The upper half of an inclined plane of inclination θ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and lower half of the plane is given by

1. μ=1/tanθ

2. μ=2/tanθ

3. μ=2tanθ

4. μ=tanθ

A block of mass m is in contact with the cart C as shown in the figure.

The coefficient of static friction between the block and the cart is $\mu .$The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies

1. $\alpha >\frac{mg}{\mu }$                                       2. $\alpha >\frac{g}{\mu m}$

3. $\alpha \ge \frac{g}{\mu }$                                          4. $\alpha <\frac{g}{\mu }$

A gramophone record is revolving with an angular velocity $\omega .$ A coin is placed at a distance r from the centre of the record. The static coefficient of friction is $\mu .$ The coin will revolve with the record if 

1. $r=\mu g{\omega }^2$                                       2. $r<\frac{{\omega }^2}{\mu g}$

3. $r\le \frac{\mu g}{{\omega }^2}$                                         4. $r\ge \frac{\mu g}{{\omega }^2}$

The mass of a lift is 2000 kg. When the tension in the supporting cable is 28000 N, its acceleration is

(a) $30 {\mathrm{ms}}^{-2} \mathrm{downwards}$                                      (b) $4 {\mathrm{ms}}^{-2} \mathrm{upwards}$

(c) $4 {\mathrm{ms}}^{-2} \mathrm{downwards}$                                        (d) $14 {\mathrm{ms}}^{-2} \mathrm{upwards}$

A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is 20m. The speed of the car at the top of the hill is between

1. 14m/s and 15m/s

2. 15m/s and 16m/s

3. 16m/s and 17m/s

4. 13m/s and 14m/s

A particle of mass m is projected with velocity v making an angle of ${45}^{°}$ with the horizontal. When the particle lands on the level ground the magnitude of the change in its momentum will be 

1.  2mv

2. mv/$\sqrt{2}$

3. mv$\sqrt{2}$

4. zero 

Sand is being dropped on a conveyor belt at the rate of M kg/s. The force necessary to keep the belt moving with a constant velocity of v m/s will be 

1. Mv newton 

2. 2 Mv newton 

3. $\frac{Mv}{2}$newton 

4. zero

Three blocks with masses m, 2m and 3m are connected by strings, as shown in the figure. After an upward force F is applied on block m, the masses move upward at constant speed v. What is the net force on the block of mass 2m? (g is the acceleration due to gravity)

1. Zero

2. 2mg

3. 3mg

4. 6mg

A system consists of three masses m₁,m₂ and m₃ connected by a string passing over a pulley P. The mass $m_1$ hangs freely and m₂ and m₃ are on a rough horizontal table (the coefficient of friction=μ) The pulley is frictionless and of negligible mass. The downward acceleration of mass m₁, is (Assume,m₁=m₂=m₃=m)


1. g(1-gμ)/9

2. 2gμ/3

3. g(1-2μ)/3

4. g(1-2μ)/2

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