A pendant having a bob of mass \(m\) is hanging in a ship sailing along the equator from east to west. When the strip is stationary with respect to water, the tension in the string is \(T_0.\) The difference between \(T_0\) and earth attraction on the bob, is:
1. \(\frac{mg+m\omega^2R}{2}\)
2. \(\frac{m\omega^2R}{3}\)
3. \(\frac{m\omega^2R}{2}\)
4. \(\frac{m\omega^2}R\)

A solid sphere is set into motion on a rough horizontal surface with a linear speed \(v\) in the forward direction and an angular speed \(\frac{v}{R}\) in the anticlockwise direction as shown in the figure. The linear speed of the sphere when it stops rotating is:

1. \(\frac{3v}{5}\)
2. \(\frac{2v}{5}\)
3. \(3v\)
4. \(\frac{6v}{5}\)

Two blocks of masses \(m_1\) and \(m_2\) are connected by a spring of spring constant \(k.\) The block of mass \(m_2\) is given a sharp impulse so that it acquires a velocity \(v_0\) towards the right. What is the maximum elongation that the spring will suffer?

1. \(\left[\frac{\mathrm{m}_1 \mathrm{~m}_2}{\mathrm{~m}_1+\mathrm{m}_2}\right]^{\frac{1}{2}} \mathrm{v}_0\)
2. \(\left[\frac{\mathrm{m}_1+\mathrm{m}_2}{\mathrm{~m}_1-\mathrm{m}_2}\right] \mathrm{v}_0\)
3. \(\left[\frac{\mathrm{m}_1+\mathrm{m}_2}{\mathrm{~m}_1-\mathrm{m}_2}\right]^{\frac{1}{2}} \mathrm{v}_0\)
4. \(\left[\frac{2 \mathrm{~m}_1+\mathrm{m}_2}{\mathrm{~m}_1 \mathrm{~m}_2}\right]^{\frac{1}{2}} \mathrm{v}_0\)

A ball of mass \(m\) hits the floor with a speed \(v\) making an angle of incidence \(e\) with the normal. The coefficient of restitution is \(e.\) The speed of the reflected ball and the angle of reflection of the ball will be:

1. \(v'=v,~\theta=\theta'\)
2. \(v'=\frac v2,~\theta=2\theta'\)
3. \(v'=2v,~\theta=2\theta'\)
4. \(v'=\frac{3v}{2},~\theta=\frac{2\theta'}3\)

A particle slides on the surface of a fixed smooth sphere starting from the  topmost point. The angle rotated by the radius through the particle, when it leaves contact with the sphere, is:
1. \(\theta=cos^{-1}\Big(\frac13\Big)\)
2. \(\theta=cos^{-1}\Big(\frac23\Big)\)
3. \(\theta=tan^{-1}\Big(\frac13\Big)\)
4. \(\theta=sin^{-1}\Big(\frac43\Big)\)

What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle \(\frac{\theta}{2}\) with the horizontal?
1. \(r=\frac{v^2\cos^2\theta}{g\cos^2\theta}\)
2. \(r=\frac{2v\sin\theta}{g\tan\theta}\)
3. \(r=\frac{v\cos\theta}{g\sin^2\frac{\theta}{2}}\)
4. \(r=\frac{3v\cos\theta}{g\cot\theta}\)

A square loop is made by a uniform conductor wire as shown in the figure,     
The net magnetic field at the centre of the loop if the side length of the square is a:
1. \(\frac{\mu_{_0}i}{2a}\)
2. zero
3. \(\frac{\mu_{_0}i^2}{a^2}\)
4. None of these