Friction and Horizontal Circular Motion Problems

Q. A car is travelling on a track which has radius of curvature $50\text{m}$. What is the maximum safe speed with which the car can travel on this track? Take $g=10{\text{m/s}}^2$. Assume coefficient of friction to be 0.8.

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

Q.

A car moving at a speed of 36 km/hr is taking a turn on a circular road of radius 50 m. A small wooden plate is kept on the seat with its plane perpendicular to the radius of the circular road (figure 7-E4). A small block of mass 100 g is kept on the seat which rests against the plate. The friction coefficient between the block and the plate is $\mu =\mathrm{0.58.}$ (a) Find the normal contact force exerted by the plate on the block. (b) The plate is slowly turned so that the angle between the normal to the plate and the radius of the road slowly increases. Find the angle at which the block will just start sliding on the plate.

Q.

A car moving on a horizontal road may be thrown out of the road in taking a turn

1. By the gravitational force

2. Due to lack of sufficient centripetal force

3. Due to rolling frictional force between tyre and road

4. Due to the reaction of the ground

Q. A stone of mass $m$ tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is

1. the same throughout the motion.
2. minimum at the highest position of the circular path.
3. minimum at the lowest position of the circular path.
4. minimum when the rope is in the horizontal position.

Q. A $1200\text{kg}$ automobile rounds a level curve of radius $200\text{m}$, on an unbanked road with a velocity of $72\text{km/h}$. What is the minimum coefficient of static friction between the tyres and road in order that the automobile may not skid? $(g=10{\text{m/s}}^2)$

1. $0.3$
2. $0.15$
3. $0.4$
4. $0.2$

Q. A turn of radius $20\text{m}$ is banked for the vehicle of mass $200\text{kg}$ going at a speed of $10\text{m/s}$. The minimum coefficient of static friction required, if the vehicle moves with a speed of $15\text{m/s}$ is given by $\frac{N}{8}$. Then, $N$ is
[Take $g=10{\text{m/s}}^2$]

Q. A car travels at a constant speed of $30\text{m/s}$ around a level circular bend of radius $100\text{m}$. What is the minimum coefficient of static friction between the tires and the road in order for the car to go round the bend without skidding?

1. $0.8$
2. $0.55$
3. $0.7$
4. $0.9$

Q. A car travelling on a level road with a curve of radius $32\text{m}$. The coefficient of static friction between the road and tyre of car is $0.2$. What speed will put the car on verge of sliding? $(g=10{\text{m/s}}^2)$

1. $10\text{m/s}$
2. $8\text{m/s}$
3. $13\text{m/s}$
4. $6\text{m/s}$

Q. A circular racing track of radius $300\text{m}$ is banked at an angle of ${17}^{\circ }$ with the horizontal. If the coefficient of static friction $\left({\mu }_s\right)$ between the wheels of the racing car and the road is $0.2$, what is the maximum permissible speed to avoid slipping? Take $g=10{\text{m/s}}^2$. Use the given data: $\tan {17}^{\circ }=0.30,\sin {17}^{\circ }=0.29,\cos {17}^{\circ }=0.96$ and $\frac{0.50}{0.94}\simeq 0.5$

1. $10\sqrt{15}\text{m/s}$
2. $3\sqrt{35}\text{m/s}$
3. $50\text{m/s}$
4. $5\sqrt{15}\text{m/s}$

Q. A train of mass $m$ moves with a velocity $v$ on the equator from east to west. If $\omega$ is the angular speed of earth about its axis and $R$ is the radius of the earth then the normal reaction acting on the train is

1. $mg[1-\frac{(\omega R-2v)\omega }{g}-\frac{v^2}{Rg}]$
2. $mg[1-2\frac{(\omega R-v)\omega }{g}-\frac{v^2}{Rg}]$
3. $mg[1-\frac{(\omega R-2v)\omega }{g}+\frac{v^2}{Rg}]$
4. $mg[1-2\frac{(\omega R-v)\omega }{g}+\frac{v^2}{Rg}]$

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

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

Q. A car of mass $800\text{kg}$ moves on a horizontal circular track of radius $40\text{m}$. If the coefficient of static friction (${\mu }_s$) is $0.5$, then the maximum velocity with which the car can move will be

1. $20\text{m/s}$
2. $25\text{m/s}$
3. $15\text{m/s}$
4. $14\text{m/s}$

Q. A car of $800kg$ negotiates a banked curve of radius $160m$ on a smooth road. If the banking angle is ${60}^{\circ }$, find the normal force between the tyres and the road. $(g=10m/s^2)$

1. $16000N$
2. $8000\sqrt{3}N$
3. $4000N$
4. $4000\sqrt{3}N$

Q. A train of mass ${10}^4\text{kg}$ rounds a curve of radius $100\text{m}$ at a speed of $10\text{m/s}$. At what angle the track be banked in order to have no thrust on the rails? ($g=10{\text{m/s}}^2$)

1. ${\tan }^{-1}\left(\frac{1}{10}\right)$
2. ${\tan }^{-1}\left(10\right)$
3. ${\tan }^{-1}\left(\frac{1}{20}\right)$
4. ${\tan }^{-1}\left(\frac{1}{100}\right)$

Q. Assertion :If the earth suddenly stops rotating about its axis, the n acceleration due to gravity will become the same at all places. Reason: The value of acceleration due to gravity is independent of rotation of the earth.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. A small mass $m$ starts sliding down a smooth and stationary semi-circular track. Which of the following graph best represents the variation of the magnitude of the force applied by the track on the mass and the angle $\theta$ ? ( assume initially $\theta =0$)

1. 
2. 
3. 
4. 

Q. A stunt driver of mass $60\text{kg}$ rides a bike inside a hollow metallic sphere of radius $45\text{m}$. Assume the bike moves in the central plane of the sphere. If the coefficient of static friction between the wheels of the bike and the metal surface is $0.8$, what must be the minimum speed of the driver so that he does not crash? Take $g=10{\text{m/s}}^2$.

1. $40\text{m/s}$
2. $23.72\text{m/s}$
3. $25.72\text{m/s}$
4. $50\text{m/s}$

Q. A train of mass ${10}^4\text{kg}$ rounds a curve of radius $100\text{m}$ at a speed of $10\text{m/s}$. At what angle the track be banked in order to have no thrust on the rails? ($g=10{\text{m/s}}^2$)

1. ${\tan }^{-1}\left(\frac{1}{10}\right)$
2. ${\tan }^{-1}\left(10\right)$
3. ${\tan }^{-1}\left(\frac{1}{20}\right)$
4. ${\tan }^{-1}\left(\frac{1}{100}\right)$

Q. A chain in the shape of circular loop of mass $m$ and radius $R$ is placed on a frictionless surface. Loop is given an angular velocity of $\omega$ as shown in the figure. Tension in the chain is

1. $m{\omega }^{2}R$
2. $\frac{m{\omega }^{2}R}{2\pi }$
3. $\frac{m{\omega }^{2}R}{2}$
4. $\frac{m{\omega }^{2}R}{\pi }$

Q. An unbanked curve has a radius of $60m$. Coefficient of friction between the tyre of truck and road is $0.75$. The distance between two front wheel of truck is $2m$. If the truck exceeds the speed of safe limit, then select the correct statement and justify your asnwer.
1) Inner wheels leave the ground first.
2) Outer wheels leave the ground first.

Q. A car is negotiating a curved road of radius $R$. The road is banked at an angle $\theta$. the coefficient of friction between the tyres of the car and the road is ${\mu }_s$. The maximum safe velocity on this road is

1. $\sqrt{g{R}^2\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }}$
2. $\sqrt{gR\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }}$
3. $\sqrt{\frac{g}{R}\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }}$
4. $\sqrt{\frac{g}{R^2}\frac{{\mu }_s+\tan \theta }{1-{\mu }_s\tan \theta }}$

Q. A turn of radius $20\text{m}$ is banked for the vehicle of mass $200\text{kg}$ going at a speed of $10\text{m/s}$. The minimum coefficient of static friction required, if the vehicle moves with a speed of $15\text{m/s}$ is given by $\frac{n}{8}$. Then, $n$ is
[Take $g=10{\text{m/s}}^2$]

Q. A car travelling on a level road with a curve of radius $32\text{m}$. The coefficient of static friction between the road and tyre of car is $0.2$. What speed will put the car on verge of sliding? $(g=10{\text{m/s}}^2)$

1. $10\text{m/s}$
2. $8\text{m/s}$
3. $13\text{m/s}$
4. $6\text{m/s}$

Q. A car is travelling on a track which has radius of curvature $50\text{m}$. What is the maximum safe speed with which the car can travel on this track? Take $g=10{\text{m/s}}^2$. Assume coefficient of friction to be 0.8.

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

Q. If a block moving up an inclined plane at ${30}^o$ with a velocity of $5m/s$, stops after $0.5s$, then coefficient of friction will be nearly

1. $0.5$
2. $0.6$
3. $0.9$
4. $1.1$

Q. The rotation of the earth about its axis speed up such that a man on the equator becomes weightless. In such a situation, what would be the duration of one day?

1. $2\pi \sqrt{R/g}$
2. $\frac{1}{2\pi }\sqrt{R/g}$
3. $\pi \sqrt{R/g}$
4. $\frac{1}{2\pi }\sqrt{Rg}$

Q. Motorcycle rider is moving in a hollow sphere in a horizontal circle of radius $5m$ with constant speed $12m/s$. Acceleration due to gravity is $9.8m/s^2$. The coefficient friction between tires of the motorcycle and the inner surface of the sphere is:

1. $2.94$
2. $0.58$
3. $0.34$
4. $0.23$

Q. A car travelling on a level road with a curve of radius $32\text{m}$. The coefficient of static friction between the road and tyre of car is $0.2$. What speed will put the car on verge of sliding? $(g=10{\text{m/s}}^2)$

1. $10\text{m/s}$
2. $8\text{m/s}$
3. $13\text{m/s}$
4. $6\text{m/s}$

Q. A particle is moving around a circular path with uniform angular speed $\left(\omega \right)$. The radius of the circular path is (r). The acceleration of the particle is?

1. $\frac{{\omega }^2}{r}$
2. $\frac{\omega }{r}$
3. $v\omega$
4. vr

Q. The work done against force of friction is: (consider ideal quantites )

1. $8.7J$
2. $10.7J$
3. $7.8J$
4. $12.7J$