Banking Angle

Q. A car of mass $1000\text{kg}$ negotiates a banked curve of radius $90\text{m}$ on a frictionless road. If the banking angle is ${45}^{\circ }$, the speed of the car is:
Take $g=10{\text{m/s}}^2$

1. $20\text{m/s}$
2. $30\text{m/s}$
3. $15\text{m/s}$
4. $25\text{m/s}$

Q.

A block is projected up to a rough plane of inclination $ 30°$. If the time of ascending is half the time for descending and the coefficient of friction is $ \mathrm{\mu } =\raisebox{1ex}{$ 3$}\!\left/ \!\raisebox{-1ex}{$5$}\right. \sqrt{\mathrm{n}} $. Then $ \mathrm{n}=$

2

Q.

The maximum speed of a car on a road–turn of radius 30 m, if the coefficient of friction between the tyres and the road is 0.4, will be

1. 6.84 m/sec

2. 10.84 m/sec

3. 9.84 m/sec

4. 8.84 m/sec

Q. A circular curve of a highway is designed for traffic moving at $72\text{km/h}$. If the radius of the curved path is $100\text{m}$, the correct angle of banking of the road should be given by
(Take $g=10{\text{m/s}}^2$)

1. ${\tan }^{-1}\left(\frac{2}{3}\right)$
2. ${\tan }^{-1}\left(\frac{3}{5}\right)$
3. ${\tan }^{-1}\left(\frac{2}{5}\right)$
4. ${\tan }^{-1}\left(\frac{1}{4}\right)$

Q. A point moves along an arc of a circle of radius R. Its velocity depends upon the distance covered `s' as $v=A\sqrt{s}$, where a is constant. The angle $\theta$ between the vector of total acceleration and tangential acceleration is:

1. $tan\theta =\sqrt{\frac{s}{R}}$
2. $tan\theta =\sqrt{\frac{s}{2R}}$
3. $tan\theta =\frac{2s}{R}$
4. $tan\theta =\frac{s}{2R}$

Q.

A cyclist riding the bicycle at a speed of $14\sqrt{3}m{s}^{-1}$ takes a turn around a circular road of radius $20\sqrt{3}$ m without skidding. Given $g=9.8m{s}^{-2}$, what is his inclination to the vertical

1. ${30}^{\circ }$

2. ${90}^{\circ }$

3. ${60}^{\circ }$

4. ${45}^{\circ }$

Q.

A cyclist goes round a circular path of circumference 34.3 m in $\sqrt{22}$ sec. the angle made by him, with the vertical, will be

1. ${42}^{\circ }$

2. ${40}^{\circ }$

3. ${48}^{\circ }$

4. ${45}^{\circ }$

Q. A vehicle is moving without slipping or skidding on a circular road, which is rough and banked. Then choose the correct option for the possible direction of friction

1. Up the incline
2. Down the incline
3. Both (a) and (b)
4. None of the above

Q. A metre gauge train is moving at a speed of $36\text{km/hr}$ along a curved track of radius $10\sqrt{3}\text{m}$. Find the height of the outer rail with respect to the inner rail, so that there is no side pressure on the rails. Take $g=10{\text{m/s}}^2$ and the width of the track to be $1\text{m}$.

1. $0.8\text{m}$
2. $0.5\text{m}$
3. $1\text{m}$
4. $0.6\text{m}$

Q. A car is moving on a circular track of radius $R$. The road is banked at an angle $\theta$. ${\mu }_s$ is the coefficient of static friction between the car and the track. Find the maximum speed with which the car can move safely on the track.

1. ${\left[\frac{rg(\cos \theta +{\mu }_s\sin \theta )}{\cos \theta -{\mu }_s\sin \theta }\right]}^{\frac{1}{2}}$
2. ${\left[\frac{rg(\sin \theta +{\mu }_s\cos \theta )}{\cos \theta -{\mu }_s\sin \theta }\right]}^{\frac{1}{2}}$
3. None
4. ${\left[\frac{rg(\sin \theta +{\mu }_s\cos \theta )}{\cos \theta +{\mu }_s\sin \theta }\right]}^{\frac{1}{2}}$

Q. The maximum safe speed for a banked road is intended to be increased by $20$%. If the angle of banking is not changed, then the radius of curvature of the road should be changed from $30\text{m}$ to $\text{m}$

Q.

A civil engineer wishes to redesign the curved roadway in such a way that a car will not have to rely on friction to round the curve without skidding. In other words, a car moving at the designated speed can negotiate the curve even when the road is covered with ice. Such a ramp is usually banked, which means that the roadway is tilted toward the inside of the curve. Suppose the designated speed for the ramp is to be 10.0 m/s and the radius of the curve is 20.0 m. At what angle should the curve be banked?

1. ${30}^{\circ }$

2. $ta{n}^{-1}\frac{1}{3}$

3. ${60}^{\circ }$

4. $ta{n}^{-1}\frac{1}{2}$

Q. A car of mass $1000\text{kg}$ negotiates a banked curve of radius $90\text{m}$ on a frictionless road. If the banking angle is ${45}^{\circ }$, the speed of car is (Take $g=10{\text{m/s}}^2$)

1. $30{\text{ms}}^{-1}$
2. $20{\text{ms}}^{-1}$
3. $10{\text{ms}}^{-1}$
4. $5{\text{ms}}^{-1}$

Q.

A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is ${10}^6$ kg. What provides the centripetal force required for this purpose - The engine or the rails? What is the angle of banking required to prevent wearing out of the rail?

Q. The angle which the bicycle and its rider must make with the vertical when going round a curve of $8.1\text{m}$ radius at $9m{s}^{-1}$ is (Take $g=10m/s^2$)

1. ${20}^{\circ }$
2. ${45}^{\circ }$
3. ${30}^{\circ }$
4. ${60}^{\circ }$

Q. A railway track is banked by making the outer rail $10\text{cm}$ higher than the inner rail. The distance between the rails is $2\text{m}$. If the speed limit for trains on this track is $72\text{km/h}$, what will be the radius of curvature of the turn for safe travelling? Take $g=10{\text{m/s}}^2$.

1. $80\text{m}$
2. $500\text{m}$
3. $800\text{m}$
   
4. $1000\text{m}$

Q.

A track consists of two circular parts ABC and CDE of equal radius 100 m and joined smoothly as shown in figure (7-E1). Each part subtends a right angle at its centre.A cycle weighing 100 kg together with the rider travels at a constant speed of 18 km/h on the track. (a) Find the normal contact force by the road on the cycle when it is at B and at D. (b) Find the force of friction exerted by the track on the tyres when the cycle is at B, C and D. (c) Find the normal force between the road and the cycle just before and just after the cycle crosses C. (d) What should be the minimum friction coefficient between the road and the tyre, which will ensure that the cyclist can move with constant speed ? Take $g=10m/s^2.$

Q.

A cyclist taking turn bends inwards while a car passenger taking same turn is thrown outwards. The reason is

1. Cyclist has to counteract the centrifugal force while in the case of car only the passenger is thrown by this force

2. Car is heavier than cycle

3. Difference in the speed of the two

4. Car has four wheels while cycle has only two

Q.

What is the normal force in vertical circular motion?

Q. Statement I : On a banked curved road, vertical component of normal reaction provides the necessary centripetal force.
Statement II : Centripetal force is always required for turning on a curved path.

1. Both the statements I and II are incorrect.
2. Statement I is incorrect and Statement II is correct.
3. Both the statements I and II are correct
4. Statement I is correct and Statement II is incorrect.

Q. A curve in a road forms an arc of radius $800\text{m}$ If the road is $39.2\text{m}$ wide, calculate the safe speed for turning, if the outer edge of the road is $0.5\text{m}$ higher than the inner edge. Take $g=9.8{\text{m/s}}^2$.

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

Q. What is the angle of banking necessary for a curved frictionless road of radius $30\text{m}$ for safe driving at a speed of $20\text{m/s}$? (Consider value of $g=10{\text{m/s}}^2$)

1. ${\tan }^{-1}\left(\frac{4}{3}\right)$
2. ${\tan }^{-1}\left(\frac{3}{4}\right)$
3. ${60}^{\circ }$
4. ${\tan }^{-1}\left(1\right)$

Q. A car has to negotiate a curve of radius $200\text{m}$. By how much should the outer side of the road be raised with respect to the inner side for a speed of $72\text{km/h}$. The width of the road is $1\text{m}$. $(g=10{\text{m/s}}^2)$

1. $0.2\text{cm}$
2. $2\text{cm}$
3. $20\text{cm}$
4. $\text{None of these}$

Q. A race track banked at an angle of ${37}^{\circ }$ has a coefficient of static friction $0.4$. What is the range of the safe velocities for the cars travelling on the track, for a turn of radius $40\text{m}$? Take $g=10{\text{m/s}}^2$.

1. $9\text{m/s}\text{to}12\text{m/s}$
2. $10\text{m/s}\text{to}20\text{m/s}$
3. $\sqrt{188.46}\text{m/s}\text{to}\sqrt{243.32}\text{m/s}$
4. $\sqrt{107.69}\text{m/s}$ to $\sqrt{657.14}\text{m/s}$

Q. Assertion : Improper banking of roads causes wear and tear of tyres.
Reason : The necessary centripetal force is provided by the force of friction between the tyres and the road.

Select the correct option.

1. Assertion is true but the reason is false.
2. Assertion and reason both are false.
3. Both assertion and reason are true but the reason is not the correct explanation for the assertion.
4. Both assertion and reason are true and the reason is the correct explanation for the assertion.
   

Q. A curve on a highway has a radius of curvature $40\text{m}$. The curved road is banked at an angle of ${45}^{\circ }$ with the horizontal. If the coefficient of static friction that the road provides is ${\mu }_s=0.20$, what is the maximum permissible speed (in $\text{m/s}$) limit for vehicles travelling on the road?
Obtain the answer as nearest integer value
Useful data : $g=10{\text{m/s}}^2$ , $\sqrt{6}=2.40$

Q. A car is moving with a speed of $60\text{m/s}$ on a curved road banked at an angle of ${45}^{\circ }$. If the coefficient of static friction between the wheels of the vehicle and the road is $0.4$, what should be the minimum radius of turn for the car so that it does not skid? Take $g=10{\text{m/s}}^2$.

1. $175.5\text{m}$
2. $200\text{m}$
3. $170\text{m}$
4. $154.3\text{m}$

Q. The maximum speed of a cyclist on a race track is $20\text{m/s}$ and he can bend to a maximum angle of ${45}^{\circ }$. The race track has a curve of radius $20\text{m}$. Is this race track safe for the racer? If not, what should be the angle of banking of the curve?

1. The track is not safe, ${15}^{\circ }$
2. The track is safe.
3. The track is not safe, ${10}^{\circ }$
4. The track is not safe, ${19}^{\circ }$

Q. If a body is at rest on a horizontal surface which is rough, then

1. frictional force between body & surface must be zero.
2. frictional force between body & surface must be non-zero.
3. frictional force must be less than or equal to ${\mu }_{s}N$ (${\mu }_s$ =coefficient of static friction, N= normal force).
4. Total contact force must be perpendicular to surface.

Q. A road is banked at an angle of $\theta ={30}^{\circ }$ and provides a coefficient of static friction ${\mu }_s=0.4$. The road gets seriously damaged if it experiences a normal reaction force of more than $10000\text{N}$. What should be the maximum mass of the vehicle allowed on this road?

1. $666\text{kg}$
2. $1000\text{kg}$
3. $5000\text{kg}$
4. $555\text{kg}$