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

(a)$\sqrt{gR\left(\frac{\tan \theta +\mu s}{1-{\mu }_s\tan \theta }\right)}$ 

(b) $\sqrt{\frac{g}{R}\left(\frac{\tan \theta +\mu s}{1-{\mu }_s\tan \theta }\right)}$

(c) $\sqrt{\frac{g}{R^2}\left(\frac{\tan \theta +\mu s}{1-{\mu }_s\tan \theta }\right)}$

(d) $\sqrt{g{R}^2\left(\frac{\tan \theta +\mu s}{1-{\mu }_s\tan \theta }\right)}$

The length of a spring is l₁ and l₂, when stretched with a force of 4 N and 5 N respectively. Its natural length is

1. l₂ + l₁

2. 2(l₂-l₁)

3. 5l₁ - 4l₂

4. 5l₂ - 4l₁

Two stones of masses \(m\) and \(2m\) are whirled in horizontal circles, the heavier one in a radius \(\frac{r}{2}\) and the lighter one in the radius \(r.\) The tangential speed of lighter stone is \(n\) times that of heavier stone when they experience the same centripetal forces. The value of \(n\) is:

One end of the string of length \(l\) is connected to a particle of mass \(m\) and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in a circle with speed \(v\), the net force on the particle (directed towards the centre) will be: (\(T\) represents the tension in the string)

A spring of force constant k is cut into lengths of ratio 1:2:3. They are connected in series and the new force constant is $k^{\text{'}}$. If they are connected in parallel and force constant is $k^{\text{'}\text{'}}, then k^{\text{'}}:k^{\text{'}\text{'}}$ is 

(1) 1:6

(2) 1:9

(3) 1:11

(4) 6:11

Three blocks A, B and C of masses 4 kg, 2 kg and 1 kg respectively, are in contact on a frictionless surface, as shown. If a force of 14 N is applied on the 4 kg block, then the contact force between A and B is
                            

1.2N

2. 6N

3. 8N

4. 18N

A block A of mass m₁ rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of table and from its other end another block B of mass m₂ is suspended. The coefficient of kinetic friction between the block and the table is  μ_k. When the block A is sliding on the table, the tension in the string is

1. (m₂+μ_km₁)g /(m₁+m₂)

2. (m₂-μ_km₁)g/(m₁+m₂)

3. m₁m₂(1+μ_k)g/(m₁+m₂)

4. m₁m₂(1-μ_k)g/(m₁+m₂)

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°, 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 will be. respectively

1. 0.6 and 0.6

2. 0.6 and 0.5

3. 0.5 and 0.6

4. 0.4 and 0.3

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 balloon with mass m is descending down with an acceleration a (where a < g). How much mass should be removed from it so that it starts moving up with an acceleration a?

1. 2ma/g+a

2. 2ma/g-a

3. ma/g+a

4. ma/g-a