrt{12g}}$
4. $\frac{1}{\sqrt{3}}$

Q. An annular ring of internal and outer radii r and R respectively oscillates in a vertical plane about a horizontal axis perpendicular to its plane and passing through a point on its outer edge. Calculate its time period and show that the length of an equivalent simple pendulum is $\frac{3R}{2}$ as $r\to 0$ and 2R as $r\to R$.

Q. A rod has mass $M$ and length $l$. Equation of motion of the rod , which is hinged at one end, when displaced by small angular displacement $\theta$, can be expressed as (Assume air friction is zero, Moment of inertia of rod about axis passing through hinged point is $I$ and acceleration due to gravity is $g$)

1. $I\frac{d^2\theta }{d{t}^2}=-Mg\frac{l}{2}\theta$
2. $I\frac{d\theta }{dt}=Mgl\theta$
3. $I\frac{d^2\theta }{d{t}^2}=-Mgl\theta$
4. $I\frac{d^2\theta }{d{t}^2}=-\frac{Mg}{l}\theta$

Q. A circular platform is mounted on a vertical frictionless axle. Its radius is $r=2m$ and its moment inertia is $I=200kg$ $m^2$. It is initially at rest. A $70kg$ man stands on the edge of the platform and begins to walk along the edge at speed $v_0=10{ms}^{-1}$ relative to the ground. When the man has walked once around the platform so that he is at his original position on it, what is his angular displacement relative to ground

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