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Tutorial No 4, Semester 1,2012/13

1. A closed pipe is vibrating with 4 nodes between itstwo ends (not counting the node at one end) andhas the same frequency as a string with fundamen-tal frequency of 270 Hz vibrating with 6 antinodesbetween its two ends. What is the frequency of thisclosed pipe when it vibrates with 5 nodes betweenits two ends (not counting the node at one end)?An open pipe with 4 nodes between its two ends isvibrating at the same frequency as the closed pipewith 5 nodes. If the closed pipe has a length of kcm, what is the length of the open pipe?

2. An open pipe labelled K has a fundamental frequencyofn Hz. This open pipe is then cut into five shorterpipes, all of equal length, which we label L1 to L5.

L1, L2 and L3 are then joined together, and one endof this combined pipe closed up to make a closedpipe which we label M. L4 is cut into half and oneof the halves joined up with L5 to make an openpipe which we label N. What are the fundamentalfrequencies of M and N? If N is joined to the openend of M to make a longer closed pipe labelled O,what would be the fundamental frequency of O?

3. An open pipe which has a fundamental frequencyof 200 Hz is vibrating with 6 nodes between its twoends, producing beats of 10 Hz when it combineswith a closed pipe which is vibrating with 2 nodes

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between its two ends (not counting the node at one

end). What are the possible values of the funda-mental frequency of the closed pipe? If the openpipe could be slightly shortened by cutting off a veryshort piece of the pipe from one end, explain how thiscould be used to determine the actual fundamentalfrequency of the closed pipe. If the beat frequencyincreases when the open pipe is thus shortened, andthe closed pipe is h cm long, what was the originallength of the open pipe before it was shortened?

4. The three levers in the action of a Cristofori pianocause the corresponding hammer to move upwards tostrike the corresponding piano string when a partic-ular key on the piano is struck downwards. In a par-ticular Cristofori piano, the first, second and thirdlevers of the action multiply the distance moved bythe downwards movement of the key by factors of1, 2.8 and 5.8 respectively, causing the hammer to

move upwards by the combined factors of the threelevers. If a key is struck downwards with a veloc-ity of 4.5 cm per second, what is the velocity of thecorresponding hammer as it is caused to move up-wards? If the third lever is replaced and has a differ-ent multiplication factor, and it is found that for thehammer to move upwards with same velocity, thedownwards velocity of the key needs to be only 3.5

cm per second, what is the multiplication factor ofthe new third lever? (Give the answer to 2 decimalplaces.)

5. A piano piece is being performed by a pianist on a

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grand piano equipped with a properly functioning

sostenuto (middle) pedal. The pianist depresses thesustain or loud (rightmost) pedal and then depressesthe keys D2 and D3, then releases the D2 and D3keys and then releases the sustain pedal. He then de-presses the sostenuto pedal and depresses the E2 andB2 keys. He then releases the E2 and B2 keys and re-leases the sostenuto pedal. Finally he depresses theG2 and G3 keys and depresses the sostenuto (mid-dle) pedal. If he now releases the G2 and G3 keys,

what notes should still remain sustained as long asthe sostenuto pedal is depressed? Would it make anydifference if the roles of the sostenuto and sustainpedals are substituted for each other in the abovesequence?