Oscillations of Spring — Revision Notes

⚡ 30-Second Revision

Hooke's Law: — $F = -kx$
Angular Frequency: — $omega = sqrt{k/m}$
Period: — $T = 2pisqrt{m/k}$
Frequency: — $f = \frac{1}{2pi}sqrt{k/m}$
Displacement: — $x(t) = Acos(omega t + phi)$
Velocity: — $v(t) = -Aomegasin(omega t + phi)$
Maximum Velocity: — $v_{max} = Aomega$
Acceleration: — $a(t) = -Aomega^2cos(omega t + phi) = -omega^2 x$
Maximum Acceleration: — $a_{max} = Aomega^2$
Potential Energy: — $U = \frac{1}{2}kx^2$
Kinetic Energy: — $K = \frac{1}{2}mv^2$
Total Mechanical Energy: — $E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$
Springs in Series: — $rac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + dots$
Springs in Parallel: — $k_{eq} = k_1 + k_2 + dots$
Period is independent of amplitude and gravity (for vertical spring, only equilibrium shifts).

2-Minute Revision

The oscillations of a spring-mass system are a prime example of Simple Harmonic Motion (SHM). The core principle is Hooke's Law, $F = -kx$, where the restoring force is proportional to displacement $x$ and acts oppositely.

This leads to the defining SHM equation $a = -omega^2 x$, with angular frequency $omega = sqrt{k/m}$. The period of oscillation, $T = 2pisqrt{m/k}$, is crucial and depends only on mass ($m$) and spring constant ($k$), *not* on amplitude or gravity (for vertical springs, gravity just shifts the equilibrium point).

Frequency $f = 1/T$. Energy is conserved, continuously converting between kinetic ($K = \frac{1}{2}mv^2$) and potential ($U = \frac{1}{2}kx^2$) forms. Total energy is constant, $E = \frac{1}{2}kA^2$. Maximum velocity ($Aomega$) occurs at equilibrium, and maximum acceleration ($Aomega^2$) at extreme positions.

Remember how spring constants combine: series ($rac{1}{k_{eq}} = sum \frac{1}{k_i}$) makes the system 'softer', parallel ($k_{eq} = sum k_i$) makes it 'stiffer'. Be wary of common traps like confusing spring and pendulum dependencies or miscalculating energy at specific points.

5-Minute Revision

Let's quickly review the essentials of spring oscillations for NEET. At its heart, a spring-mass system undergoes Simple Harmonic Motion (SHM) because the restoring force, $F = -kx$ (Hooke's Law), is directly proportional to the displacement $x$ from equilibrium and acts in the opposite direction.

This leads to an acceleration $a = -(k/m)x$, which is the hallmark of SHM. From this, we derive the angular frequency $omega = sqrt{k/m}$, the period $T = 2pisqrt{m/k}$, and the frequency $f = \frac{1}{2pi}sqrt{k/m}$.

Key takeaways for $T$ and $f$: They depend on mass and spring constant, but *not* on the amplitude of oscillation. For vertical springs, gravity shifts the equilibrium position but does not change the period. If you cut a spring, its constant changes inversely with length (e.g., half length means double $k$).

Energy conservation is vital. The total mechanical energy $E = K + U$ is constant. Kinetic energy $K = \frac{1}{2}mv^2$ is maximum at equilibrium ($x=0$), where $v = v_{max} = Aomega$. Potential energy $U = \frac{1}{2}kx^2$ is maximum at the extreme positions ($x=pm A$), where $v=0$. The total energy can be expressed as $E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$. This allows you to find velocity at any displacement: $v = omegasqrt{A^2 - x^2}$.

Example: A $0.1,\text{kg}$ mass oscillates on a spring with $k = 40,\text{N/m}$ and amplitude $0.05,\text{m}$.

1
Period: — $T = 2pisqrt{0.1/40} = 2pisqrt{1/400} = 2pi/20 = pi/10,\text{s}$.
2
Total Energy: — $E = \frac{1}{2}kA^2 = \frac{1}{2} \times 40 \times (0.05)^2 = 20 \times 0.0025 = 0.05,\text{J}$.
3
Maximum Speed: — $E = \frac{1}{2}mv_{max}^2 implies 0.05 = \frac{1}{2} \times 0.1 \times v_{max}^2 implies v_{max}^2 = 1 implies v_{max} = 1,\text{m/s}$.

Remember spring combinations: for series, $rac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$; for parallel, $k_{eq} = k_1 + k_2$. Practice these formulas and conceptual nuances to ace NEET questions.

Prelims Revision Notes

Oscillations of Spring: NEET Revision Notes

1. Simple Harmonic Motion (SHM) Basics:

Definition: — A type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.
Hooke's Law: — $F = -kx$, where $k$ is the spring constant (N/m) and $x$ is displacement.
SHM Equation: — $a = -omega^2 x$, where $omega$ is angular frequency.

2. Key Parameters & Formulas:

Angular Frequency ($omega$): — $omega = sqrt{k/m}$ (rad/s)
Period (T): — Time for one complete oscillation. $T = 2pi/omega = 2pisqrt{m/k}$ (s)

* Crucial: $T$ is independent of amplitude for ideal SHM. * Crucial: For vertical springs, $T$ is independent of gravity ($g$ only shifts equilibrium).

Frequency (f): — Number of oscillations per second. $f = 1/T = omega/(2pi) = \frac{1}{2pi}sqrt{k/m}$ (Hz)

3. Displacement, Velocity, and Acceleration:

Displacement: — $x(t) = Acos(omega t + phi)$ (or $Asin(omega t + phi)$)
Velocity: — $v(t) = -Aomegasin(omega t + phi)$. Max velocity $v_{max} = Aomega$ (at $x=0$).
Acceleration: — $a(t) = -Aomega^2cos(omega t + phi) = -omega^2 x$. Max acceleration $a_{max} = Aomega^2$ (at $x=pm A$).

4. Energy in SHM (Conservation of Mechanical Energy):

Elastic Potential Energy (U): — $U = \frac{1}{2}kx^2$. Maximum at $x=pm A$, minimum (0) at $x=0$.
Kinetic Energy (K): — $K = \frac{1}{2}mv^2$. Maximum at $x=0$, minimum (0) at $x=pm A$.
Total Mechanical Energy (E): — $E = K + U = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$ (constant).
Velocity at displacement x: — $v = pm omegasqrt{A^2 - x^2}$.
When $K=U$: — Occurs at $x = pm A/sqrt{2}$.

5. Combinations of Springs:

Series Connection: — Springs connected end-to-end. $rac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + dots$. $k_{eq}$ is smaller than individual $k$'s (softer system, longer period).
Parallel Connection: — Springs connected side-by-side, sharing displacement. $k_{eq} = k_1 + k_2 + dots$. $k_{eq}$ is larger than individual $k$'s (stiffer system, shorter period).

6. Spring Constant and Length:

For a spring, $k propto 1/L$. If a spring is cut into $n$ equal parts, each part has spring constant $nk$.

7. Mass of Spring:

If spring mass $m_s$ is considered, effective mass $m_{eff} = m_{block} + m_s/3$. Then $T = 2pisqrt{m_{eff}/k}$.

8. Common Traps:

Confusing spring-mass period dependencies with simple pendulum dependencies (e.g., effect of gravity, mass).
Incorrectly applying series/parallel formulas.
Assuming linear relationships for energy or velocity with displacement.

Vyyuha Quick Recall

To remember the period formula for a spring-mass system, think: 'Two Pi, My King!'

Two Pi — $ightarrow 2pi$
My — $ightarrow m$ (mass)
King — $ightarrow k$ (spring constant)

So, $T = 2pisqrt{\frac{\text{My}}{\text{King}}} implies T = 2pisqrt{\frac{m}{k}}$