Question 19.QE.2: Describing the motion of two objects The graphical descripti......
Describing the motion of two objects
The graphical descriptions of the vibrational motions of two objects are shown in the figure. Write mathematical descriptions (equations) for these motions.
Represent mathematically (a) Vibrating object 1 has its maximum displacement of A = 0.2 m and a period T = 8 s. At time 0 the object is at zero and then its position increases. Thus, the function that describes its motion is a sine function:
x=A \sin \left(\frac{2 \pi}{T} t\right)
(b) Vibrating object 2 has a maximum displacement A = 0.4 m at time zero and has period T = 0.6 s. Thus, the function
x=A \cos \left(\frac{2 \pi}{T} t\right)
describes its motion.
Solve and evaluate The mathematical description of vibrating object 1 is
x=(0.2 \mathrm{~m}) \sin \left(\frac{2 \pi}{8 \mathrm{~s}} t\right)
The mathematical description of vibrating object 2 is
x=(0.4 \mathrm{~m}) \cos \left(\frac{2 \pi}{0.6 \mathrm{~s}} t\right)
Try it yourself: Write a mathematical description for the displacement of an object that has amplitude 0.10 m and frequency of 5.0 Hz, and starts vibrating from position x = -A.
Answer: x=-(0.10 \mathrm{~m}) \cos \left(\frac{2 \pi}{0.20 \mathrm{~s}} t\right) . Note that the period is T=\frac{1}{f}=\frac{1}{5.0 \mathrm{~s}^{-1}}=0.20 \mathrm{~s} .