Simple Harmonic Motion (Spring)

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Goal: In this lab we study examples of simple harmonic motion. We will focus on two systems: (i) a block and spring system, and (ii) a simple pendulum oscillating in gravitational field. Theory: In simple harmonic motion, the force is proportional to the displacement from the equilibrium position. The direction of the force is always towards the equilibrium position. The position of a simple harmonic oscillator varies periodically in time according to the expression where A is the amplitude of the motion, w is the angular frequency, and f is the phase constant. The value of f depends on the initial position and velocity of the oscillator. The time T for one complete vibration is called the period of the motion: Example 1: Block - spring system If a block of mass m is attached to a spring and set to oscillate, then it describes simple harmonic motion because the restoring force is proportional to the displacement and is given by: where, k is the spring constant. The following are some properties of a block and spring system oscillating in gravitational field that we will study: 1. The period of a block and spring system is independent of the amplitude of oscillation. 2. The period of a block and spring depends on the spring constant and the mass of the block. Example 2: Simple pendulum The oscillations of a simple pendulum (point mass m suspended by a light string of length l) are governed by the following equation: For small angles of oscillations sinq = x/l, is governed by: Therefore, the simple pendulum has the following properties. 1. The period of a pendulum is independent of its own mass and the amplitude of

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