A piano has the widest range of notes of any instrument. It achieves this large range using both the length and mass density, H, of the strings. In a real piano, most of the strings have roughly the same tension. We shall assume that all the strings in our piano have a tension of T = 760 Newtons. L, L27 Bass strings L1 is the longest string L88 is the shortest string Line of strike Treble strings Grand Piano Details Grand Piano Simplified We shall assume the following regarding the longest and shortest strings in the piano: String frequency (Hz) length (m) tension (N) L1 27.5 0.163 760 L88 4186.01 0.046 760 Calculate all the following: The linear mass density of the L1 string: H1 = kg/m The linear mass density of the L88 string: H88 = kg/m As you likely calculated, the L1 string has a much higher u than the L88 string. If you have ever looked inside a piano you know this to be true. For the next calculation we are going to do something silly. We will calculate how long a string must be if we want to accomplish the following: • The frequency should match the L1 string, f= 27.5 Hz • The mass density should match the L88 string, u = H88 • The tension will remain normal, T = 760 Newtons. Calculate the length of this crazy string / meters

A piano has the widest range of notes of any instrument. It achieves this large range using both the length and mass density, H, of the strings. In a real piano, most of the strings have roughly the same tension. We shall assume that all the strings in our piano have a tension of T = 760 Newtons. L, L27 Bass strings L1 is the longest string L88 is the shortest string Line of strike Treble strings Grand Piano Details Grand Piano Simplified We shall assume the following regarding the longest and shortest strings in the piano: String frequency (Hz) length (m) tension (N) L1 27.5 0.163 760 L88 4186.01 0.046 760 Calculate all the following: The linear mass density of the L1 string: H1 = kg/m The linear mass density of the L88 string: H88 = kg/m As you likely calculated, the L1 string has a much higher u than the L88 string. If you have ever looked inside a piano you know this to be true. For the next calculation we are going to do something silly. We will calculate how long a string must be if we want to accomplish the following: • The frequency should match the L1 string, f= 27.5 Hz • The mass density should match the L88 string, u = H88 • The tension will remain normal, T = 760 Newtons. Calculate the length of this crazy string / meters