Applications of Radius of Curvature

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

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

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. 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 car has to move on a level turn of radius $45\text{m}$. If the coefficient of static friction between the tyres and the road is ${\mu }_s=0.5$, find the maximum speed the car can achieve without skidding.

1. $54\text{km/h}$
2. $18\text{km/h}$
3. $30\text{km/h}$
4. $40\text{km/h}$

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. $22m{s}^{-1}$
4. $27.11m{s}^{-1}$

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.

A body takes the following path and moves with constant speed. If $a_A$ and $a_B$ are the magnitude of its radial acceleration at A and B, then.

1. $a_A=a_B$
2. $a_A<a_B$
3. $a_A>a_B$

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

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. An engineer has designed a road with a curve of radius $12\text{m}$. The material of the road offers a maximum coefficient of friction of $0.3$. What should be the maximum permissible speed limit on this road?

1. $7\text{m/s}$
2. $8\text{m/s}$
3. $6\text{m/s}$
4. $5\text{m/s}$

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

Q. What is the centripetal force experienced by a bike of weight $800\text{N}$ moving at a velocity $v\text{m/s}$ and bending with an angle of ${60}^{\circ }$ with the ground?

1. $462.43\text{N}$
2. $400\text{N}$
3. $692.8\text{N}$
4. $565.68\text{N}$

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

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

Q. What is the centripetal force experienced by a bike of weight $800\text{N}$ moving at a velocity $v\text{m/s}$ and bending with an angle of ${60}^{\circ }$ with the ground?

1. $462.43\text{N}$
2. $400\text{N}$
3. $692.8\text{N}$
4. $565.68\text{N}$