Let $\theta$ denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of bob is m, the tension in the string is mgcos$\theta$-

1.  always 

2.  never

3.  at extreme

4.  at mean position

A motorcycle is going on an overbridge of radius R. The driver maintains a constant speed. As the motorcycle is ascending on the overbridge, the normal force on it:

1.  Increases

2.  decreases

3.  remains the same

4.  fluctuates erratically

A block of mass m at the end of a string is whirled round in a vertical circle of radius R. The critical speed of the block at the top of its swing below which the string would slacken before the block reaches the top is 

(1) Rg

(2) (Rg)²

(3) R/g

(4) $\sqrt{Rg}$

A bucket full of water tied with the help of a 2 m long string performs a vertical circular motion. The minimum angular velocity of the bucket at the uppermost point so that water will not fall will be:

1. 2$\sqrt{5}$ rad/s

2. $\sqrt{5}$ rad/s

3. 5 rad/s

4. 10 rad/s

A stone of mass \(\mathrm{m}\) tied to the end of a string revolves in a vertical circle of radius \(\mathrm{R}.\) The net forces at the lowest and highest points of the circle directed vertically downwards are:

(${\mathrm{T}}_1$ and ${\mathrm{v}}_1$ denote the tension and speed at the lowest point.${\mathrm{T}}_2$ and ${\mathrm{v}}_2$ denote corresponding values at the highest point.)

What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is \(25\) m?
1. \(16.7\) m/s
2. \(15.8\) m/s
3. \(35\) m/s
4. \(24\) m/s

A particle of mass 200 g is moving in a circle of radius 2 m. The particle is just 'looping the loop'. The speed of the particle and the tension in the string at highest point of the circular path are (g=10 ms^-2)

1. 4 ms^-1, 5 N

2. 4.47 ms^-1, zero

3. 2.47 ms^-1, zero

4. 1 ms^-1, zero

A particle of mass 200 g, is whirled into a vertical circle of radius 80 cm using a massless string. The speed of the particle when the string makes an angle of $60°$ with the vertical line is 1.5 ms^-1. The tension in the string at this position is

1. 1 N

2. 1.56 N

3. 2 N

4. 3 N

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 $\sqrt{7\mathrm{gr}}$ at the lowest point. The tension in the string at the lowest point is:
1. \(6mg\)
2. \(7mg\)
3. \(8mg\)
4. \(mg\)