f = 1/T

Introduction to Oscillations

The conditions that lead to simple harmonic motion are as follows:

- There must be a position of
*stable equilibrium*. - There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force
*F*⃗ is given by*F*⃗ = −*k**x*⃗, where*x*⃗ is the displacement from equilibrium and*k*is a constant that depends on the properties of the oscillating system. - The resistive forces in the system must be reasonably small.

Consider a block of mass *m* attached to a spring with force constant *k*, as shown in the figure (Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at *x*=0. If the block is pulled to the right a distance *A* and then released, *A* will be the *amplitude* of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

The time it takes the block to complete one cycle is called the *period*. Usually, the period is denoted *T* and is measured in seconds.

The *frequency*, denoted *f*, is the number of cycles that are completed per unit of time: *f *= 1/*T*. In SI units, *f* is measured in inverse seconds, or hertz (Hz).

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency *f*? Express your answer in hertz.

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