A point mass \(m\) is moved in a vertical circle of radius \(r\) with the help of a string. The velocity of the mass is \(\sqrt{7 g r} \) at the lowest point. The tension in the string at the lowest point will be: 
1. \(6mg\)
2. \(7mg\)
3. \(8mg\)
4. \(mg\)

As shown in the figure, two masses of \(10~\text{kg}\) and \(20~\text{kg}\), respectively are connected by a massless spring. A force of \(200~\text{N}\) acts on the \(20~\text{kg}\) mass. At the instant shown, the \(10~\text{kg}\) mass has an acceleration of \(12~\text{m/s}^2\) towards the right. The acceleration of \(20~\text{kg}\) mass at this instant is:
          
1. \(12~\text{m/s}^2\)
2. \(4~\text{m/s}^2\)
3. \(10~\text{m/s}^2\)
4. zero

A particle of mass \(m\) having speed \(v\) goes in a vertical circular motion such that its centre is at its origin, as shown in the figure. If at any instant the angle made by the string with a negative \(y\text-\)axis is \(\theta\) then the tension in the string is:
[Take radius = \(R\)]

1. \(mg\sin\theta+ \frac{mv^2}{R}\)
2. \(mg\cos\theta- \frac{mv^2}{R}\)
3. \(mg\cos\theta+ \frac{mv^2}{R}\)
4. \(mg\sin\theta- \frac{mv^2}{R}\)

Three blocks \(A\), \(B\) and \(C\) of mass \(3M\), \(2M\) and \(M\) respectively are suspended vertically with the help of springs \(\mathrm{PQ}\) and \(\mathrm{TU}\) and a string \(\mathrm{RS}\) as shown in fig. The acceleration of blocks \(A\), \(B\) and \(C\) are \(a_{1} , a_{2}~ \text{and}~ a_{3}\) respectively.
     

The value of acceleration \(a_{1}\) at the moment string \(\mathrm{RS}\) is cut will be: 
1. \(g\) downward
2. \(g\) upward
3. more than \(g\) downward
4. zero

If \(\mu\) between block \(A\) and inclined plane is \(0.5\) and that between block \(B\) and the inclined plane is \(0.8\), then the normal reaction between blocks \(A\) and \(B\) will be:
                     
1. \(180\) N
2. \(216\) N
3. \(0\)
4. None of these

Calculate the reading of the spring balance shown in the figure: (take \(g=10\) m/s²)
                

1. \(60\) N
2. \(40\) N
3. \(50\) N
4. \(80\) N

A metal sphere is suspended from a wall by a string. The forces acting on the sphere are shown in the figure. Which of the following statements is NOT correct?
   $\mathrm{}$

The angle of banking for a cyclist taking a turn at a curve is given by \(\tan\theta   =   \frac{v^{n}}{rg}\) where symbols have their usual meaning. The value of \(n\) is:

A body of mass \(m\) is moving on a concave bridge \(ABC\) of the radius of curvature \(R\) at a speed \(v\). The normal reaction of the bridge on the body at the instant it is at the lowest point of the bridge is:

                            
1. \(mg-\frac{mv^{2}}{R}\)
2. \(mg+\frac{mv^{2}}{R}\)
3. \(mg\)
4. \(\frac{mv^{2}}{R}\)

A simple pendulum hangs from the roof of a train moving on horizontal rails. If the string is inclined towards the front of the train, then the train is: