The banking angle for a curved road of radius \(490\) m for a vehicle moving at \(35\) m/s is:
1. ${\tan }^{-1}(0.25)$
2. ${\tan }^{-1}(0.55)$
3. ${\tan }^{-1}(0.45)$
4. ${\tan }^{-1}(0.75)$

A body of mass \(5\) kg is suspended by the strings making angles $60°$and $30°$with the horizontal.

Then:
a. ${\mathrm{T}}_1=25$ $\mathrm{N}$                    
b. ${\mathrm{T}}_2=25$ $\mathrm{N}$
c. ${\mathrm{T}}_1=25\sqrt{3}$ $\mathrm{N}$               
d. ${\mathrm{T}}_2=25\sqrt{3}$ $\mathrm{N}$
Choose the correct option:

A block \(A\) of mass 7 kg is placed on a frictionless table. A thread tied to it passes over a frictionless pulley and carries a body \(B\) of mass 3 kg at the other end. The acceleration of the system will be: (given \(g\) = 10 ms^–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°, 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:
    

A man of mass \(60\) kg is standing on the ground and holding a string passing over a system of ideal pulleys. A mass of \(10\) kg is hanging over a light pulley such that the system is in equilibrium. The force exerted by the ground on the man is: (\(g=\) acceleration due to gravity)

1. \(20\) g
2. \(45\) g
3. \(40\) g
4. \(60\) g

A small coin is kept at a distance r from the centre of a gramophone disc rotating at an angular speed $\omega$. The minimum coefficient of friction for which a coin will not slip is:

1.  $\frac{{\mathrm{r\omega }}^2}{\mathrm{g}}$

2.  $\frac{\mathrm{g}}{{\mathrm{r\omega }}^2}$

3.  $\frac{{\mathrm{r}}^2{\mathrm{\omega }}^2}{\mathrm{g}}$

4.  $\frac{\mathrm{r\omega }}{\mathrm{g}}$

A block of mass \(\mathrm{m}\) is in contact with the cart C as shown in the figure. 
        
The coefficient of static friction between the block and the cart is $\mathrm{\mu }$. The acceleration $\mathrm{\alpha }$ of the cart that will prevent the block from falling satisfies:
1. $\mathrm{\alpha }>\frac{\mathrm{mg}}{\mathrm{\mu }}$

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

3. $\mathrm{\alpha }\ge \frac{\mathrm{g}}{\mathrm{\mu }}$

4. $\mathrm{\alpha }<\frac{\mathrm{g}}{\mathrm{\mu }}$

A car of mass \(1000\) kg negotiates a banked curve of radius \(90\) m on a frictionless road. If the banking angle is of \(45^\circ,\) the speed of the car is:

The force \(\mathrm{F}\) acting on a particle of mass \(\mathrm{m}\) is indicated by the force-time graph shown below. The change in momentum of the particle over the time interval from \(0\) to \(8\) s is:
        

A system consists of three masses \(m_1\), \(m_2\)_, and \(m_3\) connected by a string passing over a pulley \(\mathrm{P}\). The mass \(m_1\) hangs freely, and \(m_2\) and \(m_3\) are on a rough horizontal table (the coefficient of friction = \(\mu\)). The pulley is frictionless and of negligible mass. The downward acceleration of mass \(m_1\) is: (Assume \(m_1=m_2=m_3=m\) and \(g\) is the acceleration due to gravity.) 
             
1. \(\frac{g(1-g \mu)}{9}\)
2. \(\frac{2 g \mu}{3}\)
3. \( \frac{g(1-2 \mu)}{3}\)
4. \(\frac{g(1-2 \mu)}{2}\)