Physics 1004 S2014 Class 2 clickers

# Wave or pulse v in a string with tension τ and mass

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wave or pulse, v, in a string with tension τ and mass per unit length is v v is the mass of string per unit length (kg/m) often known as the linear density Example Hiker (mass 80.0 kg) is rescued by a helicopter lowering a cable (length 15.0 m, mass 8.00 kg) and chair (70.0 kg) Helicopter accelerates upwards Hiker signals the pilot by pulling on the cable. Pulse takes 0.250 sec to travel up the cable Calculate the tension in the cable mg Tension force T a m/s 2 Hiker + chair mg T ma F Free body diagram Newton’s Second Law +y v mg ma F For waves on a string 2 2 v l m v cable cable cable cable l m Definition of linear density 1 2 3 Using equations 2 and 3 We are also given that it takes 0.250 sec for a pulse to travel up the 15.0 m cable N 10 x 92 . 1 m/s 0 . 60 m 0 . 15 kg 00 . 8 m/s 0 . 60 250 . 0 0 . 15 3 2 2 v v Speed of pulse Now calculate the tension

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7/10/2014 13 Power of a Mechanical Wave The energy carried by one wavelength takes a time T (the period) to pass by a point The power carried by the wave is (the proof is in section 16.6 of HRW) v A P 2 2 2 1  This expression is for a mechanical wave on a string Power of a Wave Rate of energy transfer for any sinusoidal wave is proportional to: The angular frequency (and hence the speed) Square of the amplitude Consequences: Higher frequency waves transmit more energy Earthquakes: Change in Amplitude at Boundaries Between Materials An earthquake wave a long distance from the epicentre can be modeled as a travelling wave transporting energy v A P 2 Power transmitted It is being transmitted through a 3D solid, so the details differ from waves on a string Severe Damage and Collapse of Nimitz Freeway in California 1989 Why did the highway collapse? Granite Speed of wave = v g Mud flat Speed of wave = v m Suppose all of the energy is transmitted across the interface from granite to mud - no reflection Note that v g = 25 v m granite granite v A P 2 mudflat mudflat v A P 2 and Hence mudflat mudflat granite granite v A v A 2 2 granite mudflat mudflat granite v v A A 2 2 25 1 2 2 mudflat granite A A
7/10/2014 14 2 2 25 mudflat granite A A granite mudflat A A 5 The amplitude of the wave gets 5 times bigger as a result of crossing the boundary from a region where the wave moves fast to one where it moves more slowly This means that one side of the boundary oscillates much more than the other, creating a shear stress Combining Two Sinusoidal Waves We can just do simple addition on the two sine of cosine functions to get the combined wave function.

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