Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7 shows a graph of the ball’s position x as a function of time t. What are the oscillation’s (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

Short answer

(a) The periodof the oscillationis 0.8 s.

(b) The frequencyof the oscillationis 1.25 Hz.

(c) The angular frequencyof the oscillationis7.885 rad/s.

(d) The amplitudeof the oscillationis 3 cm.

(e) Theforce constant of the springis 148.1 N/m.

Step-by-step solution

Definition of oscillatory motion

A motion repeating itself is referred to as periodic or oscillatory motion. An object in such motion oscillates about an equilibrium position due to a restoring force or torque.

Determine the missing parameters using graph

The amplitude is the maximum displacement of the wave, A=3 cm.

a) The period of the oscillation T=0.8 s.

b) Calculate the frequency of oscillation,

$$\begin{gathered} \mathit{f}\mathbf{=}\frac{\mathbf{1}}{\mathbf{T}} \\ \mathit{f}\mathbf{=}\frac{\mathbf{1}}{\mathbf{0}\mathbf{.}\mathbf{8}} \\ \mathit{f}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{25}\mathit{H}\mathit{z} \end{gathered}$$

c) Calculate the angular frequency of the oscillation,

$$\begin{gathered} \mathit{\omega }\mathbf{=}\mathbf{2}\underset{}{\mathbf{\prod }\mathbf{f}} \\ \omega =2\underset{}{\prod \times 1.25} \\ \omega =7.855rad/s \end{gathered}$$

d) The amplitude of the oscillation is 3 cm.

e) Calculate the force constant of the spring,

$$\begin{gathered} \mathit{\omega }\mathbf{=}\sqrt{\frac{\mathbf{k}}{\mathbf{m}}} \\ {\omega }^2=\frac{k}{m} \\ k={\omega }^{2}m \\ k={\left(7.855\right)}^2\times 2.4 \\ k=148.1N/m \end{gathered}$$

Hence, the values of period, frequency, angular frequency, amplitude and force constant of the spring are0.8 s, 1.25 Hz, 7.885 rad/s, 3 cm and148.1 N/m respectively.