Applications of Radius of Curvature

Q.

Why does a cyclist lean to one side, while going along a curve? In what direction does he lean?

Q.

Why do roads on mountains have inward inclination at sharp turns?

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.

1. 

2. none of these

3. 

4. 

Q. A car is moving on a circular road of radius of curvature $500\text{m}$. If the coefficient of friction is $0.8$ and $g=10{\text{m/s}}^2$, the maximum speed of the car can have without slipping is in $\left(\text{km/hr}\right)$.

1. $5\sqrt{10}$
2. $20\sqrt{10}$
3. $72\sqrt{10}$
4. $54\sqrt{10}$

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

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 biker is moving with a speed of $15\text{m/s}$ on a flat road. The coefficient of friction between the wheels of the bike and the road is $0.4$. If he approaches a curve, what is the minimum radius of the turn that he can take safely?

1. $56.25\text{m}$
2. $22.5\text{m}$
3. $562.5\text{m}$
4. $225\text{m}$

Q. A roller coaster is travelling on a flat track with a minimum radius of curvature of $25\text{m}$ and the train car has a maximum velocity of $15\text{m/s}$. What should be the friction coefficient between the train car and the track for the ride to be safe?
$(g=10{\text{m/s}}^2)$

1. $0.5$
2. $1.5$
3. $2.5$
4. $1.0$

Q. A car is moving on a circular level road of radius of curvature $300m$. If the coefficient of friction is $0.3$ and acceleration due to gravity $10m/s^2$, the maximum speed the car can have is:

1. $30km/h$
2. $81km/h$
3. $162km/h$
4. $108km/h$

Q. A car driven round curved path of radius $18m$ without the danger of skidding. The coefficient of friction between the tyres of the car and the surface of the curved path is $0.5$. What is the maximum speed in kmph of the car for safe driving nearly? ($g=10{ms}^{-2}$)

1. $9.3$
2. $34.48$
3. $18.42$
4. $20.15$

Q. A car is moving on a circular road of radius of curvature $500\text{m}$. If the coefficient of friction is $0.8$ and $g=10{\text{m/s}}^2$, the maximum speed of the car can have without slipping is in $\left(\text{km/hr}\right)$.

1. $5\sqrt{10}$
2. $20\sqrt{10}$
3. $72\sqrt{10}$
4. $54\sqrt{10}$

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. Two boats leave shore. One moves due north at 10 m/s speed, and the other at ${37}^o$ North of East at 10 m/s. The speed of one boat with respect to another is ($\sqrt{5}=2.236$)
Write the answer until two digits after the decimal point.

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.

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.

1. $a_A=a_B$

2. $a_A<a_B$

3. $a_A>a_B$

4. none of these

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. A van is moving with a speed of $72$ $kmph$ on a level road, where the coefficient of friction between tyres and road is $0.5$. The minimum radius of curvature, the road must have, for safe driving of van, is ($g=10m/s^2$)

1. $80$ $m$
2. $40$ $m$
3. $20$ $m$
4. $4$ $m$

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 biker is moving with a speed of $15\text{m/s}$ on a flat road. The coefficient of friction between the wheels of the bike and the road is $0.4$. If he approaches a curve, what is the minimum radius of the turn that he can take safely?

1. $56.25\text{m}$
2. $22.5\text{m}$
3. $562.5\text{m}$
4. $225\text{m}$

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 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. $1\text{km/h}$