Chapters 15 and 16 Written Homework
Be sure to show all your work, particularly for odd-numbered questions. If you end up looking at a solution please cite the source of your information.
15 โ 6: A cord of mass 0.65 ๐๐ is stretched between two supports 7.2 ๐ apart. If the tension in the cord is 120 ๐, how much time will it take a pulse to travel from one support to the other?
15 โ 31: A sinusoidal wave traveling on a cord in the negative ๐ฅ direction has amplitude 1.00 ๐๐, wavelength 3.00 ๐๐, and frequency 245 ๐ป๐ง. At ๐ก = 0, the particle of string at ๐ฅ = 0 is displaced a distance ๐ท = 0.80 ๐๐ above the origin and is moving upward.
- a) Sketch the shape of the wave at ๐ก = 0.
- b) Determine the function of ๐ฅ and ๐ก that describes the wave.
16 โ 75: A motion sensor can accurately measure the distance ๐ to an object repeatedly via the sonar technique used in Example 16 – 2. A short ultrasonic pulse is emitted and reflects from any object it encounters, creating echo pulses upon their arrival back at the senor. The sensor measures the time interval ๐ก between the emission of the original pulse and the arrival of the first echo.
- a) The smallest time interval ๐ก that can be measured with high precision is 1.0 ๐๐ . What is the smallest distance (at 20ยฐ ๐ถ) that can be measured with the motion sensor?
- b) To measure an objectโs speed the motion sensor makes 15 distance measurements every second (that is, it emits 15 sound pulses per second at evenly spaced time intervals), the measurement of ๐ก must be completed within the time interval between the emissions of successive pulses. What is the largest distance (at 20ยฐ ๐ถ) that can be measured with the motion sensor?
- c) Assume that during a lab period the roomโs temperature increases from 20ยฐ ๐ถ to 23ยฐ ๐ถ. What percent error will this introduce into the motion sensorโs distance measurements?
There is an optional bonus question on the next page.
Optional Bonus Question: Show by direct substitution that the following functions satisfy the wave equation:
- a) ๐ท(๐ฅ, ๐ก) = ๐ด ๐๐(๐ฅ + ๐ฃ๐ก)
- b) ๐ท(๐ฅ, ๐ก) = (๐ฅ โ ๐ฃ๐ก)4
Hint: See example 15-17.
With partial derivatives you treat the variables that you are NOT differentiating
with respect to as if they are constants. For example:
๐
๐๐ก [3๐ฅ๐ก2 + 2๐ฅ] = 6๐ฅ๐ก + 0
๐
๐๐ฅ [3๐ฅ๐ก2 + 2๐ฅ] = 3๐ก2 + 2