Live practice

A cylindrical cork of density ρ, base area A, and height h floats vertically in liquid density ρl. Show that a small vertical displacement produces SHM and obtain its period.

A cubical thermocol box of side \(0.30\,\mathrm{m}\) and wall thickness \(0.050\,\mathrm{m}\) contains \(4.0\,\mathrm{kg}\) ice at 0°C for \(6\,\mathrm{h}\) while outside is 45°C. Estimate ice remaining for k=0.010 W m⁻¹ K⁻¹ and Lf=335 kJ kg⁻¹.

A rocket leaves Earth's surface vertically at \(5.0\,\mathrm{km\,s^{-1}}\). With M=6.0e24 kg, R=6.4e6 m and G=6.67e-11 SI, find

An oxygen molecule has mass 5.30e-\(26\,\mathrm{kg}\), moment of inertia 1.94e-\(46\,\mathrm{kg\,m^{2}}\) and mean translational speed \(500\,\mathrm{m\,s^{-1}}\). Its rotational kinetic energy is two-thirds of its translational kinetic energy. Find mean angular speed.

A pendulum bob released from 30 degrees collides elastically and head-on with an identical stationary bob at the lowest point. How high does the first bob rise afterward?

A nucleus initially at rest splits into exactly two daughter nuclei. Prove their velocities point in opposite directions.

For r(t)=3.0t i-2.0t^2 j+4.0 k metres, find velocity and acceleration, then the speed and direction of velocity at \(2.0\,\mathrm{s}\).

Compare three equal marked time intervals on a position-time curve:

Before calculating, decide whether the Sun's mean density should resemble ordinary gas or condensed matter. Then estimate it from mass 2.0e30 kg and radius 7.0e8 m and explain the result.

A letter is selected randomly from ASSASSINATION. Find the probabilities of selecting (i) a vowel(ii) a consonant.