(II) A cord of length 1.0 m has two equal-length sections with linear densities of 0.50 kg / m a

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(II) A cord of length 1.0 m has two equal-length sections...

(II) A cord of length 1.0 $\mathrm{m}$ has two equal-length sections with linear densities of 0.50 $\mathrm{kg} / \mathrm{m}$ and 1.00 $\mathrm{kg} / \mathrm{m}$ . The tension in the entire cord is constant. The ends of the cord are oscillated so that a standing wave is set up in the cord with a single node where the two sections meet. What is the ratio of the oscillatory frequencies?

Key Concepts

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere, resulting in fixed points called nodes where the medium does not move. The pattern of nodes and antinodes is determined by the boundary conditions of the system, which are essential in analyzing resonant oscillations.

Wave Speed on a String

The speed of a wave propagating along a string is given by the formula v = ?(T/?), where T is the tension in the string and ? is the linear mass density. This relation highlights how variations in mass density or tension affect the speed at which disturbances travel along the string.

Normal Modes and Frequency Determination

Normal modes refer to the specific standing wave patterns that satisfy the boundary conditions of the system, typically resulting in discrete frequencies. The frequency of these modes is determined by the wave speed and the effective length or wavelength of the vibration pattern, generally expressed by relationships like f = n(v/2L) for strings with fixed boundaries.

Boundary Conditions at Junctions

When dealing with a string composed of sections with different physical properties, the interface must satisfy specific boundary conditions such as continuity of displacement and constant tension. These conditions ensure that a node or antinode forms at the junction, playing a crucial role in determining the overall standing wave behavior and frequency distribution.

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