The point of the needle of a sewing machine moves in SHM along the x-axis with a frequency of 2.5 Hz. At t = 0 its position and velocity components are +1.1 cm and -15 cm/s, respectively. (a) Find the acceleration component of the needle at t = 0.

Verified step by step guidance

1. Identify the given parameters: frequency (f) = 2.5 Hz, initial position (x_0) = +1.1 cm, and initial velocity (v_0) = -15 cm/s.
2. Calculate the angular frequency (\(\omega\)) using the formula \(\omega = 2\pi f\).
3. Use the formula for the position of an object in SHM, \(x(t) = A \cos(\omega t + \phi)\), and solve for the amplitude (A) and phase constant (\(\phi\)) using the initial conditions.
4. Differentiate the position function \(x(t)\) to find the velocity function \(v(t) = -A\omega \sin(\omega t + \phi)\) and verify it with the given initial velocity to check the correctness of A and \(\phi\).
5. Differentiate the velocity function to find the acceleration function \(a(t) = -A\omega^2 \cos(\omega t + \phi)\), and evaluate it at t = 0 to find the acceleration component of the needle.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal position, velocity, and acceleration functions over time. In this context, the needle's movement can be described by SHM equations, which relate its position, velocity, and acceleration.

Acceleration in SHM

In SHM, the acceleration of an object is directly related to its displacement from the equilibrium position and is given by the formula a = -ω²x, where ω is the angular frequency and x is the displacement. This negative sign indicates that the acceleration is always directed towards the equilibrium position, acting as a restoring force. Understanding this relationship is crucial for calculating the acceleration at any point in the motion.

Angular Frequency

Angular frequency (ω) is a measure of how quickly an object oscillates in SHM, defined as ω = 2πf, where f is the frequency of the motion. It is expressed in radians per second and provides insight into the speed of oscillation. In this problem, knowing the frequency of 2.5 Hz allows us to calculate the angular frequency, which is essential for determining the acceleration of the needle at a specific time.