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?

Verified step by step guidance

1. To find the period (T) of the oscillation, observe the graph and identify the time it takes for the ball to complete one full cycle. From the graph, the ball returns to the same position after 0.50 seconds, indicating the period T = 0.50 s.
2. The frequency (f) is the reciprocal of the period. Use the formula: $f=\frac{1}{T}$. Substitute the period value to find the frequency.
3. The angular frequency (ω) can be calculated using the formula: $\omega =2\pi f$. Substitute the frequency value to find the angular frequency.
4. The amplitude (A) is the maximum displacement from the equilibrium position. From the graph, the amplitude is 3 cm, as the ball moves from -3 cm to +3 cm.
5. To find the force constant (k) of the spring, use the formula for the angular frequency: $\omega =\sqrt{\frac{k}{m}}$. Rearrange to solve for k: $k=m\omega ^2$. Substitute the mass and angular frequency values to find the force constant.

Key Concepts

Oscillation

Oscillation refers to the repetitive variation, typically in time, of some measure about a central value. In the context of a mass-spring system, it describes how the mass moves back and forth around an equilibrium position due to the restoring force of the spring. The motion can be characterized by parameters such as period, frequency, and amplitude.

Period and Frequency

The period (T) of an oscillation is the time it takes to complete one full cycle of motion, while frequency (f) is the number of cycles per unit time, typically measured in Hertz (Hz). They are inversely related, expressed by the formula f = 1/T. Understanding these concepts is crucial for analyzing the motion depicted in the graph.

Spring Constant

The spring constant (k) is a measure of a spring's stiffness, defined by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from the equilibrium position (F = -kx). The spring constant is essential for determining the dynamics of the oscillating system, including the relationship between the mass, frequency, and amplitude of the oscillation.