Vibrations and Waves MP205, Assignment 3

Please hand in your solutions before 5pm Wednesday 4th March, starred questions will bedone out in tutorial and do not need to be handed up.If you have any questions about this assignment, please ask your tutor,Una N Ruairc, (uniruairc@thphys.nuim.ie), Office 1.6, Dept. Mathematical Physics.

* 1. Verify the following statements:

(a) The acceleration of a system undergoing simple harmonic motion satisfiesa = 2x

(b) For an oscillating mass on a spring =

km

2. An item of mass 20 g is hung from a spring and set in oscillatory motion. Theangular frequency is given by = 10rads1.

(a) What is the spring constant?

(b) At t = 0 the displacement is 50 cm, what is the force on the mass at this time?

(c) What is the acceleration of the mass at t = 0?

3. A mass m hangs from a uniform spring of spring constant k.

(a) What is the period of oscillations, T, in the system?

(b) What would it be if the mass m were hung so that: (see Figure below).

(1) It was attached to two identical springs hanging side by side?

(2) It was attached to the lower of two identical springs connected end to end?

Give the new period of oscillations for each system in terms of T from part (a).

k kk

k

(1)(2)

4. A platform is executing simple harmonic motion in a vertical direction with anamplitude of 5cm and a frequency of 10

pivibrations per second. A block is placed on

the platform at the lowest point of its path. At what point will the block leave theplatform? [Hint: think of the forces on the block ]

* 5. A uniform rod of length L is nailed to a post so that two thirds of its length is belowthe nail. What is the period of small oscillations of the rod?

6. A circular hoop of diameter d hangs on a nail. What is the period of its oscillationsat small amplitude?

* 7. (a) An object of mass 0.5 kg is hung from the end of a steel wire of length 2 mand of diameter 0.5 mm. (Youngs modulus = 2 1011 N/m2). What is theextension of the wire?

(b) The object is lifted through a distance h (thus allowing the wire to becomeslack) and is then dropped so that the wire receives a sudden jerk. The ultimatestrength of steel is 1.1109 N/m2. What is the largest possible value of h ifthe wire is not to break?

8. A solid steel ball is to be hung at the bottom end of a steel wire of length 2m andradius 1mm. The ultimate strength (max stress) of steel is 1.1 109N/m2. Whatare:

(a) the mass of the biggest ball the wire can bear

(b) the radius of the biggest ball the wire can bear [Hint: use the fact that thedensity of steel is given by s = 7900kgm

3]

9. A metal rod, 0.5 m long, has a rectangular cross section of area 2 mm2. With therod vertical and a mass of 60kg hung from the bottom, there is an extension of 0.25mm. What is the Youngs modulus ( N/m2) for the material of the rod?