Angle of Repose

Q. If the coefficient of friction between an insect and bowl is $\mu$ and the radius of the bowl is $R$. Then the maximum height to which the insect can crawl in the bowl is

1. $\frac{R}{\sqrt{1+{\mu }^2}}$
2. $R(1-\frac{1}{\sqrt{1+{\mu }^2}})$
3. $R\sqrt{1+{\mu }^2}$
4. $R\sqrt{1+{\mu }^2}-1$

Q. A cubical block rests on a plane of $\mu =\sqrt{3}$. The angle through which the plane be inclined to the horizontal so that the block just slides down will be

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${75}^{\circ }$

Q.

The friction coefficient between a road and the tyre of a vehicle is $\frac{4}{3}$ . Find the maximum incline the road may have so that once hard brakes are applied and the wheel starts skidding, the vehicle going down at a speed of 36 km/hr is stopped within 5 m.

Q.

Two blocks A and B of mass $m_A$ and $m_B$ respectively are kept in contact on a frictionless table. The experimenter pushes the block A from behind so that the block B, what is the force exerted by the experimenter on A ?

Q. A metallic block of mass $20kg$ is dragged with a uniform velocity of $0.5m{s}^{-1}$ on a horizontal table for $2.1s$. The coefficient of static friction between the block and the table is 0.10. What will be the maximum possible rise in temperature of the metal block, if the specific heat of the block is 0.1 C.G.S. unit? Assume $g=10m{s}^{-1}$ and uniform rise in temperature throughout the whole block. [Ignore absorption of heat by the table]

1. $0.001{}^{o}C$
2. $0.0025{}^{o}C$
3. $0.05{}^{o}C$
4. $0.0035{}^{o}C$

Q.

The angle between the resultant contact force and the normal force exerted by a body on the other is called the angle of friction. Show that, if $\lambda$ be the angle of friction and $\mu$ the coefficient fo static friction, $\lambda \le {\tan }^{-1}\mu$

Q.

A uniform ladder of length 10.0 m and mass 16.0 kg is resting against a vertical wall making an angle of 37 ${}^{\circ }$ with it. The vertical wall is frictionless but the ground is rough. An electrician weighing 60.0 kg climbs up the ladder. If he stays on the ladder at a point 8.00 m from the lower end, what will be the normal force and the force of friction on the ladder by the ground ? What should be the minimum coefficient of friction for the electrician to work safely ?

Q.

Based on dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:

1. $y=a\sin \left(\frac{2\pi t}{T}\right)$
2. $y=a\sin vt$
3. $y=\frac{a}{T}\sin \left(\frac{t}{a}\right)$
4. $y=a\sqrt{2}[\sin \left(\frac{2\pi t}{T}\right)-\cos \left(\frac{2\pi t}{T}\right)]$

Q. For the arrangement shown in figure the tension in the string is [Given : ${\tan }^{-1}\left(0.8\right)={39}^{\circ }]$

1. $6\text{N}$
2. $6.4\text{N}$
3. $0.4\text{N}$
4. zero

Q.

The coefficient of static friction between a block of mass m and an incline is ${\mu }_s$ = 0.3. What can be the maximum angle θ of the incline with the horizontal so that the block does not slip on the plane? (Also called angle of repose)

1. θ = cot^-1 (0.3)

2. θ = sin^-1 (0.3)

3. θ = cos^-1 (0.3)

4. θ = tan^-1 (0.3)

Q. A body is placed on an inclined plane. The coefficient of friction between the body and the plane is $\mu$. The plane is gradually tilted up. If $\theta$ is the inclination of the plane, then frictional force on the body is

1. decreases upto $\theta ={\tan }^{-1}\left(\mu \right)$ and constant after that
2. constant upto $\theta ={\tan }^{-1}\left(\mu \right)$ and decrease after that
3. Increases upto $\theta ={\tan }^{-1}\left(\mu \right)$ and constant after that
4. increases upto $\theta ={\tan }^{-1}\left(\mu \right)$and decrease after that

Q. Mass of upper block and lower block kept over the table is $2kg$ and $1kg$ respectively and coefficient of friction between the blocks is $0.1$. Table surface is smooth. The maximum mass $m$ for which all the three blocks move with same acceleration is $(g=10m/s^2)$

1. $1kg$
2. $2/3kg$
3. $3/4kg$
4. $1/3kg$

Q. A sphere is rotating between two rough inclined walls as shown in figure. Coefficient of friciton between each wall and the sphere is $\frac{1}{3}$. If $f_1$ amd $f_2$ are the friction forces at $P$ and $Q$, then $\frac{f_1}{f_2}$ is

1. $\frac{4}{\sqrt{3}}+1$
2. $\frac{1}{\sqrt{3}}+2$
3. $\frac{1}{2}+\sqrt{3}$
4. $1+2\sqrt{3}$

Q. A block of mass $m$ is put on a rough inclined plane of inclination $\theta$, and is tied with a light thread shown. Inclination $\theta$ is increased gradually from $\theta =0^{\circ }to{90}^{\circ }.$ Match the columns according to corresponding curve.

1. $A-q,B-s,C-r,D-p$
2. $A-s,B-p,C-r,D-q$
3. $A-r,B-p,C-p,D-q$
4. $A-p,B-q,C-s,D-r$

Q. A block rests on a rough plane whose inclination $\theta$ to the horizontal can be varied. Which of the following graphs indicates how the frictional force $F$ between the block and the plane varies as $\theta$ is increased?

1. 
2. 
3. 
4. 

Q. Four rods each of mass $m$ form a square having length of diagonal $b$, rotates about its diagonal. Its moment of
inertia is $\frac{pm{b}^2}{18}$ . Find $p$.

Q. Three rods each of mass 𝑚 and length $b$ form an equilateral triangle and rotate about the median of the triangle. Its moment of inertia is $\frac{am{b}^2}{16}$. Find $a$.

Q. Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is

1. $\frac{137}{15}M{R}^2$
2. $\frac{152}{15}M{R}^2$
3. $\frac{17}{15}M{R}^2$
4. $\frac{209}{15}M{R}^2$

Q.

If the road of the previous problem is horizontal (no banking), what should be the minimum friction coefficient so that a scooter going at 18 km/hr does not skid ?

Q. Moment of inertia of uniform triangular plate about axis passing through sides $AB,AC$ and $BC$ are $I_P,I_B$ and $I_H$ respectively and about an axis perpendicular to the plane and passing through point $C$ is $I_C$. Then

1. $I_H>I_B>I_C>I_P$
   
2. $I_C>I_P>I_B>I_H$
   
3. $I_P>I_H>I_B>I_C$
   
4. none of these
   

Q. A block slides down an incline of angle 30° with an acceleration g/4. The coefficient of Kinetic friction is? (Answer in fractional form expected)

Q.

A small mass slides down an inclined plane of inclination $\theta$ with the horizontal. The coefficient friction is $\mu ={\mu }_{o}x$ where $x$ is the distance through which the mass slides down and ${\mu }_o$​ is a positive constant. Then the distance covered by the mass before it stops is

Q. In $x-y$ plane, a force $10N$ acts at an angle ${30}^{\circ }$ to the positive direction of $x$ axis. The force can be written as force needed is

1. $5\hat{i}+5\hat{j}$
2. $5\hat{i}+5\sqrt{3}\hat{j}N$
3. none of these
   
4. $5\sqrt{3}\hat{i}+5\hat{j}N$

Q.

A block is placed on an inclined plane of inclination $\theta$. The angle of inclination is such that the block slides down the plane at a constant speed. The coefficient of kinetic friction between the block and the inclined plane is equal to

1. $sin\theta$

2. $cos\theta$

3. $tan\theta$

4. $cot\theta$

Q. The elongation in a metallic rod hinged at one end and rotating in a horizontal plane becomes four times of the initial value. The angular velocity of rotation becomes

1. 
   half of initial value
2. 
   two times the initial value
3. 
   one−third of initial value
4. 
   four times the initial value

Q. Consider a uniform square plate of side a and mass $m$. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

1. $\frac{7}{12}m{a}^2$
   
2. $\frac{1}{12}m{a}^2$
   
3. $\frac{2}{3}m{a}^2$
   
4. $\frac{5}{6}m{a}^2$
   

Q. In the figure shown, an inclined plane makes an angle ${30}^{\circ }$ with the horizontal. 𝐴 groove $OA=5m$ cut in the plane makes an angle ${30}^{\circ }$ with $OX$. A short smooth cylinder is free to slide down in the groove due to the influence of gravity. The time (in $s$ taken by the cylinder to reach from $A$ to $O$, if released from rest from $A$, is $g=10m/s^2$

Q. A block of mass $2\text{kg}$ rests on a rough inclined plane making an angle of ${30}^{\circ }$ with the horizontal. The coefficient of static friction between the block and the plane is $0.7$. The frictional force on the block is

1. $9.8\text{N}$
2. $9.8\sqrt{3}\text{N}$
3. $4.9\sqrt{3}\text{N}$
4. $19.6\text{N}$

Q. A sphere of radius $r$ is kept on a concave mirror of radius of curvature $R.$ The arrangement is kept on a horizontal table (the surface of concave mirror is frictionless and sliding not rolling). If the space displasded from its equilibrium position and left then it executes $SHM$. The period of oscillation will be

1. $2\pi \sqrt{\left(\frac{(R-r)1.4}{g}\right)}$
   
2. $2\pi \sqrt{\left(\frac{R-r}{g}\right)}$
3. $2\pi \sqrt{\left(\frac{r^R}{a}\right)}$
4. $2\pi \sqrt{\left(\frac{R}{gr}\right)}$

Q. A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches ${30}^{\circ }$, the box starts to slip and slides $4.0\text{m}$ down the plank in $4.0\text{s}$. The coeffficient of static and kinetic friction between the box and the plank will respectively be

1. $0.6$ and $0.6$
2. $0.4$ and $0.3$
3. $0.6$ and $0.5$
4. $0.5$ and $0.6$