## Elastic Moduli / SHM

Elastic Moduli

A body is elastic if it spontaneously returns to its original shape after being deformed in some way. A modulus is a physical constant. There are three main types of elastic moduli.

Young's Modulus

Y = stress / strain = (F/A) / (Dl/l)

where F is a force applied to the end of the object of study and directed towards or away from its attachment point, A is the cross-sectional area of the object, and l is the length of the object.

Shear Modulus

S = stress / strain = (perpendicular F/A) / (Dx/l)

where F is a force applied perpendicularly to the length of the object, A is the cross-sectional area of the object, D x is the lateral displacement, and l is the length of the object.

Bulk Modulus

B = stress / strain = DP / (DV/V)

where P is pressure and V is volume.

Hooke’s Law

Hooke's law states that stress is directly proportional to strain.

However, Hooke's law only holds true up to a certain limit, the proportionality limit. Between the proportionality limit and the elastic limit, stress is no longer proportional to strain, although the object will return to its original size and shape once the stress is removed. Beyond the elastic limit, the object remains permanently deformed.

Simple Harmonic Motion

Simple Harmonic Motion is oscillatory motion that occurs when a restoring force of the form F = kx acts on a body, where k is a constant and x is the displacement from the resting position. A graph of x against time shows a sinusoidal curve.

It can be shown that the frequency of oscillations is proportional to Ö (k/m), m being the mass of the body. Thus, for a body oscillating on a spring, the stiffer the spring the greater is k and the higher is the frequency. Similarly, the more massive the body, the lower the frequency.

Pendulum Motion

A pendulum exhibits simple harmonic motion. The period, which is the time taken for each cycle, is given by:

T = 2p Ö (l/g)

where l is the length of the pendulum.
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