Applications of Radius of Curvature

Q. For a given surface the Gauss's law is stated as $\oint E.ds=0$. From this we can conclude that

1. E is necessarily zero on the surface
2. The total flux through the surface is zero
3. E is perpendicular to the surface at every point
4. The flux is only going out of the surface

Q. The speed of earth's rotation about its axis is $\omega$. Its speed is increased to $x$ times to make the effective acceleration due to gravity equal to zero at the equator, then the value of $x$ is $(g=10{\text{ms}}^{-2};R=6400\text{km})$.

1. $1$
2. $8.5$
3. $17$
4. $34$

Q. A motor cyclist moving with a velocity of $72\text{km/h}$ on a flat frictionless road takes a turn on the road at a point where the radius of curvature of the road is $20\text{m}$. The acceleration due to gravity is $10{\text{m/s}}^2$. In order to avoid skidding, he must bend with respect to the vertical plane by an angle

1. ${\tan }^{-1}\left(2\right)$
2. ${\cos }^{-1}\left(0.3\right)$
3. ${\sin }^{-1}\left(0.2\right)$
4. ${\tan }^{-1}\left(3\right)$

Q. Railway tracks are banked on curves so that:

1. no frictional force is produced.
2. the train may not fall inwards.
3. necessary centripetal force may be obtained from the horizontal component of normal reaction due to the track
4. the tracks do not crack.

Q. When the road is dry and coefficient of static friction is $\mu$, the maximum speed of a car in the circular path is $10m{s}^{-1}$. If the road becomes wet and ${\mu }^{\prime }=\frac{\mu }{2}$, what is the maximum permitted speed for the car?

1. $5m{s}^{-1}$
2. $10m{s}^{-1}$
3. $10\sqrt{2}m{s}^{-1}$
4. $5\sqrt{2}m{s}^{-1}$

Q. What should be the angular speed with which the earth have to rotate on its axis so that a person on the equator would weigh $\frac{3}{5}$ as much as present ?

1. $\sqrt{\frac{2g}{5R}}$
2. $\sqrt{\frac{5g}{2R}}$
3. $\sqrt{\frac{g}{R}}$
4. $\sqrt{\frac{7g}{2R}}$

Q. A wheel is rotating freely with an angular speed $\omega$ on a shaft. The moment of inertia of the wheel is $I$ and the moment ot inertia of the shaft is negligible. Another wheel of moment of inertia $3I$ initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is

1. $\frac{5}{6}$
2. $\frac{1}{4}$
3. $0$
4. $\frac{3}{4}$

Q. The figure shows the velocity and acceleration of a point like body at the initial moment of its motion. The acceleration vector of the body remains constant. The minimum radius of curvature of trajectory of the body is (in $\text{m}$ upto two decimals.)

Q.

If the radius of curvature of the path of two particles of same masses are in the ratio 1 : 2, then in order to have constant centripetal force, their velocity, should be in the ratio of

1. $\sqrt{2}:1$

2. $4:1$

3. $1:\sqrt{2}$

4. $1:4$

Q. At certain place on railway track, the radius of curvature of railway track is $200\text{m}$. If the distance between the rails is $1.6\text{m}$, and the outer rail is raised by $0.08\text{m}$ above the inner rail, find the speed of train for which there is no side pressure of the rails.
(Take $g=10{\text{m/s}}^2$)

1. $5\text{m/s}$
2. $10\text{m/s}$
3. $15\text{m/s}$
4. $20\text{m/s}$

Q. A uniform rod is hinged, as shown in the figure. It is released from horizontal position. The angular velocity of the rod, as it passes through the vertical position, is :

1. $\sqrt{\frac{4g}{l}}$
2. $\sqrt{\frac{2g}{3l}}$
3. $\sqrt{\frac{24g}{7l}}$
4. $\sqrt{\frac{3g}{7l}}$

Q. Suppose a smooth tunnel is dug along a straight line joining two points on the surface of the earth and a particle is dropped from rest at its one end. Assume that mass of earth is uniformly distributed over its volume. Then

1. the particle will emerge from the other end with velocity $\sqrt{\frac{G{M}_e}{2R_e}}$
   where, $M_e$ and $R_e$ are earth's mass and radius respectively
   
2. the particle will come to rest at the centre of the tunnel because at this position, the particle is closest to the earth's centre.
3. potential energy of the particle will always be equal to zero at centre of the tunnel if it is along a diameter.
4. acceleration of the particle will be proportional to its distance from the mid-point of the tunnel.

Q. A stationary object is released from a point $\text{P}$ at a distance $3R$ from the center of the moon which has radius $R$ and mass $M$. Which of the following are the correct expression for speed of the object on hitting the moon?

1. ${\left(\frac{2GM}{3R}\right)}^{\frac{1}{2}}$
2. ${\left(\frac{4GM}{3R}\right)}^{\frac{1}{2}}$
3. ${\left(\frac{GM}{3R}\right)}^{\frac{1}{2}}$
4. ${\left(\frac{GM}{R}\right)}^{\frac{1}{2}}$

Q. A train $A$ runs from east to west and another train $B$ of the same mass and speed runs from west to east along the equator. $A$ presses the track with a force $F_1$ and $B$ with $F_2$. Then:

1. $F_1=F_2$
2. $F_1<F_2$
3. $F_1>F_2$
4. $F_1\le F_2$

Q. The maximum tension that an inextensible ring of radius $1\text{m}$ and mass density $0.1{\text{kgm}}^{-1}$ can bear is $40\text{N}$. The maximum angular velocity with which it can be rotated in a circular path is

1. $20\text{rad/s}$
2. $18\text{rad/s}$
3. $16\text{rad/s}$
4. $15\text{rad/s}$

Q. A motor cyclist moving with a velocity of $72\text{km/hr}$ on flat road takes a turn on the road at a point where the radius of curvature of the road is $20\text{m}$. The acceleration due to gravity is $10{\text{m/s}}^2$. In order to avoid skidding, he must not bend with respect to the vertical plane by an angle greater than

1. $\theta ={\tan }^{-1}6$
2. $\theta ={\tan }^{-1}4$
3. $\theta ={\tan }^{-1}2$
4. $\theta ={\tan }^{-1}25.92$

Q. A car is moving on a circular level road of radius of curvature $300\text{m}$. If the coefficient of friction is $0.5$ and taking $g=10{\text{m/s}}^2$, the maximum speed the car can have (in $\text{km/hr}$) is

1. $278.9\text{km/h}$
2. $0.45\text{km/h}$
3. $139.4\text{km/h}$
4. $1\text{km/h}$

Q. A uniform circular disc of mass $50\text{kg}$ and radius 0.4 m is rotating with an angular velocity of $10{\text{rad s}}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass $6.25\text{kg}$ and radius 0.2 m, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity $\left({\text{in rads}}^{-1}\right)$ of the system is

Q. If the angular momentum of a particle of mass $m$ rotating along a circular path of radius $r$ with uniform speed is $L$, the centripetal force acting on the particle is

1. $\frac{L^2}{m{r}^3}$
2. $\frac{L^2}{m{r}^2}$
3. $\frac{L}{m{r}^3}$
4. $\frac{L}{m^2{r}^3}$

Q. Two satellites $S_1$ and $S_2$ revolve around a planet in coplanar circular orbits in the opposite sense. The periods of revolutions are $T$ and $nT$ respectively. Find the angular speed of $S_2$ as observed by an astronaut in $S_1$ , when they are closest to each other.

1. $\frac{2\pi (n^{\frac{1}{3}}+1)}{T(n^{\frac{1}{3}}-1)}$
   
2. $\frac{2\pi (n^{-\frac{1}{3}}+1)}{T(n^{\frac{2}{3}}-1)}$
   
3. $\frac{\pi (n^{\frac{1}{3}}+1)}{T(n^{\frac{2}{3}}-1)}$
4. $\frac{2\pi (n^{\frac{2}{3}}+1)}{T(n^{\frac{1}{3}}-1)}$

Q. A small ball of mass $m$ starts from rest from point $A(b,c)$ on a smooth slope which is a parabola. The normal force that the ground exerts at the instant, the ball arrives at lowest point $(0,0)$ is

1. $\frac{4mg{c}^2}{b^2}$
2. $mg\left(\frac{b^2+4c^2}{b^2}\right)$
3. $mg$
4. $3mg$

Q. In the figure shown $ADB$ and $BEF$ are two fixed circular paths. A block of mass $m$ enters in the tube $ADB$ through point $A$ with minimum velocity to reach point $B$. From there the block moves on another circular path of radius $R^{{}^{\prime }}$, where it is just able to complete the circle.
Choose the correct statements.

1. Velocity at $A$ must be $\sqrt{4Rg}$
2. Velocity at $A$ must be $\sqrt{2Rg}$
3. $\frac{R^{{}^{\prime }}}{R}=\frac{2}{3}$
4. The normal reaction at point $E$ is $6\text{mg}$

Q. A cyclist riding the bicycle at a speed of $14\sqrt{3}\text{m/s}$ takes a turn around a circular road of radius $20\sqrt{3}\text{m}$ without skidding. What is his inclination to the vertical?
[Take $g=9.8{\text{m/s}}^2$]

1. ${45}^{\circ }$
2. $0^{\circ }$
3. ${60}^{\circ }$
4. ${30}^{\circ }$

Q. A cycle can bend upto a maximum angle of ${45}^{\circ }$. If the radius of the bend is $10\text{m}$, then what should be the maximum speed of the cyclist so that he does not crash?

1. $10\text{m/s}$
2. $15\text{m/s}$
3. $10\text{km/h}$
4. $15\text{km/h}$

Q. A motorcycle moving with a uniform speed of $72{\text{kmh}}^{-1}$ on a flat road takes a turn on the road, at a point where the radius of curvature of the road is $20\text{m}$. The acceleration due to gravity is $10{\text{ms}}^{-2}$. In order to avoid skidding, he must not bend with respect to the vertical plane by an angle greater than:

1. $\theta ={\tan }^{-1}\left(2\right)$
2. $\theta ={\tan }^{-1}\left(6\right)$
3. $\theta ={\tan }^{-1}\left(4\right)$
4. $\theta ={\tan }^{-1}\left(25.92\right)$

Q. A particle slides along a track with elevated ends and a flat central part, as shown in figure. The flat portion $BC$ has a length $l=3.0\text{m}$. The curved portions of the track are frictionless. For the flat part of the track the coefficient of kinetic friction is ${\mu }_k=0.20$, If the particle is released at point $A$ which is at height $h=1.5\text{m}$ above the flat part of the track, then Where does the particle finally comes to rest w.r.t $B$ ?

1. $2.5\text{m}$
2. $1.5\text{m}$
3. $3.5\text{m}$
4. $4.5\text{m}$

Q. An object slides without friction from the height $25\text{cm}$ and then goes around the vertical loop of radius $10\text{cm}$ from which a symmetrical section of angle $2\alpha$ has been removed. Find the angle $\alpha$ such that after losing contact at $A$ and flying through air, the object will reach point $B$.

1. ${90}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. A particle moves in circular path of radius $R$. If centripetal force $F$ is kept constant but the angular velocity is doubled, the new radius of the path will be

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

Q. A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius R (< L) is attached at its center to the free end of the rod. Consider two ways the disc is attached. ((case A): The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

6

Q. A railway carriage box has its geometric centre (i.e we will assume the force due to gravity to act at this point) at a height of $1\text{m}$ above the rails, which are $1.5\text{m}$ apart. The maximum safe speed at which it could travel round an unbanked curve of radius $100\text{m}$ is

1. $12m{s}^{-1}$
2. $18m{s}^{-1}$
3. $27.11m{s}^{-1}$
4. $22m{s}^{-1}$