ASSIGNMENT

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 – 6: A cord of mass 0.65 𝑘𝑔 is stretched between two supports 7.2 𝑚 apart. If
the tension in the cord is 120 𝑁, how much time will it take a pulse to travel from
one support to the other?

15 – 31: A sinusoidal wave traveling on a cord in the negative 𝑥 direction has
amplitude 1.00 𝑐𝑚, wavelength 3.00 𝑐𝑚, and frequency 245 𝐻𝑧. At 𝑡 = 0, the
particle of string at 𝑥 = 0 is displaced a distance 𝐷 = 0.80 𝑐𝑚 above the origin
and is moving upward.

a) Sketch the shape of the wave at 𝑡 = 0.

b) Determine the function of 𝑥 and 𝑡 that describes the wave.
16 – 75: A motion sensor can accurately measure the distance 𝑑 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 𝑡
between the emission of the original pulse and the arrival of the first echo.

a) The smallest time interval 𝑡 that can be measured with high precision is
1.0 𝑚𝑠. What is the smallest distance (at 20° 𝐶) that can be measured
with the motion sensor?

b) To measure an object’s 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 𝑡 must be completed
within the time interval between the emissions of successive pulses.
What is the largest distance (at 20° 𝐶) that can be measured with the
motion sensor?

c) Assume that during a lab period the room’s temperature increases from
20° 𝐶 to 23° 𝐶. What percent error will this introduce into the motion
sensor’s 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) 𝐷(𝑥, 𝑡) = 𝐴 𝑙𝑛(𝑥 + 𝑣𝑡)
b) 𝐷(𝑥, 𝑡) = (𝑥 − 𝑣𝑡)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:
𝜕
𝜕𝑡 [3𝑥𝑡2 + 2𝑥] = 6𝑥𝑡 + 0
𝜕
𝜕𝑥 [3𝑥𝑡2 + 2𝑥] = 3𝑡2 + 2