Question

A block is attachod to a ceiling by a spring of force constant kWhen palled down and released, the block undergoes simple harmonic motion. The motion of the block is shown in the time plot 0(cm) Which of the choices describes the amplitude A and 16r 24r 32 frequency fof the motion of the block? O4-2 Cm t O4-2 cm f 16r Hz O4-4m H Which of the equations of motion best descrihes the motion of the block? y 2co 4c(16st) 4 cos 2 coe(8t) It the mass of the blexck is 8.14 kg, what is the force constant & of the spring? N/m

Answer 1

E a) Amplitude A = 200m frequency f= 1/= tok YCH) = A Cosest = A Gos(22It) z 2 cos (2ax that) 22 Cos Cue W = 1 (274) = 1 h (7X HOR) M K = 0.1272 mm

Answer 2

Finding the period of oscillation is usually the easiest way to begin analyzing sine (and cosine) plots. Here, the period ?$T$ equals $16\pi$, since that is the time needed to complete the cycle. The frequency is simply the inverse of the period, so $f={\displaystyle \frac{1}{16\pi }}$.

The amplitude ?$A$ is the maximum distance of travel from the origin. Here, $A=2\text{ cm.}$

The general form of the equation of motion for simple harmonic motion is

$$y\left(t\right)=A\cos (\omega t+\varphi )$$

In this case, the phase ?=0$\varphi =0$ since the block begins at its maximum positive displacement. The angular frequency 1$\omega =\frac{1}{8}{\text{ s}}^{-1}$ since ?=2??$\omega =2\pi f$. Substitute the values into the equation of motion.

$$y\left(t\right)=2\cos \left(\frac{t}{8\text{ s}}\right)$$

For simple harmonic motion, the relationship between the force constant $k$ and the mass $m$ of the block is given by

$$\omega =\sqrt{\frac{k}{m}}$$

Solve for the force constant and evaluate.

$$k=m{\omega }^2=9.14\text{ kg}\times {\left(\frac{1}{8}{\text{ s}}^{-1}\right)}^2=0.143\text{ N/m}$$

The force constant $k$ equals 0.143 N/m.