$A$ wire of length $2.5 \ m$ is fixed at one end and a box of mass $4 \ kg$ is tied at the other end. If the wire rotates in a horizontal circle about the fixed end with $\frac{2}{\pi} \ rev/s$ rotations per second,then the tension in the wire is (in $N$)

$A$ roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is $20\, m$. The speed of the car at the top of the hill is between

Two particles of the same mass are moving in circular orbits because of a force given by $F(r) = -\frac{16}{r} - r^3$. The first particle is at a distance $r = 1$,and the second is at $r = 4$. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to:

"Momentum and changes in momentum are not always in the same direction." Explain with a suitable example.

$A$ cyclist taking a turn bends inwards while a car passenger taking the same turn is thrown outwards. The reason is:

$A$ train is moving with a speed of $12 \,m/s$ on rails which are $1.5 \,m$ apart. To negotiate a curve of radius $400 \,m$, the height by which the outer rail should be raised with respect to the inner rail is (Given, $g = 10 \,m/s^2$): (in $\,cm$)