m We can make a simple model of a string on a musical instrument by considering the string to consist of a single mass m attached to the middle of a "mass-less" string that is stretched between two supports a distance L apart. Consider displacements z of the mass m from its equilibrium position (equilibrium being when the string is in a straight line between the two end supports) that are small enough so that we may assume the tension T in the string remains constant. You may also ignore gravity throughout this problem. (Hint: If/when you come across a need to use sin and the angle is small, you may use the small angle approximation, namely sin ≈ tan ≈ 0 (where must be in radians when isolated). (a) Sketch the two tension forces exerted on the mass m by the left and right segments of the string in the above diagram. Add them graphically to show the net force acting on m. (No calculation needed - just a neat sketch.) (b) Show that the z-component of the net force acting on m can be written as Fnet =-Bz, (1) and express the constant B in terms of T and L. (c) Show that the equation describing the displacement z of the mass (away from its equilibrium position) can
m We can make a simple model of a string on a musical instrument by considering the string to consist of a single mass m attached to the middle of a "mass-less" string that is stretched between two supports a distance L apart. Consider displacements z of the mass m from its equilibrium position (equilibrium being when the string is in a straight line between the two end supports) that are small enough so that we may assume the tension T in the string remains constant. You may also ignore gravity throughout this problem. (Hint: If/when you come across a need to use sin and the angle is small, you may use the small angle approximation, namely sin ≈ tan ≈ 0 (where must be in radians when isolated). (a) Sketch the two tension forces exerted on the mass m by the left and right segments of the string in the above diagram. Add them graphically to show the net force acting on m. (No calculation needed - just a neat sketch.) (b) Show that the z-component of the net force acting on m can be written as Fnet =-Bz, (1) and express the constant B in terms of T and L. (c) Show that the equation describing the displacement z of the mass (away from its equilibrium position) can
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Transcribed Image Text:m
Z
12.3 We can make a simple model of a string on a musical instrument
by considering the string to consist of a single mass m attached to
the middle of a “mass-less” string that is stretched between two
supports a distance L apart. Consider displacements z of the mass
m from its equilibrium position (equilibrium being when the string
is in a straight line between the two end supports) that are small
enough so that we may assume the tension T in the string remains
constant. You may also ignore gravity throughout this problem.
(Hint: If/when you come across a need to use sin and the angle is small, you may use the small angle
approximation, namely sin 0 ≈ tan 0 ≈ 0 (where 0 must be in radians when isolated).
(a) Sketch the two tension forces exerted on the mass m by the left and right segments of the string in the above
diagram. Add them graphically to show the net force acting on m. (No calculation needed - just a neat sketch.)
(b) Show that the z-component of the net force acting on m can be written as
Fnet
- Bz,
(1)
=
2
and express the constant B in terms of T and L.
(c) Show that the equation describing the displacement z of the mass (away from its equilibrium position) can
be written in this form:
d²z
dt²
(2)
=-Cz.
Express the constant C in terms of the variables given in the problem. (You may use B from part (b) if you
have not solved that part yet.)
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