Simple Harmonic Motion — Revision Notes

⚡ 30-Second Revision

Defining Condition: — $F = -kx$ or $a = -\omega^2 x$
Angular Frequency: — $\omega = \sqrt{k/m}$ (spring-mass), $\omega = \sqrt{g/L}$ (simple pendulum)
Time Period: — $T = 2pi/\omega = 2pisqrt{m/k}$ (spring-mass), $T = 2pisqrt{L/g}$ (simple pendulum)
Frequency: — $f = 1/T = omega/(2\pi)$
Displacement: — $x(t) = A \sin(\omega t + \phi)$ or $A \cos(\omega t + \phi)$
Velocity: — $v(t) = Aomega \cos(\omega t + \phi)$ (max $v_{max} = Aomega$ at $x=0$)
Acceleration: — $a(t) = -Aomega^2 \sin(\omega t + \phi) = -\omega^2 x(t)$ (max $a_{max} = Aomega^2$ at $x=\pm A$)
Kinetic Energy: — $E_K = \frac{1}{2}mv^2 = \frac{1}{2}momega^2(A^2 - x^2) = \frac{1}{2}k(A^2 - x^2)$
Potential Energy: — $E_P = \frac{1}{2}kx^2$
Total Energy: — $E = E_K + E_P = \frac{1}{2}kA^2 = \frac{1}{2}momega^2 A^2$ (constant)

2-Minute Revision

Simple Harmonic Motion (SHM) is a special type of oscillatory motion where the restoring force is directly proportional to displacement and opposite in direction ($F=-kx$). This leads to acceleration $a=-\omega^2 x$, where $\omega$ is the angular frequency. Key parameters are amplitude ($A$), time period ($T=2pi/\omega$), and frequency ($f=1/T$).

Displacement, velocity, and acceleration vary sinusoidally with time. Velocity leads displacement by $pi/2$, and acceleration leads velocity by $pi/2$ (or is $\pi$ out of phase with displacement). Maximum velocity is $Aomega$ (at equilibrium), and maximum acceleration is $Aomega^2$ (at extreme positions).

Energy is conserved in ideal SHM. Total energy $E = \frac{1}{2}kA^2$ is constant, continuously converting between kinetic energy ($E_K = \frac{1}{2}k(A^2-x^2)$) and potential energy ($E_P = \frac{1}{2}kx^2$). At equilibrium, $E_K$ is max, $E_P$ is zero. At extreme positions, $E_K$ is zero, $E_P$ is max.

Common examples include spring-mass systems ($T=2pisqrt{m/k}$) and simple pendulums (for small angles, $T=2pisqrt{L/g}$). Remember the square root dependence of $T$ on $m$ or $L$, and inverse square root dependence on $k$ or $g$. Be careful with units and phase relationships.

5-Minute Revision

Simple Harmonic Motion (SHM) is a fundamental concept describing oscillations where the restoring force is linear ($F=-kx$). This implies acceleration is proportional to displacement and oppositely directed ($a=-\omega^2 x$). The angular frequency $\omega = \sqrt{k/m}$ for a spring-mass system and $\omega = \sqrt{g/L}$ for a simple pendulum (small angles). The time period $T = 2pi/\omega$ and frequency $f = 1/T$.

The position of a particle in SHM is given by $x(t) = A \sin(\omega t + \phi)$, where $A$ is amplitude and $\phi$ is initial phase. Differentiating this gives velocity $v(t) = Aomega \cos(\omega t + \phi)$ and acceleration $a(t) = -Aomega^2 \sin(\omega t + \phi)$. Notice that $v$ leads $x$ by $pi/2$, and $a$ leads $v$ by $pi/2$ (or $a$ is $\pi$ out of phase with $x$).

Key Points for Maxima/Minima:

Equilibrium ($x=0$): — Velocity is maximum ($v_{max}=Aomega$), acceleration is zero, restoring force is zero, potential energy is zero, kinetic energy is maximum ($E_K = \frac{1}{2}mA^2\omega^2$).
Extreme Positions ($x=\pm A$): — Velocity is zero, acceleration is maximum ($a_{max}=Aomega^2$), restoring force is maximum ($F_{max}=kA$), potential energy is maximum ($E_P = \frac{1}{2}kA^2$), kinetic energy is zero.

Energy Conservation: Total mechanical energy $E = E_K + E_P = \frac{1}{2}kA^2 = \frac{1}{2}momega^2 A^2$ is constant. This allows you to find velocity at any position: $v = omegasqrt{A^2-x^2}$.

Example: A mass of $0.1\,\text{kg}$ oscillates with $A=0.02\,\text{m}$ and $T=0.5\,\text{s}$.

1
$\omega = 2pi/T = 2pi/0.5 = 4pi,\text{rad/s}$.
2
$v_{max} = Aomega = 0.02 \times 4\pi = 0.08pi,\text{m/s}$.
3
$a_{max} = Aomega^2 = 0.02 \times (4\pi)^2 = 0.02 \times 16\pi^2 = 0.32\pi^2\,\text{m/s}^2$.
4
Spring constant $k = momega^2 = 0.1 \times (4\pi)^2 = 0.1 \times 16\pi^2 = 1.6\pi^2\,\text{N/m}$.
5
Total Energy $E = \frac{1}{2}kA^2 = \frac{1}{2}(1.6\pi^2)(0.02)^2 = 0.8\pi^2 \times 0.0004 = 0.00032\pi^2\,\text{J}$.

Remember to handle units carefully and understand the physical implications of each parameter.

Prelims Revision Notes

Simple Harmonic Motion (SHM) - NEET Revision Notes

1. Defining Condition:

Restoring Force: $F = -kx$ (Hooke's Law)
Acceleration: $a = -\omega^2 x$, where $\omega = \sqrt{k/m}$ (angular frequency)
The negative sign indicates force/acceleration is opposite to displacement, directed towards equilibrium.

2. Key Parameters:

Amplitude ($A$): — Maximum displacement from equilibrium.
Time Period ($T$): — Time for one complete oscillation. $T = 2pi/\omega$.
Frequency ($f$): — Number of oscillations per second. $f = 1/T = omega/(2\pi)$.
Angular Frequency ($\omega$): — $\omega = 2\pi f = 2pi/T$.
Phase ($\phi$): — Initial phase constant, determined by initial conditions.

3. Equations of Motion:

Displacement: — $x(t) = A \sin(\omega t + \phi)$ or $x(t) = A \cos(\omega t + phi')$.
Velocity: — $v(t) = \frac{dx}{dt} = Aomega \cos(\omega t + \phi)$.

* Maximum velocity: $v_{max} = Aomega$ (at $x=0$). * Velocity at any $x$: $v = \pm omegasqrt{A^2 - x^2}$.

Acceleration: — $a(t) = \frac{dv}{dt} = -Aomega^2 \sin(\omega t + \phi) = -\omega^2 x(t)$.

* Maximum acceleration: $a_{max} = Aomega^2$ (at $x=\pm A$).

4. Phase Relationships:

Velocity leads displacement by $pi/2$ (or $90^\circ$).
Acceleration leads velocity by $pi/2$ (or $90^\circ$).
Acceleration is $\pi$ (or $180^\circ$) out of phase with displacement.

5. Energy in SHM (Ideal/Undamped):

Kinetic Energy ($E_K$): — $E_K = \frac{1}{2}mv^2 = \frac{1}{2}momega^2(A^2 - x^2) = \frac{1}{2}k(A^2 - x^2)$.

* Maximum at $x=0$, zero at $x=\pm A$.

Potential Energy ($E_P$): — $E_P = \frac{1}{2}kx^2$.

* Maximum at $x=\pm A$, zero at $x=0$.

Total Mechanical Energy ($E$): — $E = E_K + E_P = \frac{1}{2}kA^2 = \frac{1}{2}momega^2 A^2 = \text{constant}$.

6. Examples of SHM:

Spring-Mass System: — $T = 2pisqrt{m/k}$. (For springs in series: $1/k_{eq} = 1/k_1 + 1/k_2$; for parallel: $k_{eq} = k_1 + k_2$).
Simple Pendulum (small angles): — $T = 2pisqrt{L/g}$. (Independent of mass and amplitude for small angles).

7. Important Points:

At equilibrium ($x=0$): $v_{max}$, $a=0$, $F=0$, $E_K$ max, $E_P=0$.
At extreme positions ($x=\pm A$): $v=0$, $a_{max}$, $F_{max}$, $E_K=0$, $E_P$ max.
All SHM is oscillatory, but not all oscillatory motion is SHM (e.g., large angle pendulum).
Average velocity over one complete cycle is zero. Average speed is $4A/T$.

Vyyuha Quick Recall

To remember the phase relationships in SHM (Displacement, Velocity, Acceleration):

Displacement Lags Velocity Lags Acceleration by $pi/2$.

Think: Don't Lag, Velocity Leads Always!

This means if displacement is a sine function, velocity is a cosine (leading by $pi/2$), and acceleration is a negative sine (leading velocity by $pi/2$, thus lagging displacement by $\pi$). Alternatively, acceleration is always opposite to displacement.