Banking Angle

Q.

A circular track of radius 600 m is to be designed for cars at an average speed of 180 km/hr. What should be the angle of banking of the track?

1. 

2. 

3. 

4. 

Q.

A Block is released from rest at the top of an inclined plane which later curves into a circular track of radius r as shown. Find the minimum height h from where it should be released so that it is able to complete the circle.

1. 

2. 

3. 2r

4. 5r

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. ${45}^{\circ }$

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

3. ${42}^{\circ }$

4. ${48}^{\circ }$

Q.

Why Are Are Banked On Curves?

Q. An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. What is the radius of the loop ?

Q. On a highway, the maximum speed of vehicles is $25\text{m/s}$. The material of the road provides a coefficient of friction of $0.2$. If the highway has a turn with a radius of curvature of $50\text{m}$, then what should be the banking angle for this turn to ensure the safety of the drivers?

1. ${\sin }^{-1}\left(0.2\right)$
2. ${\cos }^{-1}\left(0.5\right)$
3. ${\tan }^{-1}\left(0.84\right)$
4. ${\tan }^{-1}\left(1.2\right)$

Q. If the angle of banking of a road is ${32}^o$ and if a turn has radius $72m$, what is the maximum speed at which a car can turn, given the coefficient of friction between the car tyres and the road is $0.34$? (in $m/s$)

1. $29.4$
   
2. $26.3$
   
3. $21.3$
   
4. $62.3$
   

Q.

what is the cause and outcome of rotational motion?

Q. A car is moving on a road with a radius of curvature $70\text{m}$ banked at an angle of ${45}^{\circ }$. The coefficient of static friction between the road and the wheels of the car is ${\mu }_s=0.15$. Take $g=10{\text{m/s}}^2$. If the speed of the car is $20\text{m/s}$, then will the car skid ? If so, in what direction?

1. The car will not skid.
2. The car will skid up the incline.
3. The car will skid down the incline.
4. Skidding of car depends on the mass of the car.

Q. A curve has a radius of $50\text{m}$ and a banking angle ($\theta$) of ${15}^{\circ }$. What is the maximum safe speed for a car on this curve if the road does not offer any friction? Take $g=10{\text{m/s}}^2$.

1. $10\sqrt{5(4-\sqrt{3})}\text{m/s}$
2. $10\sqrt{5(2-\sqrt{3})}\text{m/s}$
3. $10\sqrt{10(2-\sqrt{5})}\text{m/s}$
4. $10\sqrt{5(3-\sqrt{3})}\text{m/s}$

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

Q. A circular road of radius $r$ is banked for a speed $v=40$ km/hr. A car of mass $m$ attempts to go on the circular road. The friction coefficient between the tyre and the road is negligible. Find the correct option(s).

1. The car cannot make a turn without skidding
2. If the car turns at a speed less than $40$ km/hr, it will slip down
3. If the car turns at a correct speed of $40$ km/hr, the force by the road on the car is equal to $\frac{m{v}^2}{r}$
4. If the car turns at a correct speed of $40$ km/hr, the force by the road on the car is greater than mg as well as greater than $\frac{m{v}^2}{r}$

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. $555\text{kg}$

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

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

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

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 m

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

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

3. ${45}^{\circ }$

4. ${60}^{\circ }$

Q. The coefficient of friction between the tyres and the road is 0.1. The maximum speed with which a cyclist can take a circular turn of radius 3m without skidding is then
(Take g= 10 $m{s}^{-2}$)

1. $\sqrt{15}m{s}^{-1}$
2. $\sqrt{3}m{s}^{-1}$
3. $\sqrt{30}m{s}^{-1}$
4. $\sqrt{10}m{s}^{-1}$

Q. Radius of curvature of a road is 60m. It is a to be banked so that no friction force is required for a car travelling on the road at 25 mt/sec. Find the angle of banking ?

1. ${45}^{\circ }$
2. ${47}^{\circ }$
3. ${90}^{\circ }$
4. ${180}^{\circ }$

Q. A turn of radius $20m$ is banked for the vehicles going at a speed of $36km/h$. If the coefficient of static friction between the road and the tyre is $0.4$, what are the possible speeds of a vehicle so that it neither slips down nor skids up?

1. $14.7km/h$ and $54km/h$
2. $14.2km/h$ and $50km$
3. $11.7km/h$ and $44km/h$
4. $17.7km/h$ and
   $34km/h$

Q. A curved road of $50\text{m}$ radius is banked at correct angle for a given maximum speed $vm/s$. If the radius of curvature of the road is changed to $200\text{m}$ keeping the same banking angle, the maximum safe speed on the road is

1. $2v\text{m/s}$
2. $0.5v\text{m/s}$
3. $3v\text{m/s}$
4. $4v\text{m/s}$

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

Q. On a highway, the maximum speed of vehicles is $50\text{m/s}$. The material of the road provides a coefficient of static friction ${\mu }_s=0.35$. If the highway has a circular turn with a radius of curvature $90\text{m}$, then what should be the banking angle $\left(\theta \right)$ for this turn to ensure safe driving ?

1. ${\tan }^{-1}\left(2.24\right)$
2. ${\tan }^{-1}\left(1.23\right)$
3. ${\sin }^{-1}\left(2.26\right)$
4. ${\cos }^{-1}\left(1.26\right)$

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 106 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.

An aircraft executes a horizontal loop with a speed of 150 m/s with its, wings banked at an angle of ${120}^{\circ }$. The radius of the loop is $(g=10m/s^2)$

1. 10.6 km

2. 9.6 km

3. 7.4 km

4. 5.8 km

Q. A vehicle is travelling along un-banked curved path. If the friction between the road and tyres suddenly disappears, then the vehicle:

1. moves along trangential direction
2. moves along radially outward direction
3. moves along a direction between tangential and radially outward direction
4. moves along the same curved path