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 time it takes for the ball to complete one full cycle on the graph. From the second image, the ball completes one cycle from 0 to 1 second, so T = 1.0 s.
2. The frequency (f) is the reciprocal of the period. Use the formula: f = 1/T. Substitute the period you found in the previous step to calculate the frequency.
3. The angular frequency (ω) is related to the frequency by the formula: ω = 2πf. Use the frequency calculated in the previous step to find the angular frequency.
4. The amplitude (A) is the maximum displacement from the equilibrium position. From the graph, the maximum displacement is 3.0 cm, so A = 3.0 cm.
5. The force constant (k) of the spring can be found using the formula: k = mω², where m is the mass of the ball (2.40 kg) and ω is the angular frequency. Substitute the 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 is periodic, characterized by parameters such as period, frequency, and amplitude.

Period and Frequency

The period (T) of an oscillation is the time taken for one complete cycle of motion, while frequency (f) is the number of cycles per unit time, typically measured in hertz (Hz). They are inversely related by the equation f = 1/T. Understanding these concepts is crucial for analyzing the motion of oscillating systems, such as the ball attached to the spring.

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 oscillation, including the angular frequency, which is given by the formula ω = √(k/m), where m is the mass attached to the spring.