WEBVTT
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A string with a linear mass density of 0.0060 kilograms per meter is tied to the ceiling, and a mass of 20 kilograms is tied to the free end of the string.
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The string is plucked, sending a pulse down the string.
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Find the speed of the pulse down the string.
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In this statement, we’re told that the linear mass density of the string is 0.0060 kilograms per meter; we’ll call that 𝜇.
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We’re told that a mass of 20 kilograms is tied to one end of the string; we’ll call that value 𝑚.
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We want to know when the string is plucked, what is the speed of the pulse that moves down the string, which we’ll call 𝑣?
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To begin, let’s draw a diagram of the situation.
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We have a string with one end tied to a ceiling and the other end fixed to a mass of value 20 kilograms.
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This mass puts the string under tension.
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When the string is plucked, a pulse travels down the string with some speed that we’ve called 𝑣, and it’s that speed we want to solve for.
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Let’s recall the relationship between speed, tension, and linear mass density, 𝜇.
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Wave speed, 𝑣, is equal to the square root of the tension force acting on a string along its length divided by its linear mass density, 𝜇.
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Applying this relationship to our scenario, wave speed 𝑣 is equal to the square root of the tension force which is created by the weight force of the mass.
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Therefore, this force is the mass itself multiplied by the acceleration due to gravity, 𝑔.
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That tension force is divided by 𝜇.
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We assume here that 𝑔 is exactly 9.8 meters per second squared.
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We’re given values of 𝑚 and 𝜇 and can plug in for the three variables in this equation.
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When we compute this square root, we find that the wave speed along the string, to two significant figures, is equal to 180 meters per second.
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That’s how fast the pulse travels down the string.