\(\theta\)(in degrees)	\(\theta\)(in radian)	sin\(\theta\)
0 5 10 15 20 25 30 35 40 45	0 0.087 0.174 0.262 0.349 0.436 0.524 0.611 0.698 0.785	0 0.087 0.174 0.256 0.342 0.422 0.500 0.574 0.643 0.707

For \(\theta\) in radian, sin \(\theta\) ≈ \(\theta\) for very small angles

This means that “for θ as large as 10 degrees, sin θ is nearly the same as θ when θ is expressed in radians”. As θ increases in value sinθ gradually becomes different from θ

Consider a U -shaped glass tube which consists of two open arms with uniform cross-sectional area A. Let us pour a non-viscous uniform incompressible liquid of density p in the U- shaped tube to a height h as shown in the figure. If the liquid and tube are not disturbed then the liquid surface will be in equilibrium position O. It means the pressure as meazured at any point on the liquid is the same and also at the surface on the arm (edge of the tube on either side), which balances with the atmospheric pressure. Due to this the level of liquid in each arm will be the same. by blowing air one can provide sufficient force in one arm, and the liquid gets disturbed from equilibrium position O which means, the pressure at blown arm is higher than the other arm. This created difference in pressure which will cause the liquid to oscillate for a very short duration of time about the mean or equilibrium position and finally comes to rest
Time period of the oscillation is
T = \(2\pi \sqrt { \frac { 1 }{ 2g } } \) second   ......(1)

Consider a system containing a block of mass m attached to a massless spring with stiffness constant of force constant or spring constant k placed on a smooth horizontal surface (frictionless surface) as shown in figure. Let x₀ be the equilibrium position or mean position of mass m when it is left undisturbed. Suppose the mass is displaced through a small displacement x towards right from its equilibrium position and then released, it will oscillate back and forth about its mean position x₀. Let F be the restoring force (due to stretching of the spring) which is proportional to the amount of displacement of block. For one dimensional motion, mathematically, we have
\(F \propto x \)
F = -kx
Where negative sign implies that the restoring force will always act opposite to the direction of the displacement. This equation is called Hooke's law. Notice that, the restoring force is linear with the displacement. This is not always true; in case if we apply a very large stretching force, then the amplitude of oscillations becomes very large (Which means, force is proportional to displacement containing higher powers of x) and therefore, the oscillation of the system is not linear and hence, it is called non-linear oscillation. We restrict ourselves only to linear oscillations throughout our discussions, which means Hooke's law is valid (force and displacement have a linear relationship). From Newton's second law, we can write the equation for the particle executing simple harmonic motion
\(m \frac{d^{2} x}{d t^{2}} =-\mathrm{kx} \)
\(\frac{d^{2} x}{d t^{2}} =-\frac{k}{m} x\) ....(1)
Comparing the equation (1) with simple harmonic motion equation we get
\(\omega^{2}=\frac{k}{m}\)
which means the angular frequency or natural frequency of the oscillator is
\(\omega=\sqrt{\frac{k}{m}} \mathrm{rad} \mathrm{s}^{-1}.\)  .....(2)
The frequency of the oscillation is
\(f=\frac{\omega}{2 \pi}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}} \text { Hertz }\) .....(3)
and the time period of the oscillation is
\(\mathrm{T}=\frac{1}{f}=2 \pi \sqrt{\frac{m}{k}} \text { seconds }\) .....(4)

S.No	Simple Harmonic Motion	Angular Harmonic Motion
1.	The displacement of the particle is measured in terms of linear displacement \(\vec { r } \).	The displacement of the particle is measured in terms of angular displacement  \(\vec { \theta } \)(also known as angle of twist).
2.	Acceleration of the particle is \(\vec { a } =-{ \omega }^{ 2 }\vec { r } \)	Angular acceleration of the particle is \(\vec { \alpha } =-{ \omega }^{ 2 }\vec { \theta } \)
3.	Force, \(\vec { F } =m\vec { a } \) where m is called mass of the particle.	Torque, \(\vec { \tau } =I\vec { \propto } \) where I is called moment of inertia of a body.
4.	The restoring force \(\vec { F } =-k\vec { r } \), where k is restoring force constant.	The restoring torque \(\vec { \tau } =-k\vec { \theta } \), where the symbol k (kappa) is called restoring torsion constant It depends on the property of a particular torsion fiber.
5.	Angular frequency \(\omega =\sqrt { \frac { k }{ m } } \)rads-1	Angular frequency, \(\omega =\sqrt { \frac { k }{ I } } \) rad s-1

Consider a particle of mass m moving with uniform speed v along the circumference of a circle of radius A is anti clockwise direction. Let us assume that the origin of the coordinate system coincides with the centre O of the circle.
Graphical represention:
The points P₁, P₂, P₃...... P₈ in the circle show the various position of the particle during its circular motion.
Consider the points P₁, P₂, P₃, ..... P₈, on the right hand side. They are the projections of the points P₁, P₂, P₃, in the circle. If we look at the motion of these projections their motion seems to be periodic and oscillatory.
This is called simple harmonic motion.