Reference no: EM13871608

1. A thin stick one metre long is suspended at one end from the ceiling by a string.  As it swings, the angle of the stick to the vertical is consistently twice that of the string.  What is the length of the string? (Show all working.)

2.  A 20 cm diameter hollow ball is suspended on a 20 cm string.  (a) How many degrees of freedom will it have?  (b) Describe the modes of oscillation. (c) What are the corresponding frequencies of oscillation?

3.  In a simulation, the stick of question 1 is to be represented by two point masses.  Where should they be placed?

4. What is the inertia tensor of a 1 metre rigid square sheet of mass M, lying diagonally in the xy plane?

5.  A fragile crystal wineglass has virtually no damping.  If the sides are squeezed with a force of 5 N, its width will be compressed by 1 mm, at which point it will break.  Its dynamics can be described by a pair of masses each 1 gram, joined by a spring.  At what frequency will it 'ping'? (Hint: Since the movement is symmetric, assume that the centre remains fixed and base your equations on a single side.)

6.  The 'ping' amplitude dies away with a time-constant of ten seconds, meaning that the sinusoidal deflection is multiplied by e^-t/10.  What is now the second-order equation that describes the deflection?

7.  An opera singer is challenged to sing at it to cause it to shatter.  If the sound is equivalent to a vibrating 'pinch' force of amplitude .001 N, will the glass be broken?  If so, how long will it take it to break? (The marks are for the calculation and reasoning, with derivations of differential equations and their solution.)